Question

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

Explanation:

Step1: Find the third angle in \(\triangle XYZ\)

By the triangle - angle sum theorem (\(m\angle X+m\angle Y + m\angle Z=180^{\circ}\)). Given \(m\angle X = 90^{\circ}\) and \(m\angle Y=30^{\circ}\), then \(m\angle Z=180^{\circ}-(90^{\circ}+30^{\circ}) = 60^{\circ}\).

Step2: Analyze congruence criteria

The congruence criteria (SSS - Side - Side - Side, SAS - Side - Angle - Side, ASA - Angle - Side - Angle, AAS - Angle - Angle - Side, HL - Hypotenuse - Leg for right - triangles) all require information about side lengths.
We know that \(\triangle XYZ\) has angles \(90^{\circ},30^{\circ},60^{\circ}\) and \(\triangle TUV\) has angles \(30^{\circ},60^{\circ}\) (and the third angle \(m\angle T=180^{\circ}-(30^{\circ}+60^{\circ}) = 90^{\circ}\)). But we have no information about the side lengths of either triangle.

Answer:

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