Trapezoids and Kites LESSON 6-5 Additional Examples

XYZW

is an isosceles trapezoid, and

m



X

Find

m



Y

,

m



Z

, and

m



W

.

= 156.

m



X

+

m



W

= 180 Two angles that share a leg of a trapezoid are supplementary.

156 +

m



W

= 180 Substitute.

m



W

= 24 Subtract 156 from each side.

Because the base angles of an isosceles trapezoid are congruent,

m



Y

=

m



X

= 156 and

m



Z

=

m



W

= 24.

Quick Check HELP GEOMETRY

Trapezoids and Kites LESSON 6-5 Additional Examples

Half of a spider’s web is shown below, formed by layers of congruent isosceles trapezoids. Find the measures of the angles in

ABDC

.

Trapezoid

ABDC

is part of an isosceles triangle whose vertex angle is at the center of the spider web. Because there are 6 adjacent congruent vertex angles at the center of the web, By the Triangle Angle-Sum Theorem,

m



A

+

m



B

= 150.

together forming a straight angle, each vertex angle measures , or 30.

6

m



A

+

m



B

+ 30 = 180, so Because

2m



A ABDC

is part of an isosceles triangle,

m



A

= 150 and

m



A

=

m



B

= 75.

=

m



B

, so

HELP GEOMETRY

Trapezoids and Kites LESSON 6-5 Additional Examples (continued)

Another way to find the measure of each acute angle is to divide the difference of 180 and the measure of the vertex angle by 2: 180 – 30 2 = 75 Because the bases of a trapezoid are parallel, the two angles that share a leg are supplementary, so

m C

=

m D

= 180 – 75 = 105.

Quick Check HELP GEOMETRY

Trapezoids and Kites LESSON 6-5 Additional Examples

Find

m

1,

m

2, and

m

3 in the kite.

HELP

m

2 = 90

RU

=

RS m

1 = 72

m

3 +

m RDU

+ 72 = 180

m RDU

= 90

m

3 + 90 + 72 = 180

m

3 + 162 = 180

m

3 = 18 Diagonals of a kite are perpendicular.

Definition of a kite Isosceles Triangle Theorem Triangle Angle –Sum Theorem Diagonals of a kite are perpendicular.

Substitute.

Simplify.

Subtract 162 from each side.

Quick Check GEOMETRY