Question

△pqr and △rst are shown below.
which statement is true?
△pqr is similar to △rst.
△pqr is not similar to △rst.
there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Find angles of $\triangle PQR$

In $\triangle PQR$, we know two angles: $\angle Q = 88^\circ$ and $\angle PRQ = 56^\circ$. The sum of angles in a triangle is $180^\circ$. So, $\angle P = 180^\circ - 88^\circ - 56^\circ = 36^\circ$.

Step2: Find angles of $\triangle RST$

In $\triangle RST$, we know $\angle T = 56^\circ$ and $\angle SRT = 36^\circ$. The sum of angles in a triangle is $180^\circ$. So, $\angle S = 180^\circ - 56^\circ - 36^\circ = 88^\circ$.

Step3: Compare angles for similarity

Now, let's list the angles of both triangles:

• $\triangle PQR$: $\angle P = 36^\circ$, $\angle Q = 88^\circ$, $\angle PRQ = 56^\circ$
• $\triangle RST$: $\angle SRT = 36^\circ$, $\angle S = 88^\circ$, $\angle T = 56^\circ$

We can see that the corresponding angles are equal (AA similarity criterion: if two angles of one triangle are equal to two angles of another triangle, the triangles are similar). So, $\triangle PQR \sim \triangle RST$ (by AA similarity, since $\angle P = \angle SRT = 36^\circ$, $\angle Q = \angle S = 88^\circ$, and the third angles will also be equal as the sum is $180^\circ$).

Answer:

$\triangle PQR$ is similar to $\triangle RST$.