How can a similarity transformation be used to determine that the AA criterion proves the dilated triangle A′B′C′ is similar to triangle ABC ?

Question

Possible Answers:
Translate the dilated triangle so one of its angles exactly matches the corresponding angle in the original triangle; repeat using a different pair of angles. Translate the dilated triangle so one of its angles exactly matches the corresponding angle in the original triangle; repeat using a different pair of angles. Reflect the triangle across the x-axis and check for symmetry among the triangles with respect to the x-axis. Reflect the triangle across the x -axis and check for symmetry among the triangles with respect to the x -axis. Find the angle measures of the original triangle and the dilated triangle, then compare proportionality of the ratio to the scale factor. Find the angle measures of the original triangle and the dilated triangle, then compare proportionality of the ratio to the scale factor. Center the original triangle inside the dilated triangle and see if the side lengths are congruent by using the scale factor. Center the original triangle inside the dilated triangle and see if the side lengths are congruent by using the scale factor.

Answer 1

To determine that the dilated triangle \( A'B'C' \) is similar to triangle \( ABC \) using the AA (Angle-Angle) criterion, we can use the following approach from the given options:

Translate the dilated triangle so one of its angles exactly matches the corresponding angle in the original triangle; repeat using a different pair of angles.

Explanation:

The AA criterion states that if in two triangles, two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

In this case:

1. We can translate triangle \( A'B'C' \) so that one of its angles (say \( A' \)) matches exactly with the corresponding angle in triangle \( ABC \) (say \( A \)).
2. Next, we can compare another pair of angles—typically, if we establish that angle \( B' \) matches angle \( B \) (from triangle \( ABC \)), we conclude that \( A' \) is congruent to \( A \) and \( B' \) is congruent to \( B \).

Since we are dealing with triangles, when we can show that two angles are identical, it automatically infers the third angle must also be congruent due to the triangle angle sum property (which states that the sum of angles in a triangle is always 180°). Therefore, triangle \( A'B'C' \) is similar to triangle \( ABC \) by the AA criterion.

Conclusion:

Thus, the correct choice is: Translate the dilated triangle so one of its angles exactly matches the corresponding angle in the original triangle; repeat using a different pair of angles.