1. The parallel sides of a trapezoid are called bases.
VOCABULARY
6-6 TRAPEZOIDS and KITES
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
The parallel sides of a trapezoid are called bases.
The nonparallel sides are called legs.
The two s that share a base of a trapezoid are called
base angles. A trapezoid has two pairs of base s.
An isosceles trapezoid is a trapezoid with legs that are ≅.
A kite is a quadrilateral with two pairs of
consecutive sides ≅ and no opposite sides ≅.
A midsegment of a trapezoid is the segment that joins the midpoints of its legs.

2. The legs of a trapezoid are the non-parallel sides.
6-6 TRAPEZOIDS and KITES
Word or Word Phrase
Defintion
Picture or Example
trapezoid
A trapezoid is a quadrilateral with one pair of parallel sides.
legs of a trapezoid
𝑻𝑷 𝒐𝒓 𝑹𝑨
bases of a trapezoid
𝑻𝑹 𝒐𝒓 𝑷𝑨
isosceles trapezoid
An isosceles trapezoid is a trapezoid with legs that are congruent.
base angles
A and B
or C and D
kite
A kite is a quadrilateral with 2 pairs of consecutive,  sides. In a kite, no opposite sides are .
midsegment
of a trapezoid
The legs of a trapezoid are the
non-parallel sides.
The bases of a trapezoid
are the parallel sides.
The base angles are the s that share the base of a trapezoid.
The midsegment of a trapezoid is the segment that joins the midpoints of the legs.

3. ∴ In each region, the s are either supplementary or ≅.
6-6 TRAPEZOIDS and KITES
OBJECTIVES: To verify and use the properties of trapezoids and kites.
Two isosceles triangles form the figure below. Each white segment is a midsegment of a triangle. What can you determine about the angles in region 2? In region 3? Explain.
The midsegment of each isosceles  is ‖ to its base,
so same-side interior s are supplementary.
Since base s in an isosceles  are ≅,
so the s sharing the midsegment of each  are ≅.
∴ In each region, the s are either supplementary or ≅.

4. If a quadrilateral is an isosceles trapezoid,
6-6 TRAPEZOIDS and KITES
OBJECTIVES: To verify and use the properties of trapezoids and kites.
Theorem 6-20
If a quadrilateral is an isosceles trapezoid,
then each pair of base angles is congruent.
Theorem 6-21
If a quadrilateral is an isosceles trapezoid,
then its diagonals are congruent.

5. If a quadrilateral is a trapezoid, then
6-6 TRAPEZOIDS and KITES
OBJECTIVES: To verify and use the properties of trapezoids and kites.
Theorem 6-22
If a quadrilateral is a trapezoid, then
(1) the midsegment is parallel to the bases, and
(2) the length of the midsegment is half the sum of the lengths of the bases.

6. Concept Summary - Relationships Among Quadrilaterals
6-6 TRAPEZOIDS and KITES
OBJECTIVES: To verify and use the properties of trapezoids and kites.
Theorem 6-23
If a quadrilateral is a kite, then
its diagonals are perpendicular.
Concept Summary Relationships Among Quadrilaterals

7. 𝑫𝑬 ‖ 𝑪𝑭 , so same-side interior s are supplementary.
6-6 TRAPEZOIDS and KITES
OBJECTIVE: To verify and use the properties of trapezoids and kites.
Finding Angle Measures in Trapezoids
a. In the diagram, PQRS is an isosceles trapezoid and mR = 106. What are mP, mQ, and mS?
𝒎𝑷=𝒎𝑸=𝟕𝟒
𝒎𝑺=𝟏𝟎𝟔
b. In Problem 1, if CDEF were not an isosceles trapezoid, would C and D still be supplementary? Explain.
𝐘𝐞𝐬;
𝑫𝑬 ‖ 𝑪𝑭 , so same-side interior s are supplementary.

