Informal+and+Formal+Proof-Logic,+Triangles,+Polygons,+Quads.

media type="custom" key="12091871" //This chapter delves into the properties of polygons and quadrilaterals, including parallelograms, rhombi, rectangles, squares, kites, and trapezoids. Students learn how to prove that different types of quadrilaterals are parallelograms.//
 * __ Polygons and Quadrilaterals __**

This chapter starts with the properties of polygons and narrows to focus on quadrilaterals. We will study several different types of quadrilaterals: parallelograms, rhombi, rectangles, squares, kites and trapezoids. Then, we will prove that different types of quadrilaterals are parallelograms.

Learning Objectives

 * Extend the concept of interior and exterior angles from triangles to convex polygons.
 * Find the sums of interior angles in convex polygons.

Review Queue

 * 1) What do the angles in a triangle add up to?
 * 2) Find the measure of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]] and [[image:http://www.ck12.org/ck12/ucs/?math=y caption="y"]]. [[image:http://www.ck12.org/ck12/images?id=323304 width="150"]]
 * 3) A linear pair adds up to _____.
 * 4) [[image:http://www.ck12.org/ck12/images?id=323290 width="130"]]
 * 5) Find [[image:http://www.ck12.org/ck12/ucs/?math=w%5E%5Ccirc%2C%20x%5E%5Ccirc%2C%20y%5E%5Ccirc caption="w^\circ, x^\circ, y^\circ"]], and [[image:http://www.ck12.org/ck12/ucs/?math=z%5E%5Ccirc caption="z^\circ"]].
 * 6) What is [[image:http://www.ck12.org/ck12/ucs/?math=w%5E%5Ccirc%20%2B%20y%5E%5Ccirc%20%2B%20z%5E%5Ccirc caption="w^\circ + y^\circ + z^\circ"]]?
 * Know What?** In nature, geometry is all around us. For example, sea stars have geometric symmetry. The common sea star, top, has five arms, but some species have over 20! To the right are two different kinds of sea stars. Name the polygon that is created by joining their arms and determine if either polygon is regular.

Interior Angles in Convex Polygons
In Chapter 4, you learned that interior angles are the angles inside a triangle and that these angles add up to. This concept will now be extended to any polygon. As you can see in the images below, a polygon has the same number of interior angles as it does sides. But, what do the angles add up to? Tools Needed: paper, pencil, ruler, colored pencils (optional) 1. Draw a quadrilateral, pentagon, and hexagon. 2. Cut each polygon into triangles by drawing all the diagonals from one vertex. Count the number of triangles. Make sure none of the triangles overlap. 3. Make a table with the information below. 4. Notice that the total number of degrees goes up by. So, if the number sides is, then the number of triangles from one vertex is. Therefore, the formula would be. The polygon has 13 sides. “Equiangular” tells us every angle is equal. So, each angle is. If, then each angle is In the Equiangular Polygon Formula, the word //equiangular// can be switched with //regular//. Write an equation and solve for.
 * Investigation 6-1: Polygon Sum Formula**
 * **//Name of Polygon//** || **//Number of Sides//** || **//Number of [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20s caption="\triangle s"]] from one vertex//** || **//(Column 3) [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctimes%20%28%5E%5Ccirc caption="\times (^\circ"]] in a [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle caption="\triangle"]])//** || **//Total Number of Degrees//** ||
 * Quadrilateral || 4 || 2 || [[image:http://www.ck12.org/ck12/ucs/?math=2%20%5Ctimes%20180%5E%5Ccirc caption="2 \times 180^\circ"]] || [[image:http://www.ck12.org/ck12/ucs/?math=360%5E%5Ccirc caption="360^\circ"]] ||
 * Pentagon || 5 || 3 || [[image:http://www.ck12.org/ck12/ucs/?math=3%20%5Ctimes%20180%5E%5Ccirc caption="3 \times 180^\circ"]] || [[image:http://www.ck12.org/ck12/ucs/?math=540%5E%5Ccirc caption="540^\circ"]] ||
 * Hexagon || 6 || 4 || [[image:http://www.ck12.org/ck12/ucs/?math=4%20%5Ctimes%20180%5E%5Ccirc caption="4 \times 180^\circ"]] || [[image:http://www.ck12.org/ck12/ucs/?math=720%5E%5Ccirc caption="720^\circ"]] ||
 * Polygon Sum Formula:** For any [[image:http://www.ck12.org/ck12/ucs/?math=n- caption="n-"]]gon, the interior angles add up to [[image:http://www.ck12.org/ck12/ucs/?math=%28n%20-%202%29%20%5Ctimes%20180%5E%5Ccirc caption="(n - 2) \times 180^\circ"]].
 * Example 1:** The interior angles of a polygon add up to [[image:http://www.ck12.org/ck12/ucs/?math=1980%5E%5Ccirc caption="1980^\circ"]]. How many sides does it have?
 * Solution:** Use the Polygon Sum Formula and solve for [[image:http://www.ck12.org/ck12/ucs/?math=n caption="n"]].
 * Example 2:** How many degrees does **//each angle//** in an __equiangular__ nonagon have?
 * Solution:** First we need to find the sum of the interior angles, set [[image:http://www.ck12.org/ck12/ucs/?math=n%20%3D%209. caption="n = 9."]]
 * Equiangular Polygon Formula:** For any __equiangular__ [[image:http://www.ck12.org/ck12/ucs/?math=n- caption="n-"]]gon, the measure of each angle is [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B%28n-2%29%20%5Ctimes%20180%5E%5Ccirc%7D%7Bn%7D caption="\frac{(n-2) \times 180^\circ}{n}"]].
 * Regular Polygon:** When a polygon is __equilateral__ **and** __equiangular__.
 * Example 3:** An interior angle in a regular polygon is [[image:http://www.ck12.org/ck12/ucs/?math=135%5E%5Ccirc caption="135^\circ"]]. How many sides does this polygon have?
 * Solution:** Here, we will set the Equiangular Polygon Formula equal to [[image:http://www.ck12.org/ck12/ucs/?math=135%5E%5Ccirc caption="135^\circ"]] and solve for [[image:http://www.ck12.org/ck12/ucs/?math=n caption="n"]].
 * Example 4:** **//Algebra Connection//** Find the measure of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]].
 * Solution:** From our investigation, we found that a quadrilateral has [[image:http://www.ck12.org/ck12/ucs/?math=360%5E%5Ccirc caption="360^\circ"]].

Exterior Angles in Convex Polygons
An exterior angle is an angle that is formed by extending a side of the polygon (Chapter 4). As you can see, there are two sets of exterior angles for any vertex on a polygon, one going around clockwise ( hexagon), and the other going around counter-clockwise ( hexagon). The angles with the same colors are vertical and congruent. The Exterior Angle Sum Theorem said the exterior angles of a triangle add up to. Let’s extend this theorem to all polygons. Tools Needed: pencil, paper, colored pencils, scissors The angles all fit around a point, meaning that the angles add up to, just like a triangle.
 * Investigation 6-2: Exterior Angle Tear-Up**
 * 1) Draw a hexagon like the ones above. Color in the exterior angles.
 * 2) Cut out each exterior angle. [[image:http://www.ck12.org/ck12/images?id=323241 width="225"]]
 * 3) Fit the six angles together by putting their vertices together. What happens? [[image:http://www.ck12.org/ck12/images?id=323173 width="75"]]
 * Exterior Angle Sum Theorem:** The sum of the exterior angles of any polygon is [[image:http://www.ck12.org/ck12/ucs/?math=360%5E%5Ccirc caption="360^\circ"]].
 * Example 5:** What is [[image:http://www.ck12.org/ck12/ucs/?math=y caption="y"]]?
 * Solution:** [[image:http://www.ck12.org/ck12/ucs/?math=y caption="y"]] is an exterior angle and all the given angles add up to [[image:http://www.ck12.org/ck12/ucs/?math=360%5E%5Ccirc caption="360^\circ"]]. Set up an equation.
 * Example 6:** What is the measure of each exterior angle of a regular heptagon?
 * Solution:** Because the polygon is regular, the interior angles are equal. It also means the exterior angles are equal. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7B360%5E%5Ccirc%7D%7B7%7D%20%5Capprox%2051.43%5E%5Ccirc caption="\frac{360^\circ}{7} \approx 51.43^\circ"]]
 * Know What? Revisited** The stars make a pentagon and an octagon. The pentagon looks to be regular, but we cannot tell without angle measurements or lengths.

