Question

1. what are the measures of \\( \angle 1 \\) and \\( \angle 2 \\)?

a. \\( m \angle 1 = 36 ^ { \circ } ; m \angle 2 = 90 ^ { \circ } \\)
b. \\( m \angle 1 = 51 ^ { \circ } ; m \angle 2 = 75 ^ { \circ } \\)
c. \\( m \angle 1 = 54 ^ { \circ } ; m \angle 2 = 54 ^ { \circ } \\)
d. \\( m \angle 1 = 75 ^ { \circ } ; m \angle 2 = 51 ^ { \circ } \\)

Explanation:

Step1: Find the measure of ∠2

Use the property of supplementary angles.
$$m\angle2 + 129^{\circ}=180^{\circ}$$
$$m\angle2=180^{\circ}- 129^{\circ}=51^{\circ}$$

Step2: Find the measure of ∠3

Use the property of supplementary angles.
$$m\angle3 + 54^{\circ}=180^{\circ}$$
$$m\angle3=180^{\circ}-54^{\circ}=126^{\circ}$$

Step3: Find the measure of ∠1

Use the triangle - angle sum theorem (\(m\angle1+m\angle3 + m\angle2=180^{\circ}\)).
Substitute \(m\angle2 = 51^{\circ}\) and \(m\angle3=126^{\circ}\) into the formula:
$$m\angle1+126^{\circ}+51^{\circ}=180^{\circ}$$
$$m\angle1=180^{\circ}-(126^{\circ}+51^{\circ})=75^{\circ}$$

Answer:

D. \(m\angle1 = 75^{\circ}\); \(m\angle2 = 51^{\circ}\)