Question

for the figure shown, find m∠1 and m∠2. m∠1 = 127° m∠2 = □° (the figure is not drawn to scale.)

Explanation:

Step1: Use angle - sum property of a triangle

The sum of interior angles of a triangle is 180°. In the triangle with angle 32° and angle 1, if we assume the third - angle of the triangle is x. Then, considering the linear - pair with the 97° angle, x = 180° - 97°=83°.

Step2: Calculate angle 1

Using the angle - sum property of a triangle (32°+x + m∠1 = 180°), substitute x = 83° into the equation: 32°+83°+m∠1 = 180°. Then m∠1=180°-(32° + 83°)=65°. But we are given m∠1 = 127°, so there might be some mis - understanding in the problem setup. If we consider the exterior - angle property. The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. Let's assume the correct way is to use the exterior - angle formed by the 97° angle. The exterior angle (97°) of the small triangle with angle 32° and part of ∠1 is related as follows: Let the non - adjacent angles of the exterior angle be 32° and y. Then 97°=32° + y, so y = 97° - 32°=65°. And if we assume ∠1 is composed of this y and another angle, and we know m∠1 = 127°, then the other part of ∠1 is 127°-65° = 62°. Now, for ∠2, we know that the sum of angles on a straight line is 180°. Since the angle adjacent to ∠2 and part of the angle - sum with 97° is 180° - 127°=53°. And considering the other part of the angle at the vertex of ∠2 from the 30° triangle, we use the angle - sum property of the large triangle formed. But a simpler way is to use the fact that the sum of angles around a point is 360°. Let's use the linear - pair and angle - sum relationships. We know that the angle adjacent to ∠2 and related to the 97° angle and ∠1. Since the angle adjacent to ∠2 and part of the same vertex as the 97° angle is 180° - 127° = 53°. And considering the 30° angle in the other triangle, we use the fact that the sum of angles in a triangle formed by these angles is 180°. The angle adjacent to ∠2 and part of the vertex with 97° and related to ∠1 forms a linear - pair with ∠2. We know that the sum of angles in the triangle with 30° and the angle related to the 97° vertex is 180°. The angle at the vertex related to 97° and ∠1 and the 30° triangle is 180°-(30° + 97°)=53°. Since the sum of angles on a straight line is 180°, m∠2=180°-(53° + 30°)=97°.

Answer:

m∠2 = 97°