Review Questions

 * Questions 1-13 are similar to Examples 1-3 and 6.
 * Questions 14-30 are similar to Examples 4 and 5.
 * 1) Fill in the table.
 * **# of sides** || **# of [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20s caption="\triangle s"]] from one vertex** || [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20s%20%5Ctimes%20180%5E%5Ccirc caption="\triangle s \times 180^\circ"]] **(sum)** || **Each angle in a** **//regular//** **[[image:http://www.ck12.org/ck12/ucs/?math=n- caption="n-"]]gon** || **Sum of the** **//exterior//** **angles** ||
 * 3 || 1 || [[image:http://www.ck12.org/ck12/ucs/?math=180%5E%5Ccirc caption="180^\circ"]] || [[image:http://www.ck12.org/ck12/ucs/?math=60%5E%5Ccirc caption="60^\circ"]] ||  ||
 * 4 || 2 || [[image:http://www.ck12.org/ck12/ucs/?math=360%5E%5Ccirc caption="360^\circ"]] || [[image:http://www.ck12.org/ck12/ucs/?math=90%5E%5Ccirc caption="90^\circ"]] ||  ||
 * 5 || 3 || [[image:http://www.ck12.org/ck12/ucs/?math=540%5E%5Ccirc caption="540^\circ"]] || [[image:http://www.ck12.org/ck12/ucs/?math=108%5E%5Ccirc caption="108^\circ"]] ||  ||
 * 6 || 4 || [[image:http://www.ck12.org/ck12/ucs/?math=720%5E%5Ccirc caption="720^\circ"]] || [[image:http://www.ck12.org/ck12/ucs/?math=120%5E%5Ccirc caption="120^\circ"]] ||  ||
 * 7 ||  ||   ||   ||   ||
 * 8 ||  ||   ||   ||   ||
 * 9 ||  ||   ||   ||   ||
 * 10 ||  ||   ||   ||   ||
 * 11 ||  ||   ||   ||   ||
 * 12 ||  ||   ||   ||   ||
 * 1) **//Writing//** Do you think the interior angles of a regular [[image:http://www.ck12.org/ck12/ucs/?math=n- caption="n-"]]gon could ever be [[image:http://www.ck12.org/ck12/ucs/?math=180%5E%5Ccirc caption="180^\circ"]]? Why or why not? What about [[image:http://www.ck12.org/ck12/ucs/?math=179%5E%5Ccirc caption="179^\circ"]]?
 * 2) What is the sum of the angles in a 15-gon?
 * 3) What is the sum of the angles in a 23-gon?
 * 4) The sum of the interior angles of a polygon is [[image:http://www.ck12.org/ck12/ucs/?math=4320%5E%5Ccirc caption="4320^\circ"]]. How many sides does the polygon have?
 * 5) The sum of the interior angles of a polygon is [[image:http://www.ck12.org/ck12/ucs/?math=3240%5E%5Ccirc caption="3240^\circ"]]. How many sides does the polygon have?
 * 6) What is the measure of each angle in a regular 16-gon?
 * 7) What is the measure of each angle in an equiangular 24-gon?
 * 8) Each interior angle in a regular polygon is [[image:http://www.ck12.org/ck12/ucs/?math=156%5E%5Ccirc caption="156^\circ"]]. How many sides does it have?
 * 9) Each interior angle in an equiangular polygon is [[image:http://www.ck12.org/ck12/ucs/?math=90%5E%5Ccirc caption="90^\circ"]]. How many sides does it have?
 * 10) What is the measure of each exterior angle of a dodecagon?
 * 11) What is the measure of each exterior angle of a 36-gon?
 * 12) What is the sum of the exterior angles of a 27-gon?
 * //Algebra Connection//** For questions 14-26, find the measure of the missing variable(s).
 * 1) [[image:http://www.ck12.org/ck12/images?id=323296 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=323179 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=323292 width="115"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323257 width="115"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323219 width="115"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=323249 width="115"]]
 * 7) [[image:http://www.ck12.org/ck12/images?id=323248 width="100"]]
 * 8) [[image:http://www.ck12.org/ck12/images?id=323193 width="115"]]
 * 9) [[image:http://www.ck12.org/ck12/images?id=323295 width="100"]]
 * 10) [[image:http://www.ck12.org/ck12/images?id=323184 width="125"]]
 * 11) [[image:http://www.ck12.org/ck12/images?id=323339 width="125"]]
 * 12) [[image:http://www.ck12.org/ck12/images?id=323303 width="125"]]
 * 13) [[image:http://www.ck12.org/ck12/images?id=323283 width="125"]]
 * 14) [[image:http://www.ck12.org/ck12/images?id=323176 width="125"]]
 * 15) The interior angles of a pentagon are [[image:http://www.ck12.org/ck12/ucs/?math=x%5E%5Ccirc%2C%20x%5E%5Ccirc%2C%202x%5E%5Ccirc%2C%202x%5E%5Ccirc%2C caption="x^\circ, x^\circ, 2x^\circ, 2x^\circ,"]] and [[image:http://www.ck12.org/ck12/ucs/?math=2x%5E%5Ccirc caption="2x^\circ"]]. What is [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]]?
 * 16) The exterior angles of a quadrilateral are [[image:http://www.ck12.org/ck12/ucs/?math=x%5E%5Ccirc%2C%202x%5E%5Ccirc%2C%203x%5E%5Ccirc%2C caption="x^\circ, 2x^\circ, 3x^\circ,"]] and [[image:http://www.ck12.org/ck12/ucs/?math=4x%5E%5Ccirc. caption="4x^\circ."]] What is [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]]?
 * 17) The interior angles of a hexagon are [[image:http://www.ck12.org/ck12/ucs/?math=x%5E%5Ccirc%2C%20%28x%20%2B%201%29%5E%5Ccirc%2C%20%28x%20%2B%202%29%5E%5Ccirc%2C%20%28x%20%2B%203%29%5E%5Ccirc%2C%20%28x%20%2B%204%29%5E%5Ccirc%2C caption="x^\circ, (x + 1)^\circ, (x + 2)^\circ, (x + 3)^\circ, (x + 4)^\circ,"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%28x%20%2B%205%29%5E%5Ccirc. caption="(x + 5)^\circ."]] What is [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]]?

Review Queue Answers

 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=180%5E%5Ccirc caption="180^\circ"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=72%5E%5Ccirc%20%2B%20%287x%2B3%29%5E%5Ccirc%20%2B%20%283x%2B5%29%5E%5Ccirc%20%3D%20180%5E%5Ccirc%5C%21%5C%5C%0A%7B%5C%3B%7D%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cquad%20%5C%2010x%20%2B%2080%5E%5Ccirc%20%3D%20180%5E%5Ccirc%5C%21%5C%5C%0A%7B%5C%3B%7D%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5C%20%5C%20%5Cquad%20%5C%2010x%20%3D%20100%5E%5Ccirc%5C%21%5C%5C%0A%7B%5C%3B%7D%5Cqquad%20%5Cqquad%20%5Cqquad%20%5Cqquad%20%5C%20%5C%20%5Cqquad%20%5C%20x%20%3D%2010%5E%5Ccirc caption="72^\circ + (7x+3)^\circ + (3x+5)^\circ = 180^\circ\!\\ {\;}\qquad \qquad \qquad \quad \ 10x + 80^\circ = 180^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \ \ \quad \ 10x = 100^\circ\!\\ {\;}\qquad \qquad \qquad \qquad \ \ \qquad \ x = 10^\circ"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=180%5E%5Ccirc caption="180^\circ"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=w%20%3D%20108%5E%5Ccirc%2C%20%5C%20x%20%3D%2049%5E%5Ccirc%2C%20%5C%20y%20%3D%20131%5E%5Ccirc%2C%20%5C%20z%20%3D%20121%5E%5Ccirc caption="w = 108^\circ, \ x = 49^\circ, \ y = 131^\circ, \ z = 121^\circ"]]
 * 5) [[image:http://www.ck12.org/ck12/ucs/?math=360%5E%5Ccirc caption="360^\circ"]]

Learning Objectives

 * Define a parallelogram.
 * Understand the properties of a parallelogram
 * Apply theorems about a parallelogram’s sides, angles and diagonals.

Review Queue

 * 1) Draw a quadrilateral with __one__ set of parallel sides.
 * 2) Draw a quadrilateral with __two__ sets of parallel sides.
 * 3) Find the measure of the missing angles in the quadrilaterals below.
 * 4) [[image:http://www.ck12.org/ck12/images?id=323250 width="150"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323284 width="100"]]
 * Know What?** A college has a parallelogram-shaped courtyard between two buildings. The school wants to build two walkways on the diagonals of the parallelogram and a fountain where they intersect. The walkways are going to be 50 feet and 68 feet long. Where would the fountain be?

Notice that each pair of sides is marked parallel. Also, recall that two lines are parallel when they are perpendicular to the same line. Parallelograms have a lot of interesting properties. Tools Needed: Paper, pencil, ruler, protractor To continue to explore the properties of a parallelogram, see the website: [] If then If then If then If then __Given__: is a parallelogram with diagonal __Prove__: The proof of the Opposite Angles Theorem is almost identical. For the last step, the //angles// are congruent by CPCTC. If, then by the Opposite Angles Theorem.
 * What is a Parallelogram?**
 * Parallelogram:** A quadrilateral with two pairs of parallel sides.
 * Investigation 6-2: Properties of Parallelograms**
 * 1) Draw a set of parallel lines by drawing a 3 inch line on either side of your ruler. [[image:http://www.ck12.org/ck12/images?id=323192 width="175"]]
 * 2) Rotate the ruler and repeat so you have a parallelogram. If you have colored pencils, outline the parallelogram in another color. [[image:http://www.ck12.org/ck12/images?id=323234 width="125"]]
 * 3) Measure the four interior angles of the parallelogram as well as the length of each side. What do you notice?
 * 4) Draw the diagonals. Measure each and then measure the lengths from the point of intersection to each vertex. [[image:http://www.ck12.org/ck12/images?id=323311 width="150"]]
 * Opposite Sides Theorem:** If a quadrilateral is a parallelogram, then the opposite sides are congruent.
 * Opposite Angles Theorem:** If a quadrilateral is a parallelogram, then the opposite angles are congruent.
 * Consecutive Angles Theorem:** If a quadrilateral is a parallelogram, then the consecutive angles are supplementary.
 * Parallelogram Diagonals Theorem:** If a quadrilateral is a parallelogram, then the diagonals bisect each other.
 * //Proof of Opposite Sides Theorem//**
 * **//Statement//** || **//Reason//** ||
 * 1. [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is a parallelogram with diagonal [[image:http://www.ck12.org/ck12/ucs/?math=%20%5Coverline%7BBD%7D caption=" \overline{BD}"]] || Given ||
 * 2. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5C%7C%20%5Coverline%7BDC%7D%2C%20%5Coverline%7BAD%7D%20%5C%7C%20%5Coverline%7BBC%7D caption="\overline{AB} \| \overline{DC}, \overline{AD} \| \overline{BC}"]] || Definition of a parallelogram ||
 * 3. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20ABD%20%5Ccong%20%5Cangle%20BDC%2C%20%5Cangle%20ADB%20%5Ccong%20%5Cangle%20DBC caption="\angle ABD \cong \angle BDC, \angle ADB \cong \angle DBC"]] || Alternate Interior Angles Theorem ||
 * 4. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BDB%7D%20%5Ccong%20%5Coverline%7BDB%7D caption="\overline{DB} \cong \overline{DB}"]] || Reflexive PoC ||
 * 5. [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20ABD%20%5Ccong%20%5Ctriangle%20CDB caption="\triangle ABD \cong \triangle CDB"]] || ASA ||
 * 6. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5Ccong%20%5Coverline%7BDC%7D%2C%20%5Coverline%7BAD%7D%20%5Ccong%20%5Coverline%7BBC%7D caption="\overline{AB} \cong \overline{DC}, \overline{AD} \cong \overline{BC}"]] || CPCTC ||
 * Example 1:** [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is a parallelogram. If [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20A%20%3D%2056%5E%5Ccirc caption="m \angle A = 56^\circ"]], find the measure of the other angles.
 * Solution:** Draw a picture. When labeling the vertices, the letters are listed, in order, clockwise.
 * Example 2:** **//Algebra Connection//** Find the values of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]] and [[image:http://www.ck12.org/ck12/ucs/?math=y caption="y"]].
 * Solution:** Opposite sides are congruent.

