GEOMETRY

XYZW is an isosceles trapezoid, and m X = 156.

Find mY, mZ, and mW.

m X + mW = 180 Two angles that share a leg of a trapezoid are supplementary.

156 + mW = 180 Substitute.

mW = 24 Subtract 156 from each side.

Because the base angles of an isosceles trapezoid are congruent, mY = m X = 156 and mZ = mW = 24.

Trapezoids and KitesLESSON 6-5

Additional Examples

GEOMETRY

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, together forming a straight angle, each vertex angle measures , or 30.180

6

By the Triangle Angle-Sum Theorem, m A + m B + 30 = 180, so m A + m B = 150.

Because ABDC is part of an isosceles triangle, m A = m B, so 2m A = 150 and m A = m B = 75.

Trapezoids and KitesLESSON 6-5

Additional Examples

GEOMETRY

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.

(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 – 302 = 75

Trapezoids and KitesLESSON 6-5

Additional Examples

GEOMETRY

Find m 1, m 2, and m 3 in the kite.

RU = RS Definition of a kite

m 3 + m RDU + 72 = 180 Triangle Angle–Sum Theorem

m 2 = 90 Diagonals of a kite are perpendicular.

m 1 = 72 Isosceles Triangle Theorem

m RDU = 90 Diagonals of a kite are perpendicular.

m 3 + 90 + 72 = 180 Substitute.

m 3 + 162 = 180 Simplify.

m 3 = 18 Subtract 162 from each side.

Trapezoids and KitesLESSON 6-5

Additional Examples