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Geometry Chapter 8
8-5: USE PROPERTIES OF TRAPEZOIDS AND KITES
Use Properties of Trapezoids and Kites
Objective: Students will be able to identify and use properties
to solve trapezoids and kites.
Agenda
Trapezoid
Kite
Examples
Trapezoid
A quadrilateral with exactly one pair of parallel sides is called a Trapezoid.
The parallel sides are called the bases.
The other sides are called the legs.
base
base
legleg
Trapezoid
A quadrilateral with exactly one pair of parallel sides is called a Trapezoid.
The parallel sides are called the bases.
The other sides are called the legs.
base
base
legleg
140° 100°
40° 80°base
base
leg
leg
70°
90° 90°
110°
Trapezoid
A quadrilateral with exactly one pair of parallel sides is called a Trapezoid.
The parallel sides are called the bases.
The other sides are called the legs.
Knowledge Connection:
What do you notice
about the angles?
base
base
legleg
140° 100°
40° 80°base
base
leg
leg
70°
90° 90°
110°
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
To see if 𝑸𝑹𝑺𝑻 is a trapezoid, we must show
that it has only 1 pair of opposite sides that
are parallel.
To do that, we must find the slope of all four
sides and compare them.
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
You can use the slope equation, or you can
count the rise and run, to find the slope of
each line.
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
Slope 𝑅𝑆
𝑚 =4 − 3
2 − 0
𝒎 =𝟏
𝟐
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
Slope 𝑅𝑆
𝑚 =4 − 3
2 − 0
𝒎 =𝟏
𝟐
Slope 𝑄𝑇
𝑚 =2 − 0
4 − 0=2
4
𝒎 =𝟏
𝟐
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
Slope 𝑅𝑆
𝑚 =4 − 3
2 − 0
𝒎 =𝟏
𝟐
Slope 𝑄𝑇
𝑚 =2 − 0
4 − 0=2
4
𝒎 =𝟏
𝟐
Same slope, thus 𝑹𝑺 ∥ 𝑸𝑻
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
Slope 𝑄𝑅
𝑚 =3 − 0
0 − 0=3
0
𝑺𝒍𝒐𝒑𝒆 𝑼𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
Slope 𝑄𝑅
𝑚 =3 − 0
0 − 0=3
0
𝑺𝒍𝒐𝒑𝒆 𝑼𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
Slope 𝑆𝑇
𝑚 =4 − 2
2 − 4=
2
−2
𝒎 = −𝟏
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
Slope 𝑄𝑅
𝑚 =3 − 0
0 − 0=3
0
𝑺𝒍𝒐𝒑𝒆 𝑼𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅
Slope 𝑆𝑇
𝑚 =4 − 2
2 − 4=
2
−2
𝒎 = −𝟏
Slopes are not the same, thus 𝑸𝑹 ∦ 𝑺𝑻
Example 1
Use the graph to show that quadrilateral 𝑸𝑹𝑺𝑻 is a trapezoid.
We showed that exactly 1 pair of parallel
opposites sides.
Thus, 𝑸𝑹𝑺𝑻 is a Trapezoid.
Example 2
Use the graph to show that quadrilateral 𝑾𝑿𝒀𝒁 is a trapezoid.
Example 2
Use the graph to show that quadrilateral 𝑾𝑿𝒀𝒁 is a trapezoid.
Slope 𝑊𝑋
𝑚 =1 − (−3)
−2 − 6=
4
−8
𝒎 = −𝟏
𝟐
Example 2
Use the graph to show that quadrilateral 𝑾𝑿𝒀𝒁 is a trapezoid.
Slope 𝑊𝑋
𝑚 =1 − (−3)
−2 − 6=
4
−8
𝒎 = −𝟏
𝟐
Slope 𝑌𝑍
𝑚 =1 − 4
5 − (−1)=−3
6
𝒎 = −𝟏
𝟐
Example 2
Use the graph to show that quadrilateral 𝑾𝑿𝒀𝒁 is a trapezoid.
Slope 𝑊𝑋
𝑚 =1 − (−3)
−2 − 6=
4
−8
𝒎 = −𝟏
𝟐
Slope 𝑌𝑍
𝑚 =1 − 4
5 − (−1)=−3
6
𝒎 = −𝟏
𝟐
Same slope, thus 𝑾𝑿 ∥ 𝒀𝒁
Example 2
Use the graph to show that quadrilateral 𝑾𝑿𝒀𝒁 is a trapezoid.
Slope 𝑋𝑌
𝑚 =4 − 1
−1 − (2)=3
1
𝒎 = 𝟑
Example 2
Use the graph to show that quadrilateral 𝑾𝑿𝒀𝒁 is a trapezoid.
