Question

in △xyz, ( mangle x = 90^{circ} ) and ( mangle y = 30^{circ} ). in △tuv, ( mangle u = 30^{circ} ) and ( mangle v = 60^{circ} ). which is true about the two triangles?
no congruency statement can be made because the side lengths are unknown.
( \triangle xyzcong\triangle tuv )
no congruency statement can be made because only two angles in each triangle are known.
( \triangle xyzcong\triangle vut )

Explanation:

Step1: Find the measure of the third angle in each triangle

In \(\triangle XYZ\), using the angle - sum property of a triangle (\(m\angle X+m\angle Y + m\angle Z=180^{\circ}\)). Given \(m\angle X = 90^{\circ}\) and \(m\angle Y=30^{\circ}\), then \(m\angle Z=180^{\circ}-(90^{\circ}+30^{\circ}) = 60^{\circ}\).
In \(\triangle TUV\), using the angle - sum property of a triangle (\(m\angle T+m\angle U + m\angle V=180^{\circ}\)). Given \(m\angle U = 30^{\circ}\) and \(m\angle V=60^{\circ}\), then \(m\angle T=180^{\circ}-(30^{\circ}+60^{\circ})=90^{\circ}\).
So, \(\angle X=\angle T = 90^{\circ}\), \(\angle Y=\angle U = 30^{\circ}\), \(\angle Z=\angle V = 60^{\circ}\).

Step2: Check the congruence criteria

For triangle congruence (e.g., ASA, AAS, SSS, SAS, HL), we need information about side lengths. Since we only know the angle measures (AAA - angle - angle - angle does not prove congruence, it proves similarity) and no side lengths are given.

Answer:

No congruency statement can be made because the side lengths are unknown.