Question

in the figure shown, $\angle prs$ and $\angle spq$ are right angles, and $m\angle spr = 40^{\circ}$.
choose all true statements.
\\(\square\\) a. $\triangle spr\sim \triangle pqr$ by aa
\\(\square\\) b. $\triangle spq\sim \triangle qrp$ by aa
\\(\square\\) c. $\triangle spq\sim \triangle pqr$ by aa
\\(\square\\) d. $\triangle spq\sim \triangle srp$ by aa
\\(\square\\) e. $\triangle pqr\sim \triangle spq$ by aa

Explanation:

Step1: Identify given right angles

$\angle PRS = 90^\circ$, $\angle SPQ = 90^\circ$, $\angle PRQ = 90^\circ$

Step2: Analyze shared angles

For $\triangle SPQ$ and $\triangle QRP$:

• $\angle SPQ = \angle QRP = 90^\circ$
• $\angle Q$ is a common angle

For $\triangle PQR$ and $\triangle SPQ$:

• $\angle PQR = \angle SPQ$ (common angle $\angle Q$, right angle)
• $\angle PRQ = \angle SPQ = 90^\circ$

Step3: Verify AA similarity

AA similarity requires 2 congruent angles.

• Option B: $\triangle SPQ \sim \triangle QRP$ (2 matching right angles + shared $\angle Q$)
• Option E: $\triangle PQR \sim \triangle SPQ$ (2 matching right angles + shared $\angle Q$, similarity is symmetric)

Other options lack 2 congruent corresponding angles.

Answer:

B. $\triangle SPQ\sim \triangle QRP$ by AA
E. $\triangle PQR\sim \triangle SPQ$ by AA