Question

△qrs and △ust are shown below. which statement is true? △qrs is similar to △ust. △qrs is not similar to △ust.

Explanation:

Step1: Find angles in $\triangle QRS$

Since $QR = RS$ (marked equal), $\angle Q=\angle RQS = 55^{\circ}$. Using the angle - sum property of a triangle ($\angle Q+\angle R+\angle QSR=180^{\circ}$), we find $\angle QSR = 180-(55 + 55)=70^{\circ}$. And $\angle QSR$ and $\angle UST$ are vertical angles, so $\angle UST = 70^{\circ}$.

Step2: Find angles in $\triangle UST$

Using the angle - sum property of a triangle ($\angle U+\angle T+\angle UST=180^{\circ}$) in $\triangle UST$ with $\angle T = 65^{\circ}$ and $\angle UST = 70^{\circ}$, we get $\angle U=180-(65 + 70)=45^{\circ}$.

Step3: Check similarity

For two triangles to be similar, their corresponding angles must be equal. In $\triangle QRS$, angles are $55^{\circ},55^{\circ},70^{\circ}$. In $\triangle UST$, angles are $45^{\circ},65^{\circ},70^{\circ}$. Corresponding angles are not equal.

Answer:

$\triangle QRS$ is not similar to $\triangle UST$.