Question

prove that △xyz ~ △pqr using aa similarity, where ∠x = ∠p and ∠z = ∠r. fill in the missing statement.

statement	reason
——	given
△xyz ~ △pqr	aa similarity criterion

options:
a. ∠y = ∠q
b. ∠z = ∠q
c. ∠z = ∠r
d. ∠r = ∠y

which transformation would map a triangle onto itself, preserving congruence?
options:
a. stretch
b. reflection
c. scaling
d. dilation

Explanation:

First Sub - Question (Proving Triangle Similarity)

To prove \(\triangle XYZ\sim\triangle PQR\) using AA (Angle - Angle) similarity, we need two pairs of corresponding angles to be equal. We are given that \(\angle X=\angle P\), and we need to find the other given angle. The AA similarity criterion requires two angles, so the missing given angle should be \(\angle Z = \angle R\) (option c). Option a (\(\angle Y=\angle Q\)) would be a result of the triangle angle - sum property and similarity, not a given angle here. Option b (\(\angle Z=\angle Q\)) is incorrect as \(Z\) and \(Q\) are not corresponding angles. Option d (\(\angle R=\angle Y\)) is also incorrect as they are not corresponding angles.

• A stretch (option a) changes the size of the figure, so it does not preserve congruence.
• A reflection (option b) is a rigid transformation. A reflection of a triangle over a line of symmetry (if it has one) or an appropriate line will map the triangle onto itself while preserving congruence (since the shape and size remain the same).
• Scaling (option c) and dilation (option d) change the size of the triangle, so they do not preserve congruence.

Answer:

c. \(\angle Z=\angle R\)

Second Sub - Question (Triangle Transformation)