Diagonals in a Parallelogram
From the Parallelogram Diagonals Theorem, we know that the diagonals of a parallelogram bisect each other. Because they are the same point, the diagonals intersect at each other’s midpoint. This means they bisect each other.
 * Example 3:** Show that the diagonals of [[image:http://www.ck12.org/ck12/ucs/?math=FGHJ caption="FGHJ"]] bisect each other.
 * Solution:** Find the midpoint of each diagonal.
 * //This is one way to show a quadrilateral is a parallelogram.//**
 * Example 4:** **//Algebra Connection//** [[image:http://www.ck12.org/ck12/ucs/?math=SAND caption="SAND"]] is a parallelogram and [[image:http://www.ck12.org/ck12/ucs/?math=SY%20%3D%204x%20-%2011 caption="SY = 4x - 11"]] and [[image:http://www.ck12.org/ck12/ucs/?math=YN%20%3D%20x%20%2B%2010 caption="YN = x + 10"]]. Solve for [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]].
 * Solution:**
 * Know What? Revisited** The diagonals bisect each other, so the fountain is going to be 34 feet from either endpoint on the 68 foot diagonal and 25 feet from either endpoint on the 50 foot diagonal.

Review Questions
is a parallelogram. Fill in the blanks below. For questions 11-19, find the measures of the variable(s). All the figures below are parallelograms. Use the parallelogram to find: Find the point of intersection of the diagonals to see if is a parallelogram. Fill in the blanks in the proofs below. __Given__: is a parallelogram with diagonal __Prove__: __Given__: is a parallelogram with diagonals  and __Prove__:
 * Questions 1-6 are similar to Examples 2 and 4.
 * Questions 7-10 are similar to Example 1.
 * Questions 11-23 are similar to Examples 2 and 4.
 * Questions 24-27 are similar to Example 3.
 * Questions 28 and 29 are similar to the proof of the Opposite Sides Theorem.
 * Question 30 is a challenge. Use the properties of parallelograms.
 * 1) If [[image:http://www.ck12.org/ck12/ucs/?math=AB%20%3D%206 caption="AB = 6"]], then [[image:http://www.ck12.org/ck12/ucs/?math=CD%20%3D caption="CD ="]] ______.
 * 2) If [[image:http://www.ck12.org/ck12/ucs/?math=AE%20%3D%204 caption="AE = 4"]], then [[image:http://www.ck12.org/ck12/ucs/?math=AC%20%3D caption="AC ="]] ______.
 * 3) If [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20ADC%20%3D%2080%5E%5Ccirc%2C%20m%20%5Cangle%20DAB caption="m \angle ADC = 80^\circ, m \angle DAB"]] = ______.
 * 4) If [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20BAC%20%3D%2045%5E%5Ccirc%2C%20m%20%5Cangle%20ACD caption="m \angle BAC = 45^\circ, m \angle ACD"]] = ______.
 * 5) If [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20CBD%20%3D%2062%5E%5Ccirc%2C%20m%20%5Cangle%20ADB caption="m \angle CBD = 62^\circ, m \angle ADB"]] = ______.
 * 6) If [[image:http://www.ck12.org/ck12/ucs/?math=DB%20%3D%2016 caption="DB = 16"]], then [[image:http://www.ck12.org/ck12/ucs/?math=DE caption="DE"]] = ______.
 * 7) If [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20B%20%3D%2072%5E%5Ccirc caption="m \angle B = 72^\circ"]] in parallelogram [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]], find the other three angles.
 * 8) If [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20S%20%3D%20143%5E%5Ccirc caption="m \angle S = 143^\circ"]] in parallelogram [[image:http://www.ck12.org/ck12/ucs/?math=PQRS caption="PQRS"]], find the other three angles.
 * 9) If [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5Cperp%20%5Coverline%7BBC%7D caption="\overline{AB} \perp \overline{BC}"]] in parallelogram [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]], find the measure of all four angles.
 * 10) If [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20F%20%3D%20x%5E%5Ccirc caption="m \angle F = x^\circ"]] in parallelogram [[image:http://www.ck12.org/ck12/ucs/?math=EFGH caption="EFGH"]], find the other three angles.
 * 1) [[image:http://www.ck12.org/ck12/images?id=323268 width="130"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=323229 width="80"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=323190 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323312 width="100"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323310 width="150"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=323305 width="150"]]
 * 7) [[image:http://www.ck12.org/ck12/images?id=323286 width="150"]]
 * 8) [[image:http://www.ck12.org/ck12/images?id=323260 width="130"]]
 * 9) [[image:http://www.ck12.org/ck12/images?id=323227 width="130"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20AWE caption="m \angle AWE"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20ESV caption="m \angle ESV"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20WEA caption="m \angle WEA"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20AVW caption="m \angle AVW"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=E%28-1%2C%203%29%2C%20F%283%2C%204%29%2C%20G%285%2C%20-1%29%2C%20H%281%2C%20-2%29 caption="E(-1, 3), F(3, 4), G(5, -1), H(1, -2)"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=E%283%2C%20-2%29%2C%20F%287%2C%200%29%2C%20G%289%2C%20-4%29%2C%20H%285%2C%20-4%29 caption="E(3, -2), F(7, 0), G(9, -4), H(5, -4)"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=E%28-6%2C%203%29%2C%20F%282%2C%205%29%2C%20G%286%2C%20-3%29%2C%20H%28-4%2C%20-5%29 caption="E(-6, 3), F(2, 5), G(6, -3), H(-4, -5)"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=E%28-2%2C%20-2%29%2C%20F%28-4%2C%20-6%29%2C%20G%28-6%2C%20-4%29%2C%20H%28-4%2C%200%29 caption="E(-2, -2), F(-4, -6), G(-6, -4), H(-4, 0)"]]
 * 1) **//Opposite Angles Theorem//**
 * **//Statement//** || **//Reason//** ||
 * 1. || Given ||
 * 2. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5C%7C%20%5Coverline%7BDC%7D%2C%5Coverline%7BAD%7D%20%5C%7C%20%5Coverline%7BBC%7D caption="\overline{AB} \| \overline{DC},\overline{AD} \| \overline{BC}"]] ||  ||
 * 3. || Alternate Interior Angles Theorem ||
 * 4. || Reflexive PoC ||
 * 5. [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20ABD%20%5Ccong%20%5Ctriangle%20CDB caption="\triangle ABD \cong \triangle CDB"]] ||  ||
 * 6. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20A%20%5Ccong%20%5Cangle%20C caption="\angle A \cong \angle C"]] ||  ||
 * 1) **//Parallelogram Diagonals Theorem//**
 * **//Statement//** || **//Reason//** ||
 * 1. ||  ||
 * 2. || Definition of a parallelogram ||
 * 3. || Alternate Interior Angles Theorem ||
 * 4. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5Ccong%20%5Coverline%7BDC%7D caption="\overline{AB} \cong \overline{DC}"]] ||  ||
 * 5. ||  ||
 * 6. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAE%7D%20%5Ccong%20%5Coverline%7BEC%7D%2C%20%5Coverline%7BDE%7D%20%5Ccong%20%5Coverline%7BEB%7D caption="\overline{AE} \cong \overline{EC}, \overline{DE} \cong \overline{EB}"]] ||  ||
 * 1) **//Challenge//** Find [[image:http://www.ck12.org/ck12/ucs/?math=x%2C%20y%5E%5Ccirc%2C caption="x, y^\circ,"]] and [[image:http://www.ck12.org/ck12/ucs/?math=z%5E%5Ccirc caption="z^\circ"]]. (The two quadrilaterals with the same side are parallelograms.)

Review Queue Answers

 * 1) [[image:http://www.ck12.org/ck12/images?id=330823 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=330824 width="115"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=3x%2Bx%2B3x%2Bx%3D360%5E%5Ccirc%5C%21%5C%5C%0A%7B%5C%3B%7D%5Cqquad%20%5Cqquad%20%5Cquad%20%5C%208x%20%3D%20360%5E%5Ccirc%5C%21%5C%5C%0A%7B%5C%3B%7D%5Cquad%20%5C%20%5Cqquad%20%5Cquad%20%5C%20%5C%20%5C%20%5C%20x%20%3D%2045%5E%5Ccirc caption="3x+x+3x+x=360^\circ\!\\ {\;}\qquad \qquad \quad \ 8x = 360^\circ\!\\ {\;}\quad \ \qquad \quad \ \ \ \ x = 45^\circ"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=4x%2B2%3D90%5E%5Ccirc%5C%21%5C%5C%0A%7B%5C%3B%7D%5Cquad%20%5C%204x%3D88%5E%5Ccirc%5C%21%5C%5C%0A%7B%5C%3B%7D%5Cqquad%20x%3D22%5E%5Ccirc caption="4x+2=90^\circ\!\\ {\;}\quad \ 4x=88^\circ\!\\ {\;}\qquad x=22^\circ"]]

Learning Objectives

 * Prove a quadrilateral is a parallelogram.
 * Show a quadrilateral is a parallelogram in the [[image:http://www.ck12.org/ck12/ucs/?math=x-y caption="x-y"]] plane.

