Question

which similarity statement is true?
○ δsrq ~ δtrs
○ δsrq ~ δrtq
○ δsrq ~ δrts
○ δsrq ~ δqrt

Explanation:

Step1: Identify Right Angles

In the diagram, $\angle SRQ = 90^\circ$ (right angle at $R$) and $\angle STR = 90^\circ$ (right angle at $T$). Also, $\angle S$ is common to $\triangle SRQ$ and $\triangle TRS$.

Step2: Apply AA Similarity Criterion

AA (Angle - Angle) similarity states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. For $\triangle SRQ$ and $\triangle TRS$:

• $\angle S=\angle S$ (common angle).
• $\angle SRQ=\angle TRS = 90^\circ$ (right angles).

So, by AA similarity, $\triangle SRQ\sim\triangle TRS$.

Let's check other options:

• For $\triangle SRQ\sim\triangle RTQ$: Angles do not match as required for similarity.
• For $\triangle SRQ\sim\triangle RTS$: Order of vertices is incorrect for angle correspondence.
• For $\triangle SRQ\sim\triangle QRT$: Angles do not match as required for similarity.

Answer:

$\boldsymbol{\triangle SRQ \sim \triangle TRS}$ (the first option: $\triangle SRQ \sim \triangle TRS$)