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?
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?
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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!
The ratios of these two specific pairs of sides must be equal!
Updated On: Jul 22, 2026
- $\frac{AC}{AB} = \frac{PR}{PQ}$
- $\frac{BC}{AC} = \frac{PR}{QR}$
- $\frac{AC}{BC} = \frac{PR}{PQ}$
- $\frac{AC}{BC} = \frac{PR}{QR}$
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
We are given that $\Delta ABC$ and $\Delta PQR$ are similar under the Side-Angle-Side (SAS) similarity criterion.
The equal corresponding angle identified is $\angle C = \angle R$.
We need to determine the correct corresponding side ratio that satisfies this SAS similarity.
Step 2: Key Formula or Approach:
The SAS (Side-Angle-Side) similarity criterion states that if one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the triangles are similar.
- For $\Delta ABC$, the sides that include the angle $\angle C$ are $AC$ and $BC$.
- For $\Delta PQR$, the sides that include the angle $\angle R$ are $PR$ and $QR$.
Step 3: Detailed Explanation:
• Identify the including sides for both of the given corresponding angles:
In $\Delta ABC$, the angle $\angle C$ is located between sides $AC$ and $BC$.
In $\Delta PQR$, the angle $\angle R$ is located between sides $PR$ and $QR$.
• According to the SAS similarity criterion, the ratio of these including sides must be equal:
\[ \frac{AC}{PR} = \frac{BC}{QR} \]
• Rearrange the terms of this proportion to match the format of the options:
Multiply both sides by $PR$ and divide both sides by $BC$:
\[ \frac{AC}{BC} = \frac{PR}{QR} \]
• Let us compare this result with the options:
- Option (A) uses $AB$ and $PQ$, which are not the sides enclosing the respective angles.
- Option (B) has mismatched term inversion.
- Option (C) incorrectly uses side $PQ$.
- Option (D) exactly matches our rearranged proportional equation: $\frac{AC}{BC} = \frac{PR}{QR}$.
Step 4: Final Answer:
The statement that was proved true is $\frac{AC}{BC} = \frac{PR}{QR}$.
We are given that $\Delta ABC$ and $\Delta PQR$ are similar under the Side-Angle-Side (SAS) similarity criterion.
The equal corresponding angle identified is $\angle C = \angle R$.
We need to determine the correct corresponding side ratio that satisfies this SAS similarity.
Step 2: Key Formula or Approach:
The SAS (Side-Angle-Side) similarity criterion states that if one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the triangles are similar.
- For $\Delta ABC$, the sides that include the angle $\angle C$ are $AC$ and $BC$.
- For $\Delta PQR$, the sides that include the angle $\angle R$ are $PR$ and $QR$.
Step 3: Detailed Explanation:
• Identify the including sides for both of the given corresponding angles:
In $\Delta ABC$, the angle $\angle C$ is located between sides $AC$ and $BC$.
In $\Delta PQR$, the angle $\angle R$ is located between sides $PR$ and $QR$.
• According to the SAS similarity criterion, the ratio of these including sides must be equal:
\[ \frac{AC}{PR} = \frac{BC}{QR} \]
• Rearrange the terms of this proportion to match the format of the options:
Multiply both sides by $PR$ and divide both sides by $BC$:
\[ \frac{AC}{BC} = \frac{PR}{QR} \]
• Let us compare this result with the options:
- Option (A) uses $AB$ and $PQ$, which are not the sides enclosing the respective angