Question

diagram: right angle at a (ab=3, ac=4, bc=5), right angle at e (de=9, ce=12, cd=15), point c on ae. options: 1. $\frac{15}{4} = \frac{12}{5} = \frac{9}{3}$ shows proportional sides, $\triangle abc sim \triangle edc$ (sas). 2. $angle dce cong angle bca$ (vertical angles), $\frac{15}{5} = \frac{12}{4}$ proportional, $\triangle abc sim \triangle edc$ (sas). 3. $angle e, angle a$ right angles (congruent), $\frac{12}{4} = \frac{9}{3}$ proportional, $\triangle abc sim \triangle edc$ (sss). 4. $angle dce cong angle cba$ (vertical angles?), $\frac{15}{3} = \frac{12}{4}$ proportional, $\triangle abc sim \triangle edc$ (sss).

Explanation:

Step1: Analyze Angle Congruence

First, identify congruent angles. $\angle E$ and $\angle A$ are right angles (so $\angle E \cong \angle A$). Also, $\angle DCE$ and $\angle BCA$ are vertical angles, so $\angle DCE \cong \angle BCA$. Let's check the second option: it says $\angle DCE \cong \angle BCA$ (vertical angles) and checks the ratio of sides around the congruent angle. For $\triangle ABC$ and $\triangle EDC$, the sides around $\angle BCA$ (in $\triangle ABC$: $BC = 5$, $AC = 4$) and around $\angle DCE$ (in $\triangle EDC$: $DC = 15$, $EC = 12$). Calculate the ratios: $\frac{DC}{BC} = \frac{15}{5} = 3$, $\frac{EC}{AC} = \frac{12}{4} = 3$. Wait, the second option has $\frac{15}{5}=\frac{12}{4}$, which is $3 = 3$, so the sides around the congruent vertical angles are proportional. By SAS Similarity (two sides proportional, included angle congruent), $\triangle ABC \sim \triangle EDC$. Let's check other options:

• First option: Ratio $\frac{15}{4}=\frac{12}{5}=\frac{3}{3}$? $\frac{15}{4}=3.75$, $\frac{12}{5}=2.4$, not equal. Wrong.
• Third option: Says SSS, but SSS needs all three sides proportional. $\frac{12}{4}=3$, $\frac{9}{3}=3$, but what about the hypotenuses? $\frac{15}{5}=3$, but it claims SSS with $\frac{12}{4}=\frac{9}{3}$, but the angle part is right angles (which is SAS, not SSS). Wrong.
• Fourth option: $\angle DCE \cong \angle CBA$? No, vertical angles are $\angle DCE$ and $\angle BCA$, not $\angle CBA$. Wrong.

Step2: Confirm SAS Similarity

The second option states $\angle DCE \cong \angle BCA$ (vertical angles) and $\frac{15}{5}=\frac{12}{4}$ (sides around the included angle proportional), so by SAS Similarity Postulate, $\triangle ABC \sim \triangle EDC$. This matches the criteria for SAS similarity (included angle congruent, two sides proportional).

Answer:

The correct option is: $\angle DCE$ is congruent to $\angle BCA$ by the Vertical Angles Theorem and $\frac{15}{5}=\frac{12}{4}$ shows the corresponding sides are proportional, therefore, $\triangle ABC \sim \triangle EDC$ by the SAS Similarity Postulate.