Question

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

Explanation:

Step1: Analyze angles in $\triangle ABC$ and $\triangle CDE$

In $\triangle ABC$, $\angle B = 90^\circ$. In $\triangle CDE$, $\angle D = 90^\circ$, so both triangles are right - angled.
In $\triangle CDE$, we know that one of the non - right angles is $59^\circ$, let $\angle ECD=59^\circ$. Then $\angle DEC = 90^\circ - 59^\circ=31^\circ$.
In $\triangle ABC$, $\angle ACB + \angle ECD+ 90^\circ=180^\circ$ (since $B$, $C$, $D$ are collinear). So $\angle ACB=180^\circ - 90^\circ - 59^\circ = 31^\circ$.

Step2: Check for similar triangles

In $\triangle ABC$ and $\triangle CDE$:

• $\angle B=\angle D = 90^\circ$ (right angles)
• $\angle ACB=\angle DEC = 31^\circ$ (calculated above)

By the AA (Angle - Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. So $\triangle ABC\sim\triangle CDE$.

Answer:

$\triangle ABC$ is similar to $\triangle CDE$.