Geometry Chapter 8rgeometry16-17.weebly.com/uploads/8/6/1/2/86123628/... · Trapezoid A...

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




base

base

legleg

140° 100°

40° 80°base

base

leg

leg

70°

90° 90°

110°

Trapezoid




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


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


You can use the slope equation, or you can

count the rise and run, to find the slope of

each line.

Example 1


Slope 𝑅𝑆

𝑚 =4 − 3

2 − 0

𝒎 =𝟏

𝟐

Example 1


Slope 𝑅𝑆

𝑚 =4 − 3

2 − 0

𝒎 =𝟏

𝟐

Slope 𝑄𝑇

𝑚 =2 − 0

4 − 0=2

4

𝒎 =𝟏

𝟐

Example 1


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


Slope 𝑄𝑅

𝑚 =3 − 0

0 − 0=3

0

𝑺𝒍𝒐𝒑𝒆 𝑼𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅

Example 1


Slope 𝑄𝑅

𝑚 =3 − 0

0 − 0=3

0


Slope 𝑆𝑇

𝑚 =4 − 2

2 − 4=

2

−2

𝒎 = −𝟏

Example 1


Slope 𝑄𝑅

𝑚 =3 − 0

0 − 0=3

0


Slope 𝑆𝑇

𝑚 =4 − 2

2 − 4=

2

−2

𝒎 = −𝟏

Slopes are not the same, thus 𝑸𝑹 ∦ 𝑺𝑻

Example 1


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


Slope 𝑊𝑋

𝑚 =1 − (−3)

−2 − 6=

4

−8

𝒎 = −𝟏

𝟐

Example 2


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


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


Slope 𝑋𝑌

𝑚 =4 − 1

−1 − (2)=3

1

𝒎 = 𝟑

Example 2


Slope 𝑋𝑌

𝑚 =4 − 1

−1 − (2)=3

1

𝒎 = 𝟑

Slope 𝑊𝑍

𝑚 =−3 − 1

6 − 5=−4

1

𝒎 = −𝟒

Example 2


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


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


𝑬 𝑭

𝑯 𝑮

a.) Answer: Yes

Since 𝑬𝑮 = 𝑭𝑯, then 𝑬𝑮 ≅ 𝑭𝑯,

making trapezoid 𝑬𝑭𝑮𝑯 isosceles by

thm 8-16

Example 3


𝑬 𝑭

𝑯 𝑮

b.) If 𝒎 < 𝑯𝑬𝑭 = 𝟕𝟎°, and 𝒎 < 𝑭𝑮𝑯 = 𝟏𝟏𝟎°, is trapezoid 𝑬𝑭𝑮𝑯 isosceles?

Example 3


𝑬 𝑭

𝑯 𝑮

𝟏𝟏𝟎°

𝟕𝟎°b.) If 𝒎 < 𝑯𝑬𝑭 = 𝟕𝟎°, and 𝒎 < 𝑭𝑮𝑯 = 𝟏𝟏𝟎°, is trapezoid 𝑬𝑭𝑮𝑯 isosceles?

Example 3


b.) We can make 𝒎 < 𝑬𝑯𝑮 = 𝟏𝟏𝟎°and 𝒎 < 𝑬𝑭𝑮 = 𝟕𝟎°. (How?)

𝑬 𝑭

𝑯 𝑮

𝟏𝟏𝟎°

𝟕𝟎°

Example 3


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


12 in.

28 in.

𝑀𝑁 =1

2(𝑃𝑄 + 𝑆𝑅)

𝑀𝑁 =1

2(12 + 28)

𝑀𝑁 =1

240 = 𝟐𝟎

Example 4


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


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


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


𝟖𝟒°

𝟏𝟒𝟎°

𝑚 < 𝐴 +𝑚 < 𝐶 + 84 + 140 = 360

Example 6


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


𝑚 < 𝐷 +𝑚 < 𝐹 + 73 + 115 = 360

Example 7


𝑚 < 𝐷 +𝑚 < 𝐹 + 73 + 115 = 360

2 𝑚 < 𝐷 + 188 = 360

2 𝑚 < 𝐷 = 172

𝒎 < 𝑫 = 𝟖𝟔°

Geometry Chapter 8rgeometry16-17.weebly.com/uploads/8/6/1/2/86123628/... · Trapezoid A...

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