Review Queue

 * 1) Plot the points [[image:http://www.ck12.org/ck12/ucs/?math=A%282%2C%202%29%2C%20B%284%2C%20-2%29%2C%20C%28-2%2C%20-4%29 caption="A(2, 2), B(4, -2), C(-2, -4)"]], and [[image:http://www.ck12.org/ck12/ucs/?math=D%28-6%2C%20-2%29 caption="D(-6, -2)"]].
 * 2) Find the slopes of [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%2C%5Coverline%7BBC%7D%2C%20%5Coverline%7BCD%7D%2C caption="\overline{AB},\overline{BC}, \overline{CD},"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAD%7D caption="\overline{AD}"]]. Is [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] a parallelogram?
 * 3) Find the point of intersection of the diagonals by finding the midpoint of each.
 * Know What?** You are marking out a baseball diamond and standing at home plate. [[image:http://www.ck12.org/ck12/ucs/?math=3%5E%7Brd%7D caption="3^{rd}"]] base is 90 feet away, [[image:http://www.ck12.org/ck12/ucs/?math=2%5E%7Bnd%7D caption="2^{nd}"]] base is 127.3 feet away, and [[image:http://www.ck12.org/ck12/ucs/?math=1%5E%7Bst%7D caption="1^{st}"]] base is also 90 feet away. The angle at home plate is [[image:http://www.ck12.org/ck12/ucs/?math=90%5E%5Ccirc caption="90^\circ"]], from [[image:http://www.ck12.org/ck12/ucs/?math=1%5E%7Bst%7D caption="1^{st}"]] to [[image:http://www.ck12.org/ck12/ucs/?math=3%5E%7Brd%7D caption="3^{rd}"]] is [[image:http://www.ck12.org/ck12/ucs/?math=90%5E%5Ccirc caption="90^\circ"]]. Find the length of the other diagonal (using the Pythagorean Theorem) and determine if the baseball diamond is a parallelogram.

Determining if a Quadrilateral is a Parallelogram
The converses of the theorems in the last section will now be used to see if a quadrilateral is a parallelogram. If then If then If then __Given__: __Prove__: is a parallelogram __Given__:, and __Prove__: is a parallelogram If then a) By the Opposite Angles Theorem Converse, is a parallelogram. b)  is not a parallelogram because the diagonals do not bisect each other.
 * Opposite Sides Theorem Converse:** If the opposite sides of a quadrilateral are congruent, then the figure is a parallelogram.
 * Opposite Angles Theorem Converse:** If the opposite angles of a quadrilateral are congruent, then the figure is a parallelogram.
 * Parallelogram Diagonals Theorem Converse:** If the diagonals of a quadrilateral bisect each other, then the figure is a parallelogram.
 * //Proof of the Opposite Sides Theorem Converse//**
 * **//Statement//** || **//Reason//** ||
 * 1. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5Ccong%20%5Coverline%7BDC%7D%2C%20%5Coverline%7BAD%7D%20%5Ccong%20%5Coverline%7BBC%7D caption="\overline{AB} \cong \overline{DC}, \overline{AD} \cong \overline{BC}"]] || Given ||
 * 2. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BDB%7D%20%5Ccong%20%5Coverline%7BDB%7D caption="\overline{DB} \cong \overline{DB}"]] || Reflexive PoC ||
 * 3. [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20ABD%20%5Ccong%20%5Ctriangle%20CDB caption="\triangle ABD \cong \triangle CDB"]] || SSS ||
 * 4. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20ABD%20%5Ccong%20%5Cangle%20BDC%2C%20%5Cangle%20ADB%20%5Ccong%20%5Cangle%20DBC caption="\angle ABD \cong \angle BDC, \angle ADB \cong \angle DBC"]] || CPCTC ||
 * 5. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5C%7C%20%5Coverline%7BDC%7D%2C%20%5Coverline%7BAD%7D%20%5C%7C%20%5Coverline%7BBC%7D caption="\overline{AB} \| \overline{DC}, \overline{AD} \| \overline{BC}"]] || Alternate Interior Angles Converse ||
 * 6. [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is a parallelogram || Definition of a parallelogram ||
 * Example 1:** Write a two-column proof.
 * Solution:**
 * **//Statement//** || **//Reason//** ||
 * 1. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5C%7C%20%5Coverline%7BDC%7D caption="\overline{AB} \| \overline{DC}"]], and [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5Ccong%20%5Coverline%7BDC%7D caption="\overline{AB} \cong \overline{DC}"]] || Given ||
 * 2. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20ABD%20%5Ccong%20%5Cangle%20BDC caption="\angle ABD \cong \angle BDC"]] || Alternate Interior Angles ||
 * 3. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BDB%7D%20%5Ccong%20%5Coverline%7BDB%7D caption="\overline{DB} \cong \overline{DB}"]] || Reflexive PoC ||
 * 4. [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20ABD%20%5Ccong%20%5Ctriangle%20CDB caption="\triangle ABD \cong \triangle CDB"]] || SAS ||
 * 5. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAD%7D%20%5Ccong%20%5Coverline%7BBC%7D caption="\overline{AD} \cong \overline{BC}"]] || CPCTC ||
 * 6. [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is a parallelogram || Opposite Sides Converse ||
 * Theorem 6-10:** If a quadrilateral has one set of parallel lines that are also congruent, then it is a parallelogram.
 * Example 2:** Is quadrilateral [[image:http://www.ck12.org/ck12/ucs/?math=EFGH caption="EFGH"]] a parallelogram? How do you know?
 * Solution:**
 * Example 3:** **//Algebra Connection//** What value of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]] would make [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] a parallelogram?
 * Solution:** [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAB%7D%20%5C%7C%20%5Coverline%7BDC%7D caption="\overline{AB} \| \overline{DC}"]]. By Theorem 6-10, [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] would be a parallelogram if [[image:http://www.ck12.org/ck12/ucs/?math=AB%20%3D%20DC caption="AB = DC"]].

Showing a Quadrilateral is a Parallelogram in the [[image:http://www.ck12.org/ck12/ucs/?math=x-y caption="x-y"]] Plane
To show that a quadrilateral is a parallelogram in the plane, you might need: Find the slopes. and the slopes are the same, is a parallelogram. is not a parallelogram because the midpoints are not the same. The diagonals are equal, so the other two sides of the diamond must also be 90 feet. The baseball diamond is a parallelogram, and more specifically, a square.
 * The Slope Formula, [[image:http://www.ck12.org/ck12/ucs/?math=%5Cfrac%7By_2%20-%20y_1%7D%7Bx_2%20-%20x_1%7D caption="\frac{y_2 - y_1}{x_2 - x_1}"]].
 * The Distance Formula, [[image:http://www.ck12.org/ck12/ucs/?math=%5Csqrt%7B%28x_2%20-%20x_1%29%5E2%20%2B%20%28y_2%20-%20y_1%29%5E2%7D caption="\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}"]].
 * The Midpoint Formula, [[image:http://www.ck12.org/ck12/ucs/?math=%5Cleft%20%28%20%5Cfrac%7Bx_1%20%2B%20x_2%20%7D%7B2%7D%20%2C%20%5Cfrac%7By_1%20%2B%20y_2%7D%7B2%7D%20%5Cright%20%29 caption="\left ( \frac{x_1 + x_2 }{2}, \frac{y_1 + y_2}{2} \right )"]].
 * Example 4:** Is the quadrilateral [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] a parallelogram?
 * Solution:** Let’s use Theorem 6-10 to see if [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is a parallelogram. First, find the length of [[image:http://www.ck12.org/ck12/ucs/?math=AB caption="AB"]] and [[image:http://www.ck12.org/ck12/ucs/?math=CD caption="CD"]].
 * Example 5:** Is the quadrilateral [[image:http://www.ck12.org/ck12/ucs/?math=RSTU caption="RSTU"]] a parallelogram?
 * Solution:** Let’s use the Parallelogram Diagonals Converse to see if [[image:http://www.ck12.org/ck12/ucs/?math=RSTU caption="RSTU"]] is a parallelogram. Find the midpoint of each diagonal.
 * Know What? Revisited** Use the Pythagorean Theorem to find the length of the second diagonal.

