Question

Given: ABCD is a trapezoid, AB=CD, MN is a midsegment, MN=30, BC=17, AB=26 Find: m?A, m?B, m?C, and m?D

Answer 1

Answer:

The measures of angle ∠A, ∠B, ∠C, and ∠D are 60, 120, 120 and 60 degree respectively.

Step-by-step explanation:

Given information:AB=CD, MN is a midsegment, MN=30, BC=17, AB=26

Since two opposite sides are equal, therefore we can say that two no parallel sides are equal.

The length of midsegment is average of length of parallel lines.

$MN=\frac{AD+BC}{2}$

$30=\frac{AD+17}{2}$

$60=AD+17$

$AD=43$

Draw perpendiculars on AD from B and C. Let angle A be θ.

Since ABCD is an isosceles trapezoid, therefore AE=FD and EF=BC

$\cos\theta=\frac{base}{hypotenuse}$

$\cos\theta=\frac{AE}{AB}$

$\cos\theta=\frac{13}{26}$

$\cos\theta=\frac{1}{2}$

$\theta=60^{\circ}$

Since ABCD is an isosceles trapezoid, therefore angles A and D are same. Angle B and C are same.

$\angle A=\angle D=60^{\circ}$

The sum of two consecutive angles of a trapezoid is 180 degree by consecutive interior angle theorem.

$\angle A+\angle B=180^{\circ}$

$60^{\circ}+\angle B=180^{\circ}$

$\angle B=120^{\circ}$

Angle B and C are same.

$\angle B=\angle C=120^{\circ}$

Therefore measures of angle ∠A, ∠B, ∠C, and ∠D are 60, 120, 120 and 60 degree respectively.