Answer: <u>Yes</u>, the triangles are similar.
The triangles are similar due to the <u> AAA </u> similarity theorem.
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Explanation:
Recall that for any triangle, the three inside angles must add to 180 degrees.
So for triangle ABC, this means that
A+B+C = 180
40+120+C = 180
160+C = 180
C = 180-160
C = 20
Similarly, we can find angle T by following these steps
R+S+T = 180
20+120+T = 180
140+T = 180
T = 180-140
T = 40
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To summarize, we found angle C to be 20 degrees, and angle T is 40 degrees.
Now note these angle pairings in which they are the same measure
- Angle A = Angle T = 40
- Angle B = Angle S = 120
- Angle C = Angle R = 20
We have three pairs of corresponding angles that are congruent. By the AAA (angle angle angle) similarity theorem, this proves the triangles are similar. We can say triangle ABC is similar to triangle TSR. The order is important so we know how the angles pair up. For instance, A and T are the first letters of ABC and TSR respectively, helping us see that A and T pair up together.
Technically the minimum amount we need is only two pairs of corresponding angles, so some math textbooks use AA instead of AAA. It's the same idea though.
Unfortunately we don't have enough information to prove the triangles congruent or not. We would need to know some information about a pair of sides. Specifically, if we had one pair of congruent sides, then we could use either ASA or AAS.