Review Questions
For questions 1-12, determine if the quadrilaterals are parallelograms. For questions 19-22, determine if is a parallelogram. Fill in the blanks in the proofs below. __Given__: __Prove__: is a parallelogram and are supplementary ||   || __Given__: __Prove__: is a parallelogram > __Prove__: is a parallelogram
 * Questions 1-12 are similar to Example 2.
 * Questions 13-15 are similar to Example 3.
 * Questions 16-22 are similar to Examples 4 and 5.
 * Questions 23-25 are similar to Example 1 and the proof of the Opposite Sides Converse.
 * 1) [[image:http://www.ck12.org/ck12/images?id=323323 width="125"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=323182 width="125"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=323270 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323347 width="100"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323327 width="125"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=323205 width="125"]]
 * 7) [[image:http://www.ck12.org/ck12/images?id=323204 width="100"]]
 * 8) [[image:http://www.ck12.org/ck12/images?id=323242 width="100"]]
 * 9) [[image:http://www.ck12.org/ck12/images?id=323346 width="100"]]
 * 10) [[image:http://www.ck12.org/ck12/images?id=323185 width="80"]]
 * 11) [[image:http://www.ck12.org/ck12/images?id=323316 width="80"]]
 * 12) [[image:http://www.ck12.org/ck12/images?id=323341 width="80"]]
 * //Algebra Connection//** For questions 13-18, determine the value of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]] and [[image:http://www.ck12.org/ck12/ucs/?math=y caption="y"]] that would make the quadrilateral a parallelogram.
 * 1) [[image:http://www.ck12.org/ck12/images?id=323175 width="125"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=323195 width="125"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=323223 width="125"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323255 width="125"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323322 width="125"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=323208 width="125"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=A%288%2C%20-1%29%2C%20B%286%2C%205%29%2C%20C%28-7%2C%202%29%2C%20D%28-5%2C%20-4%29 caption="A(8, -1), B(6, 5), C(-7, 2), D(-5, -4)"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=A%28-5%2C%208%29%2C%20B%28-2%2C%209%29%2C%20C%283%2C%204%29%2C%20D%280%2C%203%29 caption="A(-5, 8), B(-2, 9), C(3, 4), D(0, 3)"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=A%28-2%2C%206%29%2C%20B%284%2C%20-4%29%2C%20C%2813%2C%20-7%29%2C%20D%284%2C%20-10%29 caption="A(-2, 6), B(4, -4), C(13, -7), D(4, -10)"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=A%28-9%2C%20-1%29%2C%20B%28-7%2C%205%29%2C%20C%283%2C%208%29%2C%20D%281%2C%202%29 caption="A(-9, -1), B(-7, 5), C(3, 8), D(1, 2)"]]
 * 1) **//Opposite Angles Theorem Converse//**
 * **//Statement//** || **//Reason//** ||
 * 1. ||  ||
 * 2. [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20A%20%3D%20m%20%5Cangle%20C%2C%20m%20%5Cangle%20D%20%3D%20m%20%5Cangle%20B caption="m \angle A = m \angle C, m \angle D = m \angle B"]] ||  ||
 * 3. || Definition of a quadrilateral ||
 * 4. [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20A%20%2B%20m%20%5Cangle%20A%20%2B%20m%20%5Cangle%20B%20%2B%20m%20%5Cangle%20B%20%3D%20360%5E%5Ccirc caption="m \angle A + m \angle A + m \angle B + m \angle B = 360^\circ"]] ||  ||
 * 5. || Combine Like Terms ||
 * 6. || Division PoE ||
 * 7. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20A caption="\angle A"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20B caption="\angle B"]] are supplementary
 * 8. || Consecutive Interior Angles Converse ||
 * 9. [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is a parallelogram ||  ||
 * 1) **//Parallelogram Diagonals Theorem Converse//**
 * **//Statement//** || **//Reason//** ||
 * 1. ||  ||
 * 2. || Vertical Angles Theorem ||
 * 3. [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20AED%20%5Ccong%20%5Ctriangle%20CEB%5C%21%5C%5C%0A%5Ctriangle%20AEB%20%5Ccong%20%5Ctriangle%20CED caption="\triangle AED \cong \triangle CEB\!\\ \triangle AEB \cong \triangle CED"]] ||  ||
 * 4. ||  ||
 * 5. [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is a parallelogram ||  ||
 * 1) __Given__: [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20ADB%20%5Ccong%20%5Cangle%20CBD%2C%20%5Coverline%7BAD%7D%20%5Ccong%20%5Coverline%7BBC%7D caption="\angle ADB \cong \angle CBD, \overline{AD} \cong \overline{BC}"]]
 * **//Statement//** || **//Reason//** ||
 * 1. ||  ||
 * 2. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAD%7D%20%5C%7C%20%5Coverline%7BBC%7D caption="\overline{AD} \| \overline{BC}"]] ||  ||
 * 3. [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is a parallelogram ||  ||

Review Queue Answers
1. (a) is a parallelogram because the opposite sides are parallel. (b) Yes, the midpoints of the diagonals are the same, so they bisect each other.

Learning Objectives

 * Define a rectangle, rhombus, and square.
 * Determine if a parallelogram is a rectangle, rhombus, or square in the [[image:http://www.ck12.org/ck12/ucs/?math=x-y caption="x-y"]] plane.
 * Compare the diagonals of a rectangle, rhombus, and square.

Review Queue

 * 1) List five examples where you might see a square, rectangle, or rhombus in real life.
 * 2) Find the values of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]] and [[image:http://www.ck12.org/ck12/ucs/?math=y caption="y"]]that would make the quadrilateral a parallelogram.
 * 3) [[image:http://www.ck12.org/ck12/images?id=323344 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323226 width="100"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323188 width="100"]]
 * Know What?** You are designing a patio for your backyard and are marking it off with a tape measure. Two sides are **//21 feet//** long and two sides are **//28 feet//** long. Explain how you would __only__ use the tape measure to make your patio a rectangle. (You do not need to find any measurements.)

Defining Special Parallelograms
Rectangles, Rhombuses (also called Rhombi) and Squares are all more specific versions of parallelograms, also called special parallelograms. Taking the theorems we learned in the previous two sections, we have three more new theorems. is a rectangle if and only if. is a rhombus if and only if. is a square if and only if **//and//**. From the Square Theorem, we can also conclude that a **//square is a rectangle and a rhombus.//** a) b) a) All sides are congruent and one angle is, so the angles are not congruent. This is a rhombus. b) All four congruent angles and the sides are not. This is a rectangle.
 * Rectangle Theorem:** A quadrilateral is a rectangle if and only if it has four right (congruent) angles.
 * Rhombus Theorem:** A quadrilateral is a rhombus if and only if it has four congruent sides.
 * Square Theorem:** A quadrilateral is a square if and only if it has four right angles and four congruent sides.
 * Example 1:** What type of parallelogram are the ones below?
 * Solution:**
 * Example 2:** Is a rhombus SOMETIMES, ALWAYS, or NEVER a square? Explain why.
 * Solution:** A rhombus has four congruent sides and a square has four congruent sides **//and//** angles. Therefore, a rhombus is a square when it has congruent angles. This means a rhombus is SOMETIMES a square.
 * Example 3:** Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.
 * Solution:** A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.

Diagonals in Special Parallelograms
Recall from previous lessons that the **//diagonals in a parallelogram bisect each other//**. Therefore, the diagonals of a rectangle, square and rhombus also bisect each other. They also have additional properties. Tools Needed: pencil, paper, protractor, ruler is parallelogram. If, then is also a rectangle. Tools Needed: pencil, paper, protractor, ruler 1. Like with Investigation 6-2 and 6-3, draw two lines on either side of your ruler, 3 inches long. 2. Remove the ruler and mark a angle, at the left end of the bottom line drawn in Step 1. Draw the other side of the angle and make sure it intersects the top line. Measure the length of this side. 3. Mark the length found in Step 2 on the bottom line and the top line from the point of intersection with the angle. Draw in the fourth side. It will connect the two endpoints of these lengths. 4. By the way we drew this parallelogram; it is a rhombus because all the sides are equal. Draw in the diagonals. __Measure the angles__ at the point of intersection of the diagonals (4). __Measure the angles__ created by the sides and each diagonal (8). is a parallelogram. If, then is also a rhombus. is a parallelogram. If bisects  and  **__and__**  bisects, then  is also a rhombus. All the bisected angles are.
 * Investigation 6-3: Drawing a Rectangle**
 * 1) Like with Investigation 6-2, draw two lines on either side of your ruler, making them parallel. Make these lines 3 inches long. [[image:http://www.ck12.org/ck12/images?id=323192 width="175"]]
 * 2) Using the protractor, mark two [[image:http://www.ck12.org/ck12/ucs/?math=90%5E%5Ccirc caption="90^\circ"]] angles, 2.5 inches apart on the bottom line from Step 1. Extend the sides to intersect the top line. [[image:http://www.ck12.org/ck12/images?id=323259 width="200"]]
 * 3) Draw in the diagonals and measure. What do you discover? [[image:http://www.ck12.org/ck12/images?id=323278 width="100"]]
 * Theorem 6-14:** A **//parallelogram//** is a **//rectangle//** if the diagonals are congruent.
 * Investigation 6-4: Drawing a Rhombus**
 * Theorem 6-15:** A **//parallelogram//** is a **//rhombus//** if the diagonals are perpendicular.
 * Theorem 6-16:** A **//parallelogram//** is a **//rhombus//** if the diagonals bisect each angle.
 * //The converses of these three theorems are true.//** There are no theorems about the diagonals of a square. **//The diagonals of a square have the properties of a rhombus and a rectangle.//**
 * Example 4:** List **//everything//** you know about the square [[image:http://www.ck12.org/ck12/ucs/?math=SQRE caption="SQRE"]].
 * Solution:** A square has all the properties of a parallelogram, rectangle and rhombus.
 * **//Properties of a Parallelogram//** || **//Properties of a Rhombus//** || **//Properties of a Rectangle//** ||
 * * [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BSQ%7D%20%5C%7C%20%5Coverline%7BER%7D caption="\overline{SQ} \| \overline{ER}"]] || * [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BSQ%7D%20%5Ccong%20%5Coverline%7BER%7D%20%5Ccong%20%5Coverline%7BSE%7D%20%5Ccong%20%5Coverline%7BQR%7D caption="\overline{SQ} \cong \overline{ER} \cong \overline{SE} \cong \overline{QR}"]] || * [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20SER%20%3D%20m%20%5Cangle%20SQR%20%3D%20m%20%5Cangle%20QSE%20%3D%20m%20%5Cangle%20QRE%20%20%3D%2090%5E%5Ccirc caption="m \angle SER = m \angle SQR = m \angle QSE = m \angle QRE = 90^\circ"]] ||
 * * [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BSE%7D%20%5C%7C%20%5Coverline%7BQR%7D caption="\overline{SE} \| \overline{QR}"]] || * [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BSR%7D%20%5Cperp%20%5Coverline%7BQE%7D caption="\overline{SR} \perp \overline{QE}"]] ||  ||
 * || * [[image:http://www.ck12.org/ck12/ucs/?math=%20%5Cangle%20SEQ%20%5Ccong%20%5Cangle%20QER%20%5Ccong%20%5Cangle%20SQE%20%5Ccong%20%5Cangle%20EQR caption=" \angle SEQ \cong \angle QER \cong \angle SQE \cong \angle EQR"]] || * [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BSR%7D%20%5Ccong%20%5Coverline%7BQE%7D caption="\overline{SR} \cong \overline{QE}"]] ||
 * || * [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20QSR%20%5Ccong%20%5Cangle%20RSE%20%5Ccong%20%5Cangle%20QRS%20%5Ccong%20%5Cangle%20SRE caption="\angle QSR \cong \angle RSE \cong \angle QRS \cong \angle SRE"]] || * [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BSA%7D%20%5Ccong%20%5Coverline%7BAR%7D%20%5Ccong%20%5Coverline%7BQA%7D%20%5Ccong%20%5Coverline%7BAE%7D caption="\overline{SA} \cong \overline{AR} \cong \overline{QA} \cong \overline{AE}"]] ||

