Question

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

Explanation:

Step1: Identify Angles in Triangles

In $\triangle STU$, we know $\angle S = 46^\circ$. Let's assume $\angle T$ is a right angle (since the other triangle has a right angle at $V$ and the vertical lines suggest right angles, so $\angle T = 90^\circ$). Then $\angle SU T= 180^\circ - 90^\circ - 46^\circ = 44^\circ$, but we can also use the vertical angles or corresponding angles. In $\triangle WVU$, $\angle V = 90^\circ$ (right angle), $\angle WU V = 46^\circ$ (given). Now, for similarity, we can use AA (Angle - Angle) criterion. In $\triangle STU$ and $\triangle WVU$:

• $\angle S = \angle WU V = 46^\circ$ (given and vertical angles or corresponding angles? Wait, actually, $\angle T$ and $\angle V$ are both right angles (since $ST$ and $WV$ seem perpendicular to $TV$? Wait, looking at the diagram, $ST$ and $WV$ are both vertical? Wait, no, $TV$ is a vertical line? Wait, $\angle T$ and $\angle V$: $\angle T$ is at $T$, between $ST$ and $TU$, and $\angle V$ is at $V$, between $WV$ and $VU$, and both are right angles (since $\angle V$ is marked as right angle, and $ST$ and $WV$ are parallel? Wait, actually, $\angle T$ and $\angle V$ are both right angles (so $\angle T=\angle V = 90^\circ$), and $\angle S=\angle WU V = 46^\circ$. So by AA similarity (two angles equal), the triangles are similar.

Step2: Apply AA Similarity Criterion

AA (Angle - Angle) similarity states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

• In $\triangle STU$: Let's check angles. $\angle T$ is a right angle (assuming from the diagram, since $ST$ and $TU$ form a right angle? Wait, the diagram shows $\angle S = 46^\circ$, and $TU$ is vertical, $ST$ is horizontal? Wait, no, the diagram: $S$ to $T$ is horizontal, $T$ to $U$ is vertical, so $\angle T = 90^\circ$. In $\triangle WVU$, $\angle V = 90^\circ$ (right angle), and $\angle WU V = 46^\circ$ (given). Also, $\angle S = 46^\circ$ (given) and $\angle WU V = 46^\circ$, and $\angle T=\angle V = 90^\circ$. So two angles are equal, so by AA similarity, $\triangle STU \sim \triangle WVU$.

Answer:

$\triangle STU$ is similar to $\triangle WVU$.