Transcript OBJECTIVE: To verify and use the properties of

6-6 TRAPEZOIDS and
KITES
VOCABULARY
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 midsegment of a
trapezoid is the segment
that joins the midpoints
of its legs.
A kite is a
quadrilateral with
two pairs of
consecutive sides ≅
and no opposite
sides ≅.
6-6 TRAPEZOIDS and KITES
Word or Word
Phrase
trapezoid
legs of a trapezoid
bases of a trapezoid
Defintion
Picture or Example
A trapezoid is a quadrilateral with one
pair of parallel sides.
The legs of a trapezoid are the
non-parallel sides.
The bases of a trapezoid
are the parallel sides.
𝑻𝑷 𝒐𝒓 𝑹𝑨
𝑻𝑹 𝒐𝒓 𝑷𝑨
isosceles trapezoid
An isosceles trapezoid is a trapezoid
with legs that are congruent.
base angles
The base angles are the s that share
the base of a trapezoid.
kite
A kite is a quadrilateral with 2 pairs of consecutive, 
sides. In a kite, no opposite sides are .
midsegment
of a trapezoid
The midsegment of a trapezoid is the segment
that joins the midpoints of the legs.
A and B
or C and D
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 ≅.
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.
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.
6-6 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
OBJECTIVE: To verify and use the properties of trapezoids and kites.
6-6 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.
OBJECTIVE: To verify and use the properties of trapezoids and kites.
6-6 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
78
Q: What is the  measure of each one of
the 15 s meeting at the center?
𝟑𝟔𝟎°
= 𝟐𝟒°
𝟏𝟓
78
OBJECTIVE: To verify and use the properties of trapezoids and kites.
6-6 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.
OBJECTIVE: To verify and use the properties of trapezoids and kites.
6-6 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?
𝟑
How many midsegments can a trapezoid have? 𝟏
Explain. A  has 3 midsegments joining any pair of the side midpoints.
A trapezoid has 1 midsegment joining the midpoints of the two legs.
OBJECTIVE: To verify and use the properties of trapezoids and kites.
6-6 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°
OBJECTIVE: To verify and use the properties of trapezoids and kites.
6-6 TRAPEZOIDS and KITES
1. What are the measures of the numbered angles?
𝒎𝟏 = 𝟕𝟖°
𝒎𝟐 = 𝟗𝟎°
𝒎𝟏 = 𝟗𝟒°
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.
OBJECTIVE: To verify and use the properties of trapezoids and kites.
6-6 TRAPEZOIDS and KITES
3. Find the length of the perimeter of trapezoid LMNP with midsegment 𝑄𝑅.
7
8
Solve for PN :
𝟏
𝑸𝑹 = 𝑳𝑴 + 𝑷𝑵
𝟐
𝟏
 𝟐𝟓 = 𝟏𝟔 + 𝑷𝑵
𝟐
𝟓𝟎 = 16 + PN
𝟑𝟒 = PN
Perimeter of 𝑳𝑴𝑵𝑷 = 𝟐 𝟖 + 𝟏𝟔 + 𝟐 𝟕 + 𝟑𝟒
Perimeter of 𝑳𝑴𝑵𝑷 = 𝟖𝟎
OBJECTIVE: To verify and use the properties of trapezoids and kites.
6-6 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.