Parallelograms in the Coordinate Plane
If the diagonals are also perpendicular, then is a square. The slope of slope of, so  is a rectangle. __Steps to determine if a quadrilateral is a parallelogram, rectangle, rhombus, or square.__ 1. Graph the four points on **//graph paper.//** 2. See if the **//diagonals bisect each other.//** (midpoint formula) 3. See if the **//diagonals are equal.//** (distance formula) 4. See if the **//sides are congruent.//** (distance formula) 5. See if the **//diagonals are perpendicular.//** (find slopes)
 * Example 4:** Determine what type of parallelogram [[image:http://www.ck12.org/ck12/ucs/?math=TUNE caption="TUNE"]] is: [[image:http://www.ck12.org/ck12/ucs/?math=T%280%2C%2010%29%2C%20U%284%2C%202%29%2C%20N%28-2%2C%20-1%29 caption="T(0, 10), U(4, 2), N(-2, -1)"]], and [[image:http://www.ck12.org/ck12/ucs/?math=E%28-6%2C%207%29 caption="E(-6, 7)"]].
 * Solution:** Let’s see if the diagonals are equal. If they are, then [[image:http://www.ck12.org/ck12/ucs/?math=TUNE caption="TUNE"]] is a rectangle.
 * Yes:** **//Parallelogram//**, continue to #2. **No:** A **//quadrilateral//**, done.
 * Yes:** **//Rectangle,//** skip to #4. **No:** Could be a rhombus, continue to #3.
 * Yes:** **//Rhombus,//** done. **No:** **//Parallelogram,//** done.
 * Yes:** **//Square,//** done. **No:** **//Rectangle,//** done.
 * Know What? Revisited** In order for the patio to be a rectangle, the opposite sides must be congruent (see picture). To ensure that the parallelogram is a rectangle //without// measuring the angles, the diagonals must be equal.

Review Questions
> > > For questions 4-15, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none. For questions 16-21 determine if the following are ALWAYS, SOMETIME, or NEVER true. Explain your reasoning. For questions 26-29, determine what type of quadrilateral is. Use Example 4 and the steps following it to help you. >
 * Questions 1-3 are similar to #2 in the Review Queue and Example 1.
 * Questions 4-15 are similar to Example 1.
 * Questions 16-21 are similar to Examples 2 and 3.
 * Questions 22-25 are similar to Investigations 6-3 and 6-4.
 * Questions 26-29 are similar to Example 4.
 * Question 30 is a challenge.
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=RACE caption="RACE"]]is a rectangle. Find:
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=RG caption="RG"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=AE caption="AE"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=AC caption="AC"]]
 * 5) [[image:http://www.ck12.org/ck12/ucs/?math=EC caption="EC"]]
 * 6) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20RAC caption="m \angle RAC"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=DIAM caption="DIAM"]]is a rhombus. Find:
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=MA caption="MA"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=MI caption="MI"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=DA caption="DA"]]
 * 5) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20DIA caption="m \angle DIA"]]
 * 6) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20MOA caption="m \angle MOA"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=CUBE caption="CUBE"]]is a square. Find:
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20UCE caption="m \angle UCE"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20EYB caption="m \angle EYB"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20UBY caption="m \angle UBY"]]
 * 5) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20UEB caption="m \angle UEB"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=323321 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=323336 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=323246 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323330 width="100"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323220 width="100"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=323272 width="100"]]
 * 7) [[image:http://www.ck12.org/ck12/images?id=323258 width="100"]]
 * 8) [[image:http://www.ck12.org/ck12/images?id=323333 width="80"]]
 * 9) [[image:http://www.ck12.org/ck12/images?id=323299 width="100"]]
 * 10) [[image:http://www.ck12.org/ck12/images?id=323331 width="100"]]
 * 11) [[image:http://www.ck12.org/ck12/images?id=323225 width="100"]]
 * 12) [[image:http://www.ck12.org/ck12/images?id=323274 width="100"]]
 * 1) A rectangle is a rhombus.
 * 2) A square is a parallelogram.
 * 3) A parallelogram is regular.
 * 4) A square is a rectangle.
 * 5) A rhombus is equiangular.
 * 6) A quadrilateral is a pentagon.
 * //Construction//** Draw or construct the following quadrilaterals.
 * 1) A quadrilateral with congruent diagonals that is not a rectangle.
 * 2) A quadrilateral with perpendicular diagonals that is not a rhombus or square.
 * 3) A rhombus with a 6 cm diagonal and an 8 cm diagonal.
 * 4) A square with 2 inch sides.
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=A%28-2%2C%204%29%2C%20B%28-1%2C%202%29%2C%20C%28-3%2C%201%29%2C%20D%28-4%2C%203%29 caption="A(-2, 4), B(-1, 2), C(-3, 1), D(-4, 3)"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=A%28-2%2C%203%29%2C%20B%283%2C%204%29%2C%20C%282%2C%20-1%29%2C%20D%28-3%2C%20-2%29 caption="A(-2, 3), B(3, 4), C(2, -1), D(-3, -2)"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=A%281%2C%20-1%29%2C%20B%287%2C%201%29%2C%20C%288%2C%20-2%29%2C%20D%282%2C%20-4%29 caption="A(1, -1), B(7, 1), C(8, -2), D(2, -4)"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=A%2810%2C%204%29%2C%20B%288%2C%20-2%29%2C%20C%282%2C%202%29%2C%20D%284%2C%208%29 caption="A(10, 4), B(8, -2), C(2, 2), D(4, 8)"]]
 * 5) **//Challenge//** [[image:http://www.ck12.org/ck12/ucs/?math=SRUE caption="SRUE"]] is a rectangle and [[image:http://www.ck12.org/ck12/ucs/?math=PRUC caption="PRUC"]]is a square.
 * 6) What type of quadrilateral is [[image:http://www.ck12.org/ck12/ucs/?math=SPCE caption="SPCE"]]?
 * 7) If [[image:http://www.ck12.org/ck12/ucs/?math=SR%20%3D%2020 caption="SR = 20"]] and [[image:http://www.ck12.org/ck12/ucs/?math=RU%20%3D%2012 caption="RU = 12"]], find [[image:http://www.ck12.org/ck12/ucs/?math=CE caption="CE"]].
 * 8) Find [[image:http://www.ck12.org/ck12/ucs/?math=SC caption="SC"]] and [[image:http://www.ck12.org/ck12/ucs/?math=RC caption="RC"]] based on the information from part b. Round your answers to the nearest hundredth.

Review Queue Answers

 * 1) Possibilities: picture frame, door, baseball diamond, windows, walls, floor tiles, book cover, pages/paper, table/desk top, black/white board, the diamond suit (in a deck of cards).
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=x%20%3D%2011%2C%20%5C%20y%20%3D%206 caption="x = 11, \ y = 6"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=x%20%3D%20y%20%3D%2090%5E%5Ccirc caption="x = y = 90^\circ"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=x%20%3D%209%2C%20%5C%20y%20%3D%20133%5E%5Ccirc caption="x = 9, \ y = 133^\circ"]]

Learning Objectives

 * Define trapezoids, isosceles trapezoids, and kites.
 * Define the midsegments of trapezoids.
 * Plot trapezoids, isosceles trapezoids, and kites in the [[image:http://www.ck12.org/ck12/ucs/?math=x-y caption="x-y"]] plane.

Review Queue

 * 1) Draw a quadrilateral with __one__ set of parallel lines.
 * 2) Draw a quadrilateral with __one__ set of parallel lines __and__ two right angles.
 * 3) Draw a quadrilateral with __one__ set of parallel lines __and__ two congruent sides.
 * 4) Draw a quadrilateral with __one__ set of parallel lines __and__ three congruent sides.
 * Know What?** A kite, seen at the right, is made by placing two pieces of wood perpendicular to each other and one piece of wood is bisected by the other. The typical dimensions are included in the picture. If you have two pieces of wood, 36 inches and 54 inches, determine the values of [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]] and [[image:http://www.ck12.org/ck12/ucs/?math=2x caption="2x"]].

Trapezoids

 * Trapezoid:** A quadrilateral with exactly one pair of parallel sides.
 * Isosceles Trapezoid:** A trapezoid where the non-parallel sides are congruent.

