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: Find the measure of \(\angle ECD\)

Since \(\angle BCD = 180^{\circ}\) (straight - line angle), and \(\angle ECD=59^{\circ}\), then \(\angle ACB = 180^{\circ}-90^{\circ}-59^{\circ}=31^{\circ}\)

Step2: Check the angles of \(\triangle ABC\) and \(\triangle CDE\)

In \(\triangle ABC\), \(\angle B = 90^{\circ}\), \(\angle ACB = 31^{\circ}\), so \(\angle A=180^{\circ}-\angle B-\angle ACB=180^{\circ}-90^{\circ}-31^{\circ} = 59^{\circ}\)
In \(\triangle CDE\), \(\angle D = 90^{\circ}\), \(\angle ECD = 31^{\circ}\), so \(\angle E=180^{\circ}-\angle D-\angle ECD=180^{\circ}-90^{\circ}-31^{\circ}=59^{\circ}\)

Since \(\angle A=\angle E = 59^{\circ}\) and \(\angle B=\angle D = 90^{\circ}\), by the AA (Angle - Angle) similarity criterion, \(\triangle ABC\sim\triangle CDE\)

Answer:

\(\triangle ABC\) is similar to \(\triangle CDE\)