Question

find the measure of each numbered angle and name the theorems that you used to justify your work. ( mangle5 = mangle6 ) a) ( mangle5 = mangle6 = 45^{circ} ) (supplements theorem) b) ( mangle6 = mangle7 = 90^{circ} ) (supplements theorem) c) ( mangle5 = mangle6 = 45^{circ} ) (≅ complements theorem) d) ( mangle5 = mangle6 = 120^{circ} ) (supplements theorem)

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles add up to \(180^{\circ}\). Let \(m\angle5 = m\angle6=x\). The three - angle sum (the right - angle \(= 90^{\circ}\), \(\angle5\), and \(\angle6\)) forms a straight - line (supplementary angles). So, \(x + x+90^{\circ}=180^{\circ}\).

Step2: Solve the equation for \(x\)

Combine like terms: \(2x=180^{\circ}- 90^{\circ}\). Then \(2x = 90^{\circ}\), and \(x = 45^{\circ}\). The \(\cong\) Complements Theorem states that if two angles are complementary to the same angle (or to congruent angles), then they are congruent. Here, \(\angle5\) and \(\angle6\) are complementary to the same right - angle (in the sense of forming a straight - line with the right - angle as part of the angle addition on a straight - line).

Answer:

C. \(m\angle5 = m\angle6 = 45^{\circ}\) (\(\cong\) Complements Theorem)