Question
In the figure shown, angle P R S and angle S P Q are right angles, and the measure of angle S P R equals 40°.
The figure shows triangle S P Q. Line P R extends from vertex P to the opposite side, dividing triangle S P Q into 2 triangles.
Question
Choose ,begin emphasis,all,end emphasis, true statements.
Answer options with 5 options
A.
triangle S P R is similar to triangle P Q R by A A
B.
triangle S P Q is similar to triangle Q R P by A A
C.
triangle S P Q is similar to triangle P Q R by A A
D.
triangle S P Q is similar to triangle S R P by A A
E.
triangle P Q R is similar to triangle S P Q by A A
The figure shows triangle S P Q. Line P R extends from vertex P to the opposite side, dividing triangle S P Q into 2 triangles.
Question
Choose ,begin emphasis,all,end emphasis, true statements.
Answer options with 5 options
A.
triangle S P R is similar to triangle P Q R by A A
B.
triangle S P Q is similar to triangle Q R P by A A
C.
triangle S P Q is similar to triangle P Q R by A A
D.
triangle S P Q is similar to triangle S R P by A A
E.
triangle P Q R is similar to triangle S P Q by A A
Answers
GPT-4o mini
Given the information provided, we have the following angles:
- \( \angle PRS = 90° \) (right angle)
- \( \angle SPQ = 90° \) (right angle)
- \( \angle SPR = 40° \)
Using these angles, we can analyze the triangles:
1. **Triangle \( SPR \)**:
- Angles: \( \angle SPR = 40° \), \( \angle PRS = 90° \), so \( \angle S = 50° \) (since the angles in a triangle sum up to 180°).
2. **Triangle \( SQR \)**:
- Angles: \( \angle SPQ = 90° \), \( \angle SPR = 40° \), so \( \angle Q = 50° \) (since the angles in a triangle sum up to 180°).
3. **Triangle \( PQR \)**:
- Given that \( \angle PRS = 90° \) and \( \angle SQR = 50° \), then \( \angle PQR = 40° \).
Now, let's evaluate the statements:
A. **Triangle \( SPR \) is similar to triangle \( PQR \) by AA**:
- Yes, both triangles have \( \angle SPR = 40° \) and \( \angle PRS = 90° \). (True)
B. **Triangle \( SPQ \) is similar to triangle \( QRP \) by AA**:
- Not true, as we don't have enough angles to match \( \triangle SPQ \) with \( \triangle QRP \). (False)
C. **Triangle \( SPQ \) is similar to triangle \( PQR \) by AA**:
- Yes, \( \angle SPQ = 90° \) and \( \angle QRP = 50° \); both triangles also have an angle of \( 40° \). (True)
D. **Triangle \( SPQ \) is similar to triangle \( SRP \) by AA**:
- Not true, as the triangles do not have matching angles. (False)
E. **Triangle \( PQR \) is similar to triangle \( SPQ \) by AA**:
- Yes, this is true as both triangles have \( \angle QRP = 50° \) and \( \angle SPQ = 90° \). (True)
Therefore, the true statements are:
- **A**
- **C**
- **E**
- \( \angle PRS = 90° \) (right angle)
- \( \angle SPQ = 90° \) (right angle)
- \( \angle SPR = 40° \)
Using these angles, we can analyze the triangles:
1. **Triangle \( SPR \)**:
- Angles: \( \angle SPR = 40° \), \( \angle PRS = 90° \), so \( \angle S = 50° \) (since the angles in a triangle sum up to 180°).
2. **Triangle \( SQR \)**:
- Angles: \( \angle SPQ = 90° \), \( \angle SPR = 40° \), so \( \angle Q = 50° \) (since the angles in a triangle sum up to 180°).
3. **Triangle \( PQR \)**:
- Given that \( \angle PRS = 90° \) and \( \angle SQR = 50° \), then \( \angle PQR = 40° \).
Now, let's evaluate the statements:
A. **Triangle \( SPR \) is similar to triangle \( PQR \) by AA**:
- Yes, both triangles have \( \angle SPR = 40° \) and \( \angle PRS = 90° \). (True)
B. **Triangle \( SPQ \) is similar to triangle \( QRP \) by AA**:
- Not true, as we don't have enough angles to match \( \triangle SPQ \) with \( \triangle QRP \). (False)
C. **Triangle \( SPQ \) is similar to triangle \( PQR \) by AA**:
- Yes, \( \angle SPQ = 90° \) and \( \angle QRP = 50° \); both triangles also have an angle of \( 40° \). (True)
D. **Triangle \( SPQ \) is similar to triangle \( SRP \) by AA**:
- Not true, as the triangles do not have matching angles. (False)
E. **Triangle \( PQR \) is similar to triangle \( SPQ \) by AA**:
- Yes, this is true as both triangles have \( \angle QRP = 50° \) and \( \angle SPQ = 90° \). (True)
Therefore, the true statements are:
- **A**
- **C**
- **E**