Question

for the figure shown, find ( mangle1 ) and ( mangle2 ).

Explanation:

Step1: Find \(m\angle1\)

Use the exterior - angle property of a triangle. The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
For the larger triangle, \(m\angle1=112^{\circ}\) (since \(m\angle1\) and \(112^{\circ}\) are vertical angles. Vertical angles are equal).

Step2: Find \(m\angle2\)

First, find the third angle of the smaller triangle. Let's call it \(x\).
We know that the sum of angles in a triangle is \(180^{\circ}\).
In the smaller triangle, one angle is \(74^{\circ}\), another is \(m\angle1 = 112^{\circ}\) (from step 1), and the third angle \(x\) and \(m\angle2\) are related.
We also know that the exterior angle of the smaller triangle (the angle adjacent to \(29^{\circ}\)):
The sum of angles on a straight line is \(180^{\circ}\). The angle adjacent to \(29^{\circ}\) in the context of the smaller triangle is \(180 - 29=151^{\circ}\).
Using the angle - sum property of a triangle (\(74 + m\angle2+151=180\) is wrong. Let's use another approach).
We know that \(m\angle1\) is an exterior angle for the smaller triangle.
By the exterior - angle property of a triangle: \(m\angle1=74^{\circ}+m\angle2 + 29^{\circ}\)
Substitute \(m\angle1 = 112^{\circ}\) into the equation:
\(112=74 + m\angle2+29\)
\(m\angle2=112-(74 + 29)\)
\(m\angle2=112 - 103\)
\(m\angle2 = 9^{\circ}\)

Answer:

\(m\angle1 = 112^{\circ}\), \(m\angle2=9^{\circ}\)