Triangles P Q R and X Y Z are congruent, as shown.
Question
Which sequence of transformations proves that triangle P Q R is congruent to triangle X Y Z using congruence criteria A S A?
Answer options with 5 options
A.
Translate P Q R so that R aligns with z, and then reflect the result across segment P Q.
B.
Reflect P Q R across a horizontal line through R, and then translate the image so R aligns with Z.
C.
Reflect P Q R across a vertical line through R, and then translate the image so segment P Q aligns with segment X Y.
D.
Translate P Q R so that P aligns with X, and then reflect the result across a horizontal line through the image of R.
E.
Rotate P Q R by 180° counterclockwise around the point R, and then translate so that the image of P aligns with X.
The correct answer is D.
Explanation:
The congruence criteria A S A (angle-side-angle) states that if two angles and the included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent.
In this case, to prove that triangle PQR is congruent to triangle XYZ using the A S A criteria, we need to show that angle PRQ is congruent to angle ZYX, angle PQR is congruent to angle XYZ, and side PQ is congruent to side XY.
The correct sequence of transformations that achieves this is:
1. Translate PQR so that P aligns with X.
2. Reflect the result across a horizontal line through the image of R.
The translation in step 1 ensures that P aligns with X. Then, the reflection in step 2 ensures that the image of P aligns with X and that side PQ is congruent to side XY. The angles PRQ and ZYX do not change during these transformations, so they remain congruent.
Therefore, option D is the correct answer.