Question

Devansh proved that \(\triangle ABC \sim \triangle PQR\) 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:

Remember that for SAS, the sides must be the ones that actually form the angle. Just looking at the letters: \(\angle C\) involves sides with \(C\) (\(AC, BC\)); \(\angle R\) involves sides with \(R\) (\(PR, QR\)).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
SAS (Side-Angle-Side) similarity criterion states that two triangles are similar if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal.
Step 2: Detailed Explanation:
In \(\triangle ABC\), the angle is \(\angle C\). The sides forming this angle are \(AC\) and \(BC\).
In \(\triangle PQR\), the corresponding angle is \(\angle R\). The sides forming this angle are \(PR\) and \(QR\).
For SAS similarity to hold with \(\angle C = \angle R\):
\[ \frac{AC}{PR} = \frac{BC}{QR} \]
Rearranging the terms (alternando):
\[ \frac{AC}{BC} = \frac{PR}{QR} \]
This matches option (D).
Step 3: Final Answer:
The required condition is \(\frac{AC}{BC} = \frac{PR}{QR}\).