Slope 𝑋𝑌
𝑚 =4 − 1
−1 − (2)=3
1
𝒎 = 𝟑
Slope 𝑊𝑍
𝑚 =−3 − 1
6 − 5=−4
1
𝒎 = −𝟒
Example 2
Use the graph to show that quadrilateral 𝑾𝑿𝒀𝒁 is a trapezoid.
Slope 𝑋𝑌
𝑚 =4 − 1
−1 − (2)=3
1
𝒎 = 𝟑
Slope 𝑊𝑍
𝑚 =−3 − 1
6 − 5=−4
1
𝒎 = −𝟒
Slopes are not the same, thus 𝑿𝒀 ∦ 𝑾𝒁
Example 2
Use the graph to show that quadrilateral 𝑾𝑿𝒀𝒁 is a trapezoid.
We showed had exactly 1 pair of parallel
opposites sides.
Thus, 𝑾𝑿𝒀𝒁 is a Trapezoid.
Isosceles Trapezoids
A Trapezoid with congruent legs is known as an
Isosceles Trapezoid.
Theorem 8.14
Theorem 8.14: If a trapezoid is isosceles, then each pair of
base angles is congruent.
If trapezoid 𝑨𝑩𝑪𝑫 is isosceles, then
< 𝑨 ≅< 𝑫 and < 𝑩 ≅< 𝑪
𝑩 𝑪
𝑨 𝑫
Theorem 8.15
Theorem 8.15: If a trapezoid has a pair of congruent base
angles, then it is an isosceles trapezoid.
In trapezoid 𝑨𝑩𝑪𝑫,
If < 𝑨 ≅< 𝑫 (or < 𝑩 ≅< 𝑪)
Then 𝑨𝑩𝑪𝑫 is an isosceles trapezoid
𝑩 𝑪
𝑨 𝑫
Theorem 8.16
Theorem 8.16: A trapezoid is isosceles if and only if its
diagonals are congruent.
Trapezoid 𝑨𝑩𝑪𝑫 is isosceles iff
𝑨𝑪 ≅ 𝑩𝑫
𝑩 𝑪
𝑨 𝑫
Example 3
Use Trapezoid 𝑬𝑭𝑮𝑯 to answer the following.
𝑬 𝑭
𝑯 𝑮
a.) If 𝑬𝑮 = 𝑭𝑯, is trapezoid 𝑬𝑭𝑮𝑯 isosceles?
Example 3
Use Trapezoid 𝑬𝑭𝑮𝑯 to answer the following.
𝑬 𝑭
𝑯 𝑮
a.) Answer: Yes
Since 𝑬𝑮 = 𝑭𝑯, then 𝑬𝑮 ≅ 𝑭𝑯,
making trapezoid 𝑬𝑭𝑮𝑯 isosceles by
thm 8-16
Example 3
Use Trapezoid 𝑬𝑭𝑮𝑯 to answer the following.
𝑬 𝑭
𝑯 𝑮
b.) If 𝒎 < 𝑯𝑬𝑭 = 𝟕𝟎°, and 𝒎 < 𝑭𝑮𝑯 = 𝟏𝟏𝟎°, is trapezoid 𝑬𝑭𝑮𝑯 isosceles?
Example 3
Use Trapezoid 𝑬𝑭𝑮𝑯 to answer the following.
𝑬 𝑭
𝑯 𝑮
𝟏𝟏𝟎°
𝟕𝟎°b.) If 𝒎 < 𝑯𝑬𝑭 = 𝟕𝟎°, and 𝒎 < 𝑭𝑮𝑯 = 𝟏𝟏𝟎°, is trapezoid 𝑬𝑭𝑮𝑯 isosceles?
Example 3
Use Trapezoid 𝑬𝑭𝑮𝑯 to answer the following.
b.) We can make 𝒎 < 𝑬𝑯𝑮 = 𝟏𝟏𝟎°and 𝒎 < 𝑬𝑭𝑮 = 𝟕𝟎°. (How?)
𝑬 𝑭
𝑯 𝑮
𝟏𝟏𝟎°
𝟕𝟎°
Example 3
Use Trapezoid 𝑬𝑭𝑮𝑯 to answer the following.
b.) Answer: Yes
We will have < 𝑯𝑬𝑭 ≅< 𝑬𝑭𝑮 and
< 𝑭𝑮𝑯 ≅ 𝑬𝑯𝑮, making trapezoid
𝑬𝑭𝑮𝑯 isosceles by thm 8.15
𝑬 𝑭
𝑯 𝑮
𝟏𝟏𝟎°
𝟕𝟎°
Midsegment
The Midsegment of a trapezoid is the segment that
connects the midpoints of its legs.
Midsegment
Theorem 8.17
Theorem 8.17 – Midsegment Theorem for Trapezoids: The midsegment of
a trapezoid is parallel to each base and its length is one half the sum of
the lengths of the bases.
