Question

If ΔPRQ
has ∠P=30
and ∠R=127
and ΔFGH
has ∠H=30
and ∠G=23
, what criterion has been met to prove the triangles similar?(1 point)
Responses

ASA
ASA

SAS
SAS

AA
AA

SSS

Answer 1

To determine which criterion shows that the triangles ΔPRQ and ΔFGH are similar, we need to check their angle measures.

For triangle ΔPRQ, the angles are:
- ∠P = 30°
- ∠R = 127°

To find the third angle (∠Q), we use the fact that the sum of the angles in a triangle is 180°:

\[
∠Q = 180° - ∠P - ∠R = 180° - 30° - 127° = 23°
\]

So, the angles in triangle ΔPRQ are:
- ∠P = 30°
- ∠R = 127°
- ∠Q = 23°

For triangle ΔFGH, the angles are:
- ∠H = 30°
- ∠G = 23°

To find the third angle (∠F), we again use the angle sum property:

\[
∠F = 180° - ∠H - ∠G = 180° - 30° - 23° = 127°
\]

So, the angles in triangle ΔFGH are:
- ∠H = 30°
- ∠G = 23°
- ∠F = 127°

Now we have the angle measures for both triangles:
- Triangle ΔPRQ: 30°, 127°, 23°
- Triangle ΔFGH: 30°, 23°, 127°

We see that both triangles have two angles that are equal:
- ∠P = ∠H = 30°
- ∠G = ∠Q = 23°

Both triangles satisfy the AA (Angle-Angle) criterion for similarity since two angles in one triangle are equal to two angles in the other triangle.

Therefore, the correct response is:

**AA**