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

Given that \(\triangle ABC \sim \triangle PQR\) using the SAS similarity criterion, it implies that two sides and the included angle of \(\triangle ABC\) are proportional and equal, respectively, to two sides and the included angle of \(\triangle PQR\). Devansh found that \(\angle C = \angle R\), establishing that the angles opposite the sides being compared are equal.

The SAS similarity criterion states:

• Two sides of one triangle are proportional to two sides of another triangle.
• The included angles of the corresponding sides are equal.

Thus, in \(\triangle ABC \sim \triangle PQR\) by SAS similarity, we can deduce:

• The side \(\overline{AC}\) corresponds to \(\overline{PR}\), and the side \(\overline{BC}\) corresponds to \(\overline{QR}\).
• The corresponding angles \(\angle C = \angle R\).

Based on this information, the ratio of corresponding sides should be equal:

• \(\frac{AC}{BC} = \frac{PR}{QR}\)

This is because, with \(\angle C = \angle R\) being the included angle, the proportions of the sides \(\overline{AC}\) to \(\overline{BC}\) and \(\overline{PR}\) to \(\overline{QR}\) must remain consistent due to similarity.

Therefore, the option that was proved true is:

\(\frac{AC}{BC} = \frac{PR}{QR}\)

Let's rule out the other options:

• \(\frac{AC}{AB} = \frac{PR}{PQ}\): Incorrect because \(\overline{AB}\) and \(\overline{PQ}\) are not the sides directly included with angle \(\angle C\) and \(\angle R\).
• \(\frac{BC}{AC} = \frac{PR}{QR}\): Incorrect ratio setup within triangle similarity context.
• \(\frac{AC}{BC} = \frac{PR}{PQ}\): PQ is not the correctly corresponding side with respect to \(\overline{BC}\).