Question

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

Explanation:

Step1: Calculate angles in \(\triangle DCB\)

In \(\triangle DCB\), using the angle - sum property of a triangle (\(\angle C+\angle D+\angle CBD = 180^{\circ}\)). Given \(\angle C = 85^{\circ}\), \(\angle D=52^{\circ}\), then \(\angle CBD=180^{\circ}-(85^{\circ} + 52^{\circ})=43^{\circ}\).

Step2: Check for similarity using AA (Angle - Angle) criterion

In \(\triangle ACE\) and \(\triangle DCB\), \(\angle C\) is common (\(\angle C=\angle C\)) and \(\angle CBD=\angle E = 43^{\circ}\). 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.

Answer:

\(\triangle ACE\) is similar to \(\triangle DCB\).