Question

prove that δabc and δedc are similar.

Explanation:

Step1: Check angle - equality

$\angle BAC=\angle DEC = 90^{\circ}$ (Given right - angles).

Step2: Calculate side - ratios

For $\triangle ABC$ and $\triangle EDC$, we find the ratios of corresponding sides. $\frac{AC}{EC}=\frac{4}{12}=\frac{1}{3}$, $\frac{AB}{ED}=\frac{3}{9}=\frac{1}{3}$, $\frac{BC}{DC}=\frac{5}{15}=\frac{1}{3}$.

Step3: Apply similarity criterion

Since $\angle BAC=\angle DEC$ and $\frac{AC}{EC}=\frac{AB}{ED}=\frac{BC}{DC}$, by the Side - Angle - Side (SAS) similarity criterion, $\triangle ABC\sim\triangle EDC$.

Answer:

$\triangle ABC$ and $\triangle EDC$ are similar by the SAS similarity criterion.