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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

1 answer

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