8. Finding Angle Measures in Isosceles Trapezoids
6-6 TRAPEZOIDS and KITES
OBJECTIVE: To verify and use the properties of trapezoids and kites.
Finding Angle Measures in Isosceles Trapezoids
A fan like the one in Problem 2 has 15 congruent angles meeting at the center.
What are the measures of the base angles of the trapezoids in its second ring?
24
acute angles measure 𝟕𝟖
obtuse angles measure 𝟏𝟎𝟐
102
102
78
78
Q: What is the  measure of each one of
the 15 s meeting at the center?
𝟑𝟔𝟎° 𝟏𝟓 =𝟐𝟒°

9. Investigating the Diagonals of Isosceles Trapezoids
6-6 TRAPEZOIDS and KITES
OBJECTIVE: To verify and use the properties of trapezoids and kites.
Investigating the Diagonals of Isosceles Trapezoids
Choose from a variety of tools (such as a protractor, a ruler, or a compass) to investigate patterns in the diagonals of isosceles trapezoid PQRS. Explain your choice. Do your observations support your conjecture in Problem 3? Explain your reasoning.
In Problem 3 (HH): Use a protractor to measure the s formed by the diagonals and a compass to check if the diagonals are ≅.
Answer: Use a ruler to measure the segments. 𝑷𝑹=𝑸𝑺, 𝐭𝐡𝐮𝐬 𝑷𝑹 ≅ 𝑸𝑺.
This supports the conjecture that if a quadrilateral is an isosceles
trapezoid, then the diagonals are congruent.

10. b. How many midsegments can a triangle have?
6-6 TRAPEZOIDS and KITES
OBJECTIVE: To verify and use the properties of trapezoids and kites.
Using the Midsegment of a Trapezoid
a. 𝑴𝑵 is the midsegment of trapezoid PQRS. What is x? What is MN?
𝒙=𝟔
𝑴𝑵=23
b. How many midsegments can a triangle have?
How many midsegments can a trapezoid have?
Explain. 𝟑 𝟏
A  has 3 midsegments joining any pair of the side midpoints.
A trapezoid has 1 midsegment joining the midpoints of the two legs.

11. Finding Angle Measures in Kites
6-6 TRAPEZOIDS and KITES
OBJECTIVE: To verify and use the properties of trapezoids and kites.
Finding Angle Measures in Kites
Quadrilateral KLMN is a kite. What are m1, m2, and m3?
𝒎𝟏=𝟗𝟎°
𝒎𝟑=𝟑𝟔°
𝒎𝟐=𝟓𝟒°

12. 1. What are the measures of the numbered angles?
6-6 TRAPEZOIDS and KITES
OBJECTIVE: To verify and use the properties of trapezoids and kites.
1. What are the measures of the numbered angles?
𝒎𝟏=𝟕𝟖°
𝒎𝟏=𝟗𝟒°
𝒎𝟐=𝟗𝟎°
𝒎𝟐=𝟏𝟑𝟐°
𝒎𝟑=𝟏𝟐°
2. Quadrilateral WXYZ is an isosceles trapezoid. Are the two trapezoids formed
by drawing midsegment QR isosceles trapezoids? Explain.
Yes, the midsegment is ‖ to both bases and bisects each of the two congruent legs.

13. 6-6 TRAPEZOIDS and KITES
OBJECTIVE: To verify and use the properties of trapezoids and kites.
3. Find the length of the perimeter of trapezoid LMNP with midsegment 𝑄𝑅 .
Solve for PN :  7  
𝑸𝑹= 𝟏 𝟐 𝑳𝑴+𝑷𝑵
 𝟐𝟓= 𝟏 𝟐 𝟏𝟔+𝑷𝑵 8  
𝟓𝟎= 16 + PN
𝟑𝟒= PN
Perimeter of 𝑳𝑴𝑵𝑷=𝟐 𝟖 +𝟏𝟔+𝟐 𝟕 +𝟑𝟒
Perimeter of 𝑳𝑴𝑵𝑷=𝟖𝟎

14. No. No, a kite’s opposite sides are not ‖ or ≅ .
6-6 TRAPEZOIDS and KITES
OBJECTIVE: To verify and use the properties of trapezoids and kites.
4. Vocabulary Is a kite a parallelogram? Explain.
No, a kite’s opposite sides are not ‖ or ≅ .
5. Analyze Mathematical Relationships (1)(F)
How is a kite similar to a rhombus? How is it different? Explain.
Similar: Their diagonals are ⏊.
Different: Only one diagonal of a kite bisects opposite s;
a rhombus has all sides ≅.
6. Evaluate Reasonableness (1)(B) Since a parallelogram has two pairs of parallel sides, it certainly has one pair of parallel sides. Therefore, a parallelogram must also be a trapezoid. Is this reasoning correct? Explain.
No.
A trapezoid is defined as a quad. with exactly 1 pair of ‖ sides and a parallelogram has exactly 2 pairs of ‖ sides, so a parallelogram is not a trapezoid.