Question

solve for the missing angle measures.
a.
b.

Explanation:

Step1: Find the base - angle of the first isosceles triangle

The adjacent angle to \(120^{\circ}\) is \(180^{\circ}-120^{\circ} = 60^{\circ}\). Since the triangle is isosceles, the other base - angle is also \(60^{\circ}\).

Step2: Use the property of parallel lines and transversals (alternate - interior angles)

The base - angle of the second isosceles triangle (due to parallel lines and the transversal) is \(60^{\circ}\). Since the second triangle is isosceles, \(x = 60^{\circ}\) (vertical - angle property).

For part b:

Step1: Use the property of congruent triangles (SSS - Side - Side - Side)

The two triangles are congruent (marked with equal sides). The angle adjacent to \(28^{\circ}\) in the right - angled triangle (other than \(90^{\circ}\)) is \(90^{\circ}-28^{\circ}=62^{\circ}\).

Step2: Find the value of \(x\)

Since the triangles are congruent, \(x = 62^{\circ}-28^{\circ}=34^{\circ}\)

Answer:

a. \(60^{\circ}\)
b. \(34^{\circ}\)