Question

Devansh proved that $\Delta ABC \sim \Delta PQR$ using SAS similarity criteria. If he found $\angle C = \angle R$, then which of the following was proved true?

• A. $\frac{AC}{AB} = \frac{PR}{PQ}$
• B. $\frac{BC}{AC} = \frac{PR}{QR}$
• C. $\frac{AC}{BC} = \frac{PR}{PQ}$
• D. $\frac{AC}{BC} = \frac{PR}{QR}$

Hint:

For SAS similarity, always identify the angle vertex first, then trace the two sides meeting at that vertex.
The ratio of these two specific pairs of sides must be equal!

Answer 1

The Correct Option is D

Step 1: Recall what SAS similarity needs.
SAS similarity requires one equal angle plus the two sides forming that angle to be in the same ratio, in matching order.
Step 2: Locate the sides around the equal angle in each triangle.
Angle $C$ in $\triangle ABC$ is formed by sides $CA$ and $CB$; angle $R$ in $\triangle PQR$ is formed by sides $RP$ and $RQ$.
Step 3: Match them up in the same order to write the proportion.
Matching $CA \leftrightarrow RP$ and $CB \leftrightarrow RQ$ gives
\[ \frac{AC}{PR} = \frac{BC}{QR} \implies \frac{AC}{BC} = \frac{PR}{QR} \]
Step 4: Conclude.
This is exactly option (4).
\[ \boxed{\frac{AC}{BC} = \frac{PR}{QR}} \]