Question

in δxyz, m∠x = 90° and m∠y = 30°. in δtuv, m∠u = 30° and m∠v = 60°. which is true about the two triangles? δxyz ≅ δtuv δxyz ≅ δvut 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 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, \(\angle X\cong\angle T\), \(\angle Y\cong\angle U\), \(\angle Z\cong\angle V\).

Step2: Recall the 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 - angled triangles) require information about side lengths. Just knowing that the three angles of one triangle are equal to the three angles of another triangle (AA - Angle - Angle similarity is for similarity, not congruence) is not sufficient for congruence.

Answer:

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