Question

in the figure shown, $angle prs$ and $angle spq$ are right - angles, and $mangle spr = 40^{circ}$. choose all true statements. a. $\triangle sprsim\triangle pqr$ by aa b. $\triangle spqsim\triangle qrp$ by aa c. $\triangle spqsim\triangle pqr$ by aa d. $\triangle spqsim\triangle srp$ by aa e. $\triangle pqrsim\triangle spq$ by aa

Explanation:

Step1: Identify right - angles

Given $\angle PRS = 90^{\circ}$ and $\angle SPQ=90^{\circ}$.

Step2: Analyze angle - angle similarity

In $\triangle SPQ$ and $\triangle SRP$:

• $\angle SPQ=\angle SRP = 90^{\circ}$ (given right - angles).
• $\angle S$ is common to both triangles. So, $\triangle SPQ\sim\triangle SRP$ by AA (Angle - Angle) similarity criterion.

In $\triangle SPQ$ and $\triangle PQR$:

• $\angle SPQ=\angle PRQ = 90^{\circ}$ (given $\angle PRS = 90^{\circ}$ and $\angle PRQ$ is supplementary to $\angle PRS$).
• $\angle Q$ is common to both triangles. So, $\triangle SPQ\sim\triangle PQR$ by AA similarity criterion. Also, $\triangle PQR\sim\triangle SPQ$ (similarity is symmetric).

Answer:

D. $\triangle SPQ\sim\triangle SRP$ by AA
C. $\triangle SPQ\sim\triangle PQR$ by AA
E. $\triangle PQR\sim\triangle SPQ$ by AA