Question
The triangles below are similar.
Triangle S R P. Angle S is 54 degrees, R is 41 degrees, P is 85 degrees. Triangle X Y Z. Angle X is 54 degrees, Z is 41 degrees, and Y is 85 degrees.
Which similarity statements describe the relationship between the two triangles? Check all that apply.
Triangle P R S is similar to triangle X Y Z
Triangle R S P is similar to triangle Z X Y
Triangle S R P is similar to triangle X Z Y
Triangle P S R is similar to Triangle Z Y X
Triangle R P S is similar to triangle Z Y X
Triangle S P R is similar to triangle X Z Y
Triangle S R P. Angle S is 54 degrees, R is 41 degrees, P is 85 degrees. Triangle X Y Z. Angle X is 54 degrees, Z is 41 degrees, and Y is 85 degrees.
Which similarity statements describe the relationship between the two triangles? Check all that apply.
Triangle P R S is similar to triangle X Y Z
Triangle R S P is similar to triangle Z X Y
Triangle S R P is similar to triangle X Z Y
Triangle P S R is similar to Triangle Z Y X
Triangle R P S is similar to triangle Z Y X
Triangle S P R is similar to triangle X Z Y
Answers
GPT-4o mini
To determine the similarity of the triangles, we can use the corresponding angles of the triangles S R P and X Y Z. Since the angles in both triangles are the same, we can establish the following similarity statements:
1. **Triangle P R S is similar to triangle X Y Z**: This statement is **true** because the angles match: P (85°) corresponds to Y (85°), R (41°) corresponds to Z (41°), and S (54°) corresponds to X (54°).
2. **Triangle R S P is similar to triangle Z X Y**: This statement is **true** because the angles match in the same order: R (41°) corresponds to Z (41°), S (54°) corresponds to X (54°), and P (85°) corresponds to Y (85°).
3. **Triangle S R P is similar to triangle X Z Y**: This statement is **false** because the corresponding angles do not match in that order. Z (41°) would need to correspond to R (41°), but in this similarity statement, it claims Z corresponds to S (54°).
4. **Triangle P S R is similar to Triangle Z Y X**: This statement is **false** because the angles do not correspond correctly.
5. **Triangle R P S is similar to triangle Z Y X**: This statement is **false**; the angle correspondence does not match.
6. **Triangle S P R is similar to triangle X Z Y**: This statement is **false**; the angle correspondence does not match.
In summary, the following similarity statements are correct:
- Triangle P R S is similar to triangle X Y Z
- Triangle R S P is similar to triangle Z X Y
1. **Triangle P R S is similar to triangle X Y Z**: This statement is **true** because the angles match: P (85°) corresponds to Y (85°), R (41°) corresponds to Z (41°), and S (54°) corresponds to X (54°).
2. **Triangle R S P is similar to triangle Z X Y**: This statement is **true** because the angles match in the same order: R (41°) corresponds to Z (41°), S (54°) corresponds to X (54°), and P (85°) corresponds to Y (85°).
3. **Triangle S R P is similar to triangle X Z Y**: This statement is **false** because the corresponding angles do not match in that order. Z (41°) would need to correspond to R (41°), but in this similarity statement, it claims Z corresponds to S (54°).
4. **Triangle P S R is similar to Triangle Z Y X**: This statement is **false** because the angles do not correspond correctly.
5. **Triangle R P S is similar to triangle Z Y X**: This statement is **false**; the angle correspondence does not match.
6. **Triangle S P R is similar to triangle X Z Y**: This statement is **false**; the angle correspondence does not match.
In summary, the following similarity statements are correct:
- Triangle P R S is similar to triangle X Y Z
- Triangle R S P is similar to triangle Z X Y