In isosceles trapezoid ABCD, the parallel bases are AB and CD. If the measure of interior … — ACT

In isosceles trapezoid $ABCD$ , the parallel bases are $AB$ and $CD$ . If the measure of interior angle $A$ is $70^\circ$ , what is the measure, in degrees, of interior angle $C$ ?

Answer: 110 degrees

Answer

The measure of interior angle

C

110

degrees.

Since the trapezoid is isosceles with bases

AB

and

CD

, the base angles

\angle A

and

\angle B

are congruent, so

\angle B = 70^\circ

. The consecutive interior angles along the leg

BC

are supplementary because

AB \parallel CD

, which means

\angle B + \angle C = 180^\circ

. Solving for

\angle C

gives

180^\circ - 70^\circ = 110^\circ

Step-by-Step Solution

Find the measure of angle

B

using the properties of an isosceles trapezoid.

\angle B = 70^\circ

In an isosceles trapezoid, the angles sharing a base are congruent. Since

AB

is a base,

\angle A = \angle B = 70^\circ

Calculate the measure of angle

C

using the parallel lines property.

\angle C = 110^\circ

Because the bases

AB

and

CD

are parallel, the consecutive interior angles

\angle B

and

\angle C

must sum to

180^\circ

. Therefore,

\angle C = 180^\circ - 70^\circ = 110^\circ

Key Concept

Properties of an isosceles trapezoid

Alternative Method

Since the sum of interior angles in any quadrilateral is

360^\circ

, and in an isosceles trapezoid the base angles are equal (

\angle A = \angle B = 70^\circ

and

\angle C = \angle D

), we can write

70^\circ + 70^\circ + \angle C + \angle D = 360^\circ

. Since

\angle C = \angle D

, this simplifies to

140^\circ + 2\angle C = 360^\circ \implies 2\angle C = 220^\circ \implies \angle C = 110^\circ

Estimated Time:45s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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