Question

isosceles trapezoid
isosceles trapezoid a b c d, where
angle d c b is 107 degrees. altitude (90 degrees) is constructed from vertex
d to meet the base a b; angle a d b = angle 1.

1. determine these values.

a. \\( \angle 1 \\)
b. \\( \angle a \\)
c. \\( \angle b \\)

Explanation:

Step1: Use properties of isosceles trapezoid

In an isosceles trapezoid \(ABCD\) (\(AD = BC\)), \(AB\parallel CD\). The sum of adjacent angles in a trapezoid is \(180^{\circ}\). So \(\angle C+\angle B = 180^{\circ}\), \(\angle D+\angle A=180^{\circ}\). Given \(\angle C = 107^{\circ}\), then \(\angle B=180^{\circ}-\angle C=180 - 107=73^{\circ}\). Also, in an isosceles trapezoid \(\angle A=\angle B\).

Step2: Find \(\angle 1\)

Let's consider the right - angled triangle formed by the altitude from \(D\) to \(AB\). We know that \(\angle A = 73^{\circ}\), and in the right - angled triangle (where one angle is \(90^{\circ}\)), \(\angle A=\angle 1 + 90^{\circ}\) (sum of angles in a triangle is \(180^{\circ}\)). So \(\angle 1=\angle A - 90^{\circ}\)

Answer:

a. \(\angle1 = 73^{\circ}-90^{\circ}=17^{\circ}\)
b. \(\angle A = 73^{\circ}\)
c. \(\angle B=73^{\circ}\)