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 \(m\angle2\)

Use the property that adjacent angles forming a linear - pair sum to \(180^{\circ}\).
If one of the adjacent angles is \(129^{\circ}\), then \(m\angle2 + 129^{\circ}=180^{\circ}\).
Subtract \(129^{\circ}\) from both sides: \(m\angle2=180^{\circ}- 129^{\circ}=51^{\circ}\).

Step2: Find \(m\angle1\)

Use the property that the sum of angles in a triangle is \(180^{\circ}\).
Let the third angle in the triangle be \(54^{\circ}\). So, \(m\angle1 + m\angle2+54^{\circ}=180^{\circ}\).
Substitute \(m\angle2 = 51^{\circ}\) into the equation: \(m\angle1+51^{\circ}+54^{\circ}=180^{\circ}\).
Combine like terms: \(m\angle1+(51 + 54)^{\circ}=180^{\circ}\), i.e., \(m\angle1+105^{\circ}=180^{\circ}\).
Subtract \(105^{\circ}\) from both sides: \(m\angle1=180^{\circ}-105^{\circ}=75^{\circ}\).

Answer:

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