6.5: Trapezoids and Kites
Learning Objectives
 Define trapezoids, isosceles trapezoids, and kites.
 Define the midsegments of trapezoids.
 Plot trapezoids, isosceles trapezoids, and kites in the plane.
Review Queue
 Draw a quadrilateral with one set of parallel lines.
 Draw a quadrilateral with one set of parallel lines and two right angles.
 Draw a quadrilateral with one set of parallel lines and two congruent sides.
 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 and .
Trapezoids
Trapezoid: A quadrilateral with exactly one pair of parallel sides.
Isosceles Trapezoid: A trapezoid where the nonparallel 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.
Theorem 617: The base angles of an isosceles trapezoid are congruent.
If is an isosceles trapezoid, then and .
Example 1: Look at trapezoid below. What is ?
Solution: is an isosceles trapezoid. also.
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 617 Converse: If a trapezoid has congruent base angles, then it is an isosceles trapezoid.
Example 2: Is an isosceles trapezoid? How do you know?
Solution: , is not an isosceles trapezoid.
Isosceles Trapezoid Diagonals Theorem: The diagonals of an isosceles trapezoid are congruent.
Example 3: Show .
Solution: Use the distance formula to show .
Midsegment of a Trapezoid
Midsegment (of a trapezoid): A line segment that connects the midpoints of the nonparallel sides.
There is only one midsegment in a trapezoid. It will be parallel to the bases because it is located halfway between them.
Investigation 65: Midsegment Property
Tools Needed: graph paper, pencil, ruler
1. Plot and and connect them. This is NOT an isosceles trapezoid.
2. Find the midpoint of the nonparallel 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?
Midsegment Theorem: The length of the midsegment of a trapezoid is the average of the lengths of the bases.
If is the midsegment, then .
Example 4: Algebra Connection Find . All figures are trapezoids with the midsegment.
a)
b)
c)
Solution:
a) is the average of 12 and 26.
b) 24 is the average of and 35.
c) 20 is the average of and .
Kites
The last quadrilateral to study is a kite. Like you might think, it looks like a kite that flies in the air.
Kite: A quadrilateral with two sets of adjacent congruent sides.
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 nonvertex angles. If we draw the diagonal through the vertex angles, we would have two congruent triangles.
Given: with and
Prove:
Statement  Reason 

1. and  Given 
2.  Reflexive PoC 
3.  SSS 
4.  CPCTC 
Theorem 621: The nonvertex angles of a kite are congruent.
If is a kite, then .
Theorem 622: The diagonal through the vertex angles is the angle bisector for both angles.
If is a kite, then and .
The proof of Theorem 622 is very similar to the proof above for Theorem 621.
Kite Diagonals Theorem: The diagonals of a kite are perpendicular.
and triangles are isosceles triangles, so is the perpendicular bisector of (Isosceles Triangle Theorem, Chapter 4).
Example 5: Find the missing measures in the kites below.
a)
b)
Solution:
a) The two angles left are the nonvertex angles, which are congruent.
b) The other nonvertex 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.
Example 6: Use the Pythagorean Theorem to find the length of the sides of the kite.
Solution: Recall that the Pythagorean Theorem is , where is the hypotenuse. In this kite, the sides are the hypotenuses.
Kites and Trapezoids in the Coordinate Plane
Example 7: Determine what type of quadrilateral is.
Solution: Find the lengths of all the sides.
From this we see that the adjacent sides are congruent. Therefore, is a kite.
Example 8: Determine what type of quadrilateral is. .
Solution: First, graph . 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 and to see if they are parallel.
Slope of
Slope of
, so is a trapezoid. To determine if it is an isosceles trapezoid, find and .
, therefore this is only a trapezoid.
Example 9: Determine what type of quadrilateral is.
Solution: To contrast with Example 8, we will not graph this example. Let’s find the length of all four sides.
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.
Know What? Revisited If the diagonals (pieces of wood) are 36 inches and 54 inches, is half of 36, or 18 inches. Then, is 36.
Review Questions
 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 510 are similar to Example 4.
 Questions 1116 are similar to Examples 5 and 6.
 Questions 1722 are similar to Examples 46.
 Questions 23 and 24 are similar to Example 3.
 Questions 2528 are similar to Examples 79.
 Questions 29 and 30 are similar to the proof of Theorem 621.

an isosceles trapezoid. Find:

is a kite. Find:
 Writing Can the parallel sides of a trapezoid be congruent? Why or why not?
 Writing Besides a kite and a rhombus, can you find another quadrilateral with perpendicular diagonals? Explain and draw a picture.
For questions 510, find the length of the midsegment or missing side.
For questions 1116, find the value of the missing variable(s). All figures are kites.
Algebra Connection For questions 1722, find the value of the missing variable(s).
Find the lengths of the diagonals of the trapezoids below to determine if it is isosceles.
For questions 2528, 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.
 Given: and Prove: is the angle bisector of and
Statement  Reason 

1. and  
2.  
3.  
4.  CPCTC 
5. is the angle bisector of and 
 Given: Prove:
Statement  Reason 

1. and  
2.  Definition of isosceles triangles 
3. is the angle bisector of and  
4.  Isosceles Triangle Theorem 
5. 