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: Restate what SAS similarity needs.
For $\Delta ABC \sim \Delta PQR$ proved by SAS at the equal angle $\angle C = \angle R$, the two sides that form this angle in each triangle must be proportional, matched in the correct order.
Step 2: Identify the two sides at C and R.
At vertex $C$ the sides forming the angle are $AC$ and $BC$. At vertex $R$ the matching sides are $PR$ and $QR$. Since $C$ corresponds to $R$, $A$ corresponds to $P$ and $B$ corresponds to $Q$, so $AC$ must pair with $PR$, and $BC$ must pair with $QR$.
Step 3: Write the ratio and check it against the choices.
This pairing gives $\frac{AC}{PR} = \frac{BC}{QR}$, which rearranges to $\frac{AC}{BC} = \frac{PR}{QR}$. Checking the four options, only option (D) has exactly this form; the rest bring in $AB$ or $PQ$, sides that were never part of the SAS condition at $C$ and $R$.
\[ \boxed{\frac{AC}{BC} = \frac{PR}{QR}} \]