Isosceles Trapezoids
Previously, we introduced the Base Angles Theorem with isosceles triangles, which says, the two base angles are congruent. This property holds true for isosceles trapezoids. **//The two angles along the same base in an isosceles trapezoid are congruent.//** If is an isosceles trapezoid, then  and. To find, set up an equation. Notice that. These angles will always be supplementary because of the Consecutive Interior Angles Theorem from Chapter 3.
 * Theorem 6-17:** The base angles of an isosceles trapezoid are congruent.
 * Example 1:** Look at trapezoid [[image:http://www.ck12.org/ck12/ucs/?math=TRAP caption="TRAP"]] below. What is [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20A caption="m \angle A"]]?
 * Solution:** [[image:http://www.ck12.org/ck12/ucs/?math=TRAP caption="TRAP"]] is an isosceles trapezoid. [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20R%20%3D%20115%5E%5Ccirc caption="m \angle R = 115^\circ"]] also.
 * Theorem 6-17 Converse:** If a trapezoid has congruent base angles, then it is an isosceles trapezoid.
 * Example 2:** Is [[image:http://www.ck12.org/ck12/ucs/?math=ZOID caption="ZOID"]] an isosceles trapezoid? How do you know?
 * Solution:** [[image:http://www.ck12.org/ck12/ucs/?math=40%5E%5Ccirc%20%5Cneq%2035%5E%5Ccirc caption="40^\circ \neq 35^\circ"]], [[image:http://www.ck12.org/ck12/ucs/?math=ZOID caption="ZOID"]] is not an isosceles trapezoid.
 * Isosceles Trapezoid Diagonals Theorem:** The diagonals of an isosceles trapezoid are congruent.
 * Example 3:** Show [[image:http://www.ck12.org/ck12/ucs/?math=TA%20%3D%20RP caption="TA = RP"]].
 * Solution:** Use the distance formula to show [[image:http://www.ck12.org/ck12/ucs/?math=TA%20%3D%20RP caption="TA = RP"]].

Midsegment of a Trapezoid
There is only one midsegment in a trapezoid. It will be parallel to the bases because it is located halfway between them. Tools Needed: graph paper, pencil, ruler 1. Plot and  and connect them. This is NOT an isosceles trapezoid. 2. Find the midpoint of the non-parallel sides by using the midpoint formula. Label them and. Connect the midpoints to create the midsegment. 3. Find the lengths of, , and. What do you notice? If is the midsegment, then. a) b) c) a) is the average of 12 and 26. b) 24 is the average of and 35. c) 20 is the average of  and.
 * Midsegment (of a trapezoid):** A line segment that connects the midpoints of the non-parallel sides.
 * Investigation 6-5: Midsegment Property**
 * Midsegment Theorem:** The length of the midsegment of a trapezoid is the average of the lengths of the bases.
 * Example 4:** **//Algebra Connection//** Find [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]]. All figures are trapezoids with the midsegment.
 * Solution:**

Kites
The last quadrilateral to study is a kite. Like you might think, it looks like a kite that flies in the air. From the definition, a kite could be concave. If a kite is concave, it is called a **//dart.//** The angles between the congruent sides are called **//vertex angles.//** The other angles are called **//non-vertex angles.//** If we draw the diagonal through the vertex angles, we would have two congruent triangles. __Given__: with  and __Prove__: If is a kite, then. If is a kite, then  and. The proof of Theorem 6-22 is very similar to the proof above for Theorem 6-21. and triangles are isosceles triangles, so  is the perpendicular bisector of  (Isosceles Triangle Theorem, Chapter 4). a) b) a) The two angles left are the non-vertex angles, which are congruent. b) The other non-vertex angle is also. To find the fourth angle, subtract the other three angles from. Be careful with the definition of a kite. The congruent pairs are distinct, which means that **//a rhombus and square cannot be a kite.//**
 * Kite:** A quadrilateral with two sets of adjacent congruent sides.
 * **//Statement//** || **//Reason//** ||
 * 1. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKE%7D%20%5Ccong%20%5Coverline%7BTE%7D caption="\overline{KE} \cong \overline{TE}"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKI%7D%20%5Ccong%20%5Coverline%7BTI%7D caption="\overline{KI} \cong \overline{TI}"]] || Given ||
 * 2. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BEI%7D%20%5Ccong%20%5Coverline%7BEI%7D caption="\overline{EI} \cong \overline{EI}"]] || Reflexive PoC ||
 * 3. [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20EKI%20%5Ccong%20%5Ctriangle%20ETI caption="\triangle EKI \cong \triangle ETI"]] || SSS ||
 * 4. [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20K%20%5Ccong%20%5Cangle%20T caption="\angle K \cong \angle T"]] || CPCTC ||
 * Theorem 6-21:** The non-vertex angles of a kite are congruent.
 * Theorem 6-22:** The diagonal through the vertex angles is the angle bisector for both angles.
 * Kite Diagonals Theorem:** The diagonals of a kite are perpendicular.
 * Example 5:** Find the missing measures in the kites below.
 * Solution:**
 * Example 6:** Use the Pythagorean Theorem to find the length of the sides of the kite.
 * Solution:** Recall that the Pythagorean Theorem is [[image:http://www.ck12.org/ck12/ucs/?math=a%5E2%20%2B%20b%5E2%20%3D%20c%5E2 caption="a^2 + b^2 = c^2"]], where [[image:http://www.ck12.org/ck12/ucs/?math=c caption="c"]] is the hypotenuse. In this kite, the sides are the hypotenuses.

Kites and Trapezoids in the Coordinate Plane
From this we see that the adjacent sides are congruent. Therefore, is a kite. Slope of Slope of, so is a trapezoid. To determine if it is an isosceles trapezoid, find and. , therefore this is only a trapezoid. All four sides are equal. This quadrilateral is either a **//rhombus//** or a **//square//**. Let’s find the length of the diagonals. The diagonals are not congruent, so is a rhombus.
 * Example 7:** Determine what type of quadrilateral [[image:http://www.ck12.org/ck12/ucs/?math=RSTV caption="RSTV"]] is.
 * Solution:** Find the lengths of all the sides.
 * Example 8:** Determine what type of quadrilateral [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]] is. [[image:http://www.ck12.org/ck12/ucs/?math=A%28-3%2C%203%29%2C%20B%281%2C%205%29%2C%20C%284%2C%20-1%29%2C%20D%281%2C%20-5%29 caption="A(-3, 3), B(1, 5), C(4, -1), D(1, -5)"]].
 * Solution:** First, graph [[image:http://www.ck12.org/ck12/ucs/?math=ABCD caption="ABCD"]]. This will make it easier to figure out what type of quadrilateral it is. From the graph, we can tell this is __not__ a parallelogram. Find the slopes of [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BBC%7D caption="\overline{BC}"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BAD%7D caption="\overline{AD}"]] to see if they are parallel.
 * Example 9:** Determine what type of quadrilateral [[image:http://www.ck12.org/ck12/ucs/?math=EFGH caption="EFGH"]] is. [[image:http://www.ck12.org/ck12/ucs/?math=E%285%2C%20-1%29%2C%20F%2811%2C%20-3%29%2C%20G%285%2C%20-5%29%2C%20H%28-1%2C%20-3%29 caption="E(5, -1), F(11, -3), G(5, -5), H(-1, -3)"]]
 * Solution:** To contrast with Example 8, we will not graph this example. Let’s find the length of all four sides.
 * Know What? Revisited** If the diagonals (pieces of wood) are 36 inches and 54 inches, [[image:http://www.ck12.org/ck12/ucs/?math=x caption="x"]] is half of 36, or 18 inches. Then, [[image:http://www.ck12.org/ck12/ucs/?math=2x caption="2x"]] is 36.

