Answer :
Given that ∠ABC = ∠XYZ and ∠C measures 32°, we need to determine which of the statements must be true. Let's break down the information step-by-step:
1. The notation ∠ABC = ∠XYZ implies that the two triangles are congruent, meaning all corresponding angles are equal.
2. Specifically, ∠C in triangle ABC corresponds to ∠Z in triangle XYZ.
3. Since the measure of ∠C is given as 32°, the measure of the corresponding angle in the other triangle (∠Z) must also be 32°.
Given this correspondence, let's evaluate the statements provided:
A. ∠A = 32°
- This statement is not necessarily true because it does not directly specify a corresponding relationship involving ∠C or ∠Z.
B. ∠Y = 32°
- This statement is not necessarily true because it does not directly specify a corresponding relationship involving ∠C or ∠Z.
C. ∠Z = 32°
- This statement must be true because ∠C in triangle ABC corresponds to ∠Z in triangle XYZ, and both have the same measure of 32°.
D. ∠X = 32°
- This statement is not necessarily true because it does not directly specify a corresponding relationship involving ∠C or ∠Z.
Therefore, the correct answer is:
C. ∠Z = 32°
1. The notation ∠ABC = ∠XYZ implies that the two triangles are congruent, meaning all corresponding angles are equal.
2. Specifically, ∠C in triangle ABC corresponds to ∠Z in triangle XYZ.
3. Since the measure of ∠C is given as 32°, the measure of the corresponding angle in the other triangle (∠Z) must also be 32°.
Given this correspondence, let's evaluate the statements provided:
A. ∠A = 32°
- This statement is not necessarily true because it does not directly specify a corresponding relationship involving ∠C or ∠Z.
B. ∠Y = 32°
- This statement is not necessarily true because it does not directly specify a corresponding relationship involving ∠C or ∠Z.
C. ∠Z = 32°
- This statement must be true because ∠C in triangle ABC corresponds to ∠Z in triangle XYZ, and both have the same measure of 32°.
D. ∠X = 32°
- This statement is not necessarily true because it does not directly specify a corresponding relationship involving ∠C or ∠Z.
Therefore, the correct answer is:
C. ∠Z = 32°
