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 highlight the angle first, then write down the two sides meeting at that vertex.
The ratios of these two specific pairs of sides must be equal!

Answer 1

The Correct Option is D

Step 1: Recall what SAS similarity actually requires.
SAS similarity needs one equal angle, with the two sides forming (including) that angle proportional. Since the equal angle given is $\angle C = \angle R$, the sides that must be compared are exactly the ones touching $C$ and $R$.
Step 2: Identify the including sides at C and R.
In $\Delta ABC$, the sides meeting at $C$ are $AC$ and $BC$. In $\Delta PQR$, the sides meeting at $R$ are $PR$ and $QR$. These are the only sides SAS similarity can relate here.
Step 3: Eliminate options that use the wrong sides.
Option (A) uses $AB$ and $PQ$, which do not touch $C$ or $R$, so it is ruled out. Options (B) and (C) mix an including side with a non-including side. Only option (D), $\frac{AC}{BC} = \frac{PR}{QR}$, pairs the two sides that actually include the equal angles.
\[ \boxed{\frac{AC}{BC} = \frac{PR}{QR}} \]