If 𝑀𝑁 is the midsegment of trapezoid
𝑨𝑩𝑪𝑫,
Then 𝑴𝑵 ∥ 𝑨𝑩, 𝑴𝑵 ∥ 𝑫𝑪, and
𝑴𝑵 =𝟏
𝟐(𝑨𝑩 + 𝑪𝑫)
𝑩
𝑪
𝑨
𝑫
𝑵𝑴
Example 4
In the diagram, 𝑴𝑵 is the midsegment of the trapezoid 𝑷𝑸𝑹𝑺. Find 𝑴𝑵.
12 in.
28 in.
Example 4
In the diagram, 𝑴𝑵 is the midsegment of the trapezoid 𝑷𝑸𝑹𝑺. Find 𝑴𝑵.
12 in.
28 in.
𝑀𝑁 =1
2(𝑃𝑄 + 𝑆𝑅)
𝑀𝑁 =1
2(12 + 28)
𝑀𝑁 =1
240 = 𝟐𝟎
Example 4
In the diagram, 𝑴𝑵 is the midsegment of the trapezoid 𝑷𝑸𝑹𝑺. Find 𝑴𝑵.
12 in.
28 in.
𝑀𝑁 =1
2(𝑃𝑄 + 𝑆𝑅)
𝑀𝑁 =1
2(12 + 28)
𝑀𝑁 =1
240 = 𝟐𝟎
The length of 𝑴𝑵 is
20 inches.
Example 5
In the diagram, 𝑯𝑲 is the midsegment of the trapezoid 𝑫𝑬𝑭𝑮. Find 𝑯𝑲.
𝑬
𝑭
𝑫
𝑮
𝑲𝑯
6 in.
18 in.
Example 5
In the diagram, 𝑯𝑲 is the midsegment of the trapezoid 𝑫𝑬𝑭𝑮. Find 𝑯𝑲.
𝐻𝐾 =1
2(𝐷𝐸 + 𝐺𝐹)
𝐻𝐾 =1
2(6 + 18)
𝐻𝐾 =1
224 = 𝟏𝟐
𝑬
𝑭
𝑫
𝑮
𝑲𝑯
6 in.
18 in.
Example 5
In the diagram, 𝑯𝑲 is the midsegment of the trapezoid 𝑫𝑬𝑭𝑮. Find 𝑯𝑲.
𝐻𝐾 =1
2(𝐷𝐸 + 𝐺𝐹)
𝐻𝐾 =1
2(6 + 18)
𝐻𝐾 =1
224 = 𝟏𝟐
The length of 𝑯𝑲 is 12
inches.
𝑬
𝑭
𝑫
𝑮
𝑲𝑯
6 in.
18 in.
Kite
A Kite is a quadrilateral with one pair of congruent
consecutive sides, but no opposite sides are congruent.
𝑩
𝑪
𝑨
𝑫
Theorem 8.18
Theorem 8.18: If a quadrilateral is a kite, then its diagonals are
perpendicular.
If 𝑨𝑩𝑪𝑫 is a kite,
then 𝑨𝑪 ⊥ 𝑩𝑫𝑩
𝑪
𝑨
𝑫
Theorem 8.19
Theorem 8.19: If a quadrilateral is a kite, then exactly one pair
of opposite angles is congruent.
If 𝑨𝑩𝑪𝑫 is a kite,
then < 𝑨 ≅< 𝑪
(Or < 𝑩 ≅< 𝑫)𝑩
𝑪
𝑨
𝑫
Example 6
Find 𝑚 < 𝐶 in the kite shown.
𝟖𝟒°
𝟏𝟒𝟎°
Example 6
Find 𝑚 < 𝐶 in the kite shown.
𝟖𝟒°
𝟏𝟒𝟎°
𝑚 < 𝐴 +𝑚 < 𝐶 + 84 + 140 = 360
Example 6
Find 𝑚 < 𝐶 in the kite shown.
𝟖𝟒°
𝟏𝟒𝟎°
𝑚 < 𝐴 +𝑚 < 𝐶 + 84 + 140 = 360
2 𝑚 < 𝐶 + 224 = 360
2 𝑚 < 𝐶 = 136
𝒎 < 𝑪 = 𝟔𝟖°
Example 7
Find 𝒎 < 𝑫 in the kite shown.
Example 7
Find 𝒎 < 𝑫 in the kite shown.
𝑚 < 𝐷 +𝑚 < 𝐹 + 73 + 115 = 360
Example 7
Find 𝒎 < 𝑫 in the kite shown.
𝑚 < 𝐷 +𝑚 < 𝐹 + 73 + 115 = 360
2 𝑚 < 𝐷 + 188 = 360
2 𝑚 < 𝐷 = 172
𝒎 < 𝑫 = 𝟖𝟔°
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