Question

angles and congruence
in △xyz, m∠x = 90°, and m∠y = 30°. in △tuv, m∠u = 30°, and m∠v = 60°. which is true about the two triangles?

• △xyz ≅ △vut
• no congruence statement can be made because the side lengths are unknown.
• no congruence statement can be made because only two angles in each triangle are known.
• △xyz ≅ △tuv

Explanation:

Step1: Analyze angles in △XYZ

In △XYZ, \( m\angle X = 90^\circ \), \( m\angle Y = 30^\circ \). So \( m\angle Z = 180^\circ - 90^\circ - 30^\circ = 60^\circ \).

Step2: Analyze angles in △TUV

In △TUV, \( m\angle U = 30^\circ \), \( m\angle V = 60^\circ \). So \( m\angle T = 180^\circ - 30^\circ - 60^\circ = 90^\circ \).

Step3: Check congruence conditions

Congruent triangles need corresponding sides and angles equal. Here, we only know angle measures (AA similarity for similarity, but congruence needs SSS, SAS, ASA, AAS with side info). Since side lengths are unknown, we can't confirm congruence. The second option says "No congruency statement can be made because the side lengths are unknown" which matches. The first option claims congruence (wrong, no side info). The third option's reason is wrong (we know three angles, but congruence needs sides). The fourth option's congruence statement has wrong correspondence (△XYZ and △TUV, not △VUT).

Answer:

The second option (No congruency statement can be made because the side lengths are unknown)