Question

write a proof of the angle - angle similarity theorem.
if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
given: ∠t ≅ ∠x, ∠u ≅ ∠y
prove: △tuv ~ △xyz
dilate △xyz by the scale factor
(there is a triangle diagram at the bottom with triangles xyz and tuv, and angle markings indicating congruent angles)

Explanation:

Step1: Use the Angle - Angle Similarity Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Given \(\angle T\cong\angle X\) and \(\angle U\cong\angle Y\).
In \(\triangle TUV\) and \(\triangle XYZ\), we know that the sum of the interior angles of a triangle is \(180^{\circ}\). So, \(\angle V=180^{\circ}-\angle T - \angle U\) and \(\angle Z=180^{\circ}-\angle X-\angle Y\).
Since \(\angle T\cong\angle X\) and \(\angle U\cong\angle Y\), we have \(\angle V\cong\angle Z\) (by substitution).

Step2: State the similarity

By the Angle - Angle (AA) similarity criterion (if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar), we can conclude that \(\triangle TUV\sim\triangle XYZ\).

Answer:

\(\triangle TUV\sim\triangle XYZ\) (by the Angle - Angle similarity theorem)