Question

challenge -- find all of the missing angles - a, b, c, and d.
hint: the three angles of a triangle add together to make a sum of 180 degrees.

Explanation:

Step1: Find angle \(a\)

Since the sum of angles on a straight line is \(180^{\circ}\), we have \(a + 60^{\circ}+55^{\circ}=180^{\circ}\).

$$a=180^{\circ}-(60^{\circ} + 55^{\circ})=180^{\circ}-115^{\circ}=65^{\circ}$$

Step2: Find angle \(b\)

In the right - angled triangle (with one angle \(90^{\circ}\)), using the sum of angles in a triangle (\(180^{\circ}\)), we know that \(b + a+90^{\circ}=180^{\circ}\). Substitute \(a = 65^{\circ}\) into the equation.

$$b=180^{\circ}-(90^{\circ}+65^{\circ})=180^{\circ}-155^{\circ}=25^{\circ}$$

Step3: Find angle \(c\)

\(c\) and \(b\) are vertical angles. Vertical angles are equal. So \(c = b\).

$$c = 25^{\circ}$$

Step4: Find angle \(d\)

In the triangle with angle \(15^{\circ}\) and right - angle (\(90^{\circ}\)), using the sum of angles in a triangle (\(180^{\circ}\)), we have \(d+15^{\circ}+90^{\circ}=180^{\circ}\).

$$d=180^{\circ}-(90^{\circ}+15^{\circ})=180^{\circ}-105^{\circ}=75^{\circ}$$

Answer:

\(\angle a = 65^{\circ}\), \(\angle b=25^{\circ}\), \(\angle c = 25^{\circ}\), \(\angle d=75^{\circ}\)