Question

Are the two triangles similar? How do you know? Explain your reasoning thoroughly.

Answer 1

Yes the triangle are similar

Answer 2

Yes, the triangles are similar (
`\triangle HMG \sim \triangle KMJ`).

Explanation:

We're given that the two triangles share at least one angle, angles H and K. The two angles in the triangles at point M,
`\angle HMG` and
`\angle KMJ` are vertical angles, meaning they are on opposite sides of a point of intersection between two lines. Since vertical angles are always equal, these two angles are also equal.

If two triangles share two angles, they must share all three, because all triangles have a total sum of interior angles of 180 degrees. Therefore, the triangles share all 3 angles. AAA (Angle-Angle-Angle) is a proof of similarity, where two triangles share all three of their angles. Thus, the two triangles are similar.

Similarity statement (vertices should correspond):

`\triangle HMG \sim \triangle KMJ`