Question

$\triangle rst$ and $\triangle ruv$ are shown below.

which statement is true?
$\triangle rst$ is similar to $\triangle ruv$.
$\triangle rst$ is not similar to $\triangle ruv$.
there is not enough information to determine whether the triangles are similar.

Explanation:

Step1: Calculate ∠S in △RST

First, find the missing angle in △RST. The sum of angles in a triangle is $180^\circ$.
$\angle S = 180^\circ - 86^\circ - 41^\circ = 53^\circ$

Step2: Match corresponding angles

In △RST, angles are $86^\circ$, $53^\circ$, $41^\circ$. In △RUV, angles are $86^\circ$ (shared ∠R), $53^\circ$ (∠U), and $180^\circ - 86^\circ - 53^\circ = 41^\circ$ (∠V). All corresponding angles are equal.

Step3: Verify similarity rule

By AA (Angle-Angle) similarity criterion, if two pairs of corresponding angles are equal, the triangles are similar. Here all three angles match, so the triangles are similar.

Answer:

△RST is similar to △RUV.