Show the similarity between a triangle with angles A

Question

, B
, and C
, where m∠A=47°
and m∠B=36°
, and another triangle with angles G
, H
, and J
, where m∠G=36°
and m∠J=97°
.

△ABC∼△HJG
triangle upper A upper B upper C similar to triangle upper H upper J upper G

△ABC∼△HGJ
triangle upper A upper B upper C similar to triangle upper H upper G upper J

△ABC∼△GHJ
triangle upper A upper B upper C similar to triangle upper G upper H upper J

△ABC∼△GJH

Answer 1

To determine the similarity between the triangles \( \triangle ABC \) and \( \triangle HJG \) (or any of the mentioned triangles) based on their angle measures, we start by examining each triangle's angles.

1. For \( \triangle ABC \):

   • \( m\angle A = 47^\circ \)
   • \( m\angle B = 36^\circ \)
   • To find \( m\angle C \): \[ m\angle C = 180^\circ - m\angle A - m\angle B = 180^\circ - 47^\circ - 36^\circ = 97^\circ \]

2. For \( \triangle HJG \):

   • \( m\angle G = 36^\circ \)
   • \( m\angle J = 97^\circ \)
   • To find \( m\angle H \): \[ m\angle H = 180^\circ - m\angle G - m\angle J = 180^\circ - 36^\circ - 97^\circ = 47^\circ \]

Now we summarize the angles of both triangles:

• \( \triangle ABC: A = 47^\circ, B = 36^\circ, C = 97^\circ \)
• \( \triangle HJG: G = 36^\circ, H = 47^\circ, J = 97^\circ \)

Since the triangles have the same angles (specifically \( 47^\circ, 36^\circ, \) and \( 97^\circ \)), we can conclude: \[ \triangle ABC \sim \triangle HJG \]

Therefore, the correct similarity statement is: \[ \triangle ABC \sim \triangle HJG \]

This conclusion is valid because similar triangles have corresponding angles that are equal.