Review Questions
> > For questions 5-10, find the length of the midsegment or missing side. For questions 11-16, find the value of the missing variable(s). All figures are kites. Find the lengths of the diagonals of the trapezoids below to determine if it is isosceles. For questions 25-28, determine what type of quadrilateral is. could be any quadrilateral that we have learned in this chapter. If it is none of these, write none. Fill in the blanks to the proofs below. > __Prove__: is the angle bisector of  and > __Prove__:
 * Questions 1 and 2 are similar to Examples 1, 2, 5 and 6.
 * Questions 3 and 4 use the definitions of trapezoids and kites.
 * Questions 5-10 are similar to Example 4.
 * Questions 11-16 are similar to Examples 5 and 6.
 * Questions 17-22 are similar to Examples 4-6.
 * Questions 23 and 24 are similar to Example 3.
 * Questions 25-28 are similar to Examples 7-9.
 * Questions 29 and 30 are similar to the proof of Theorem 6-21.
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=TRAP caption="TRAP"]]an isosceles trapezoid. Find:
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20TPA caption="m \angle TPA"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20PTR caption="m \angle PTR"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20ZRA caption="m \angle ZRA"]]
 * 5) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20PZA caption="m \angle PZA"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=KITE caption="KITE"]]is a kite. Find:
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20ETS caption="m \angle ETS"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20KIT caption="m \angle KIT"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20IST caption="m \angle IST"]]
 * 5) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20SIT caption="m \angle SIT"]]
 * 6) [[image:http://www.ck12.org/ck12/ucs/?math=m%20%5Cangle%20ETI caption="m \angle ETI"]]
 * 1) **//Writing//** Can the parallel sides of a trapezoid be congruent? Why or why not?
 * 2) **//Writing//** Besides a kite and a rhombus, can you find another quadrilateral with perpendicular diagonals? Explain and draw a picture.
 * 1) [[image:http://www.ck12.org/ck12/images?id=323239 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=323210 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=323253 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323318 width="100"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323254 width="100"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=323291 width="100"]]
 * 1) [[image:http://www.ck12.org/ck12/images?id=323308 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=323233 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=323324 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323289 width="100"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323288 width="100"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=323348 width="100"]]
 * //Algebra Connection//** For questions 17-22, find the value of the missing variable(s).
 * 1) [[image:http://www.ck12.org/ck12/images?id=323302 width="80"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=323228 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=323181 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=323309 width="125"]]
 * 5) [[image:http://www.ck12.org/ck12/images?id=323338 width="150"]]
 * 6) [[image:http://www.ck12.org/ck12/images?id=323335 width="150"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=A%28-3%2C%202%29%2C%20B%281%2C%203%29%2C%20C%283%2C%20-1%29%2C%20D%28-4%2C%20-2%29 caption="A(-3, 2), B(1, 3), C(3, -1), D(-4, -2)"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=A%28-3%2C%203%29%2C%20B%282%2C%20-2%29%2C%20C%28-6%2C%20-6%29%2C%20D%28-7%2C%201%29 caption="A(-3, 3), B(2, -2), C(-6, -6), D(-7, 1)"]]
 * 1) [[image:http://www.ck12.org/ck12/ucs/?math=A%281%2C%20-2%29%2C%20B%287%2C%20-5%29%2C%20C%284%2C%20-8%29%2C%20D%28-2%2C%20-5%29 caption="A(1, -2), B(7, -5), C(4, -8), D(-2, -5)"]]
 * 2) [[image:http://www.ck12.org/ck12/ucs/?math=A%286%2C%206%29%2C%20B%2810%2C%208%29%2C%20C%2812%2C%204%29%2C%20D%288%2C%202%29 caption="A(6, 6), B(10, 8), C(12, 4), D(8, 2)"]]
 * 3) [[image:http://www.ck12.org/ck12/ucs/?math=A%28-1%2C%208%29%2C%20B%281%2C%204%29%2C%20C%28-5%2C%20-4%29%2C%20D%28-5%2C%206%29 caption="A(-1, 8), B(1, 4), C(-5, -4), D(-5, 6)"]]
 * 4) [[image:http://www.ck12.org/ck12/ucs/?math=A%285%2C%20-1%29%2C%20B%289%2C%20-4%29%2C%20C%286%2C%20-10%29%2C%20D%283%2C%20-5%29 caption="A(5, -1), B(9, -4), C(6, -10), D(3, -5)"]]
 * 1) __Given__: [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKE%7D%20%5Ccong%20%5Coverline%7BTE%7D caption="\overline{KE} \cong \overline{TE}"]]and [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKI%7D%20%5Ccong%20%5Coverline%7BTI%7D caption="\overline{KI} \cong \overline{TI}"]]
 * **//Statement//** || **//Reason//** ||
 * 1. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKE%7D%20%5Ccong%20%5Coverline%7BTE%7D caption="\overline{KE} \cong \overline{TE}"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKI%7D%20%5Ccong%20%5Coverline%7BTI%7D caption="\overline{KI} \cong \overline{TI}"]] ||  ||
 * 2. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BEI%7D%20%5Ccong%20%5Coverline%7BEI%7D caption="\overline{EI} \cong \overline{EI}"]] ||  ||
 * 3. [[image:http://www.ck12.org/ck12/ucs/?math=%5Ctriangle%20EKI%20%5Ccong%20%5Ctriangle%20ETI caption="\triangle EKI \cong \triangle ETI"]] ||  ||
 * 4. || CPCTC ||
 * 5. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BEI%7D caption="\overline{EI}"]] is the angle bisector of [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20KET caption="\angle KET"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20KIT caption="\angle KIT"]] ||  ||
 * 1) __Given__: [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BEK%7D%20%5Ccong%20%5Coverline%7BET%7D%2C%20%5Coverline%7BKI%7D%20%5Ccong%20%5Coverline%7BIT%7D caption="\overline{EK} \cong \overline{ET}, \overline{KI} \cong \overline{IT}"]]
 * **//Statement//** || **//Reason//** ||
 * 1. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKE%7D%20%5Ccong%20%5Coverline%7BTE%7D caption="\overline{KE} \cong \overline{TE}"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKI%7D%20%5Ccong%20%5Coverline%7BTI%7D caption="\overline{KI} \cong \overline{TI}"]] ||  ||
 * 2. || Definition of isosceles triangles ||
 * 3. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BEI%7D caption="\overline{EI}"]] is the angle bisector of [[image:http://www.ck12.org/ck12/ucs/?math=%5Cangle%20KET caption="\angle KET"]] and [[image:http://www.ck12.org/ck12/ucs/?math=%20%5Cangle%20KIT caption=" \angle KIT"]] ||  ||
 * 4. || Isosceles Triangle Theorem ||
 * 5. [[image:http://www.ck12.org/ck12/ucs/?math=%5Coverline%7BKT%7D%20%5Cbot%20%5Coverline%7BEI%7D caption="\overline{KT} \bot \overline{EI}"]] ||  ||

Review Queue Answers

 * 1) [[image:http://www.ck12.org/ck12/images?id=330827 width="100"]]
 * 2) [[image:http://www.ck12.org/ck12/images?id=330821 width="100"]]
 * 3) [[image:http://www.ck12.org/ck12/images?id=330825 width="100"]]
 * 4) [[image:http://www.ck12.org/ck12/images?id=330826 width="100"]]

Keywords and Theorems

 * Angles in Polygons**
 * Polygon Sum Formula
 * Equiangular Polygon Formula
 * Regular Polygon
 * Exterior Angle Sum Theorem
 * Properties of Parallelograms**
 * Parallelogram
 * Opposite Sides Theorem
 * Opposite Angles Theorem
 * Consecutive Angles Theorem
 * Parallelogram Diagonals Theorem
 * Proving Quadrilaterals are Parallelograms**
 * Opposite Sides Theorem Converse
 * Opposite Angles Theorem Converse
 * Consecutive Angles Theorem Converse
 * Parallelogram Diagonals Theorem Converse
 * Theorem 6-10
 * Rectangles, Rhombuses, and Squares**
 * Rectangle Theorem
 * Rhombus Theorem
 * Square Theorem
 * Theorem 6-14
 * Theorem 6-15
 * Theorem 6-16
 * Trapezoids and Kites**
 * Trapezoid
 * Isosceles Trapezoid
 * Theorem 6-17
 * Theorem 6-17 Converse
 * Isosceles Trapezoid Diagonals Theorem
 * Midsegment (of a trapezoid)
 * Midsegment Theorem
 * Kite
 * Theorem 6-21
 * Theorem 6-22
 * Kite Diagonals Theorem

Quadrilateral Flow Chart
Fill in the flow chart according to what you know about the quadrilaterals we have learned in this chapter. Determine if each quadrilateral has the given properties. If so, write yes or state how many sides (or angles) are congruent, parallel, or perpendicular.
 * Table Summary**
 * || **Opposite sides [[image:http://www.ck12.org/ck12/ucs/?math=%5C%7C caption="\|"]]** || **Diagonals bisect each other** || **Diagonals [[image:http://www.ck12.org/ck12/ucs/?math=%5Cbot caption="\bot"]]** || **Opposite sides [[image:http://www.ck12.org/ck12/ucs/?math=%5Ccong caption="\cong"]]** || **Opposite angles [[image:http://www.ck12.org/ck12/ucs/?math=%5Ccong caption="\cong"]]** || **Consecutive Angles add up to [[image:http://www.ck12.org/ck12/ucs/?math=180%5E%5Ccirc caption="180^\circ"]]** ||
 * Trapezoid ||  ||   ||   ||   ||   ||   ||
 * Isosceles Trapezoid ||  ||   ||   ||   ||   ||   ||
 * Kite ||  ||   ||   ||   ||   ||   ||
 * Parallelogram ||  ||   ||   ||   ||   ||   ||
 * Rectangle ||  ||   ||   ||   ||   ||   ||
 * Rhombus ||  ||   ||   ||   ||   ||
 * Square ||  ||   ||   ||   ||   ||   ||
 * 1) How many degrees are in a:
 * 2) triangle
 * 3) quadrilateral
 * 4) pentagon
 * 5) hexagon
 * 6) Find the measure of all the lettered angles below. The missing angle in the pentagon (at the bottom of the drawing), is [[image:http://www.ck12.org/ck12/ucs/?math=138%5E%5Ccirc caption="138^\circ"]].

Texas Instruments Resources

 * //In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See [].//**

Study Guide
Keywords: Define, write theorems, and/or draw a diagram for each word below. **Section: Angles in Polygons** Polygon Sum Formula Equiangular Polygon Formula Regular Polygon Exterior Angle Sum Theorem **Section: Properties of Parallelograms** Parallelogram Opposite Sides Theorem Opposite Angles Theorem Consecutive Angles Theorem Parallelogram Diagonals Theorem **Section: Proving Quadrilaterals are Parallelograms** Opposite Sides Theorem Converse Opposite Angles Theorem Converse Consecutive Angles Theorem Converse Parallelogram Diagonals Theorem Converse Theorem 6-10 **Section: Rectangles, Rhombuses, and Squares** Rectangle Theorem Rhombus Theorem Square Theorem Theorem 6-14 Theorem 6-15 Theorem 6-16 **Section: Trapezoids and Kites** Trapezoid Isosceles Trapezoid Theorem 6-17 Theorem 6-17 Converse Isosceles Trapezoid Diagonals Theorem Midsegment (of a trapezoid) Midsegment Theorem Kite Theorem 6-21 Theorem 6-22 Kite Diagonals Theorem
 * Homework:**
 * Homework:**
 * Homework:**
 * Homework:**
 * Homework:**

Need definitions - Conditional Statements, Inverse, Converse, Contrapositive, Open Sentences

Cool links to Logic? I know there is a lot of interesting links to "Alice in Wonderland" scenarios use to introduce high school Geometry logic concepts.

An E-How video on How to Solve Geometry Proofs: []

Annenberg teacher resources :insights into Algebra -- full lesson and a full video on quadratic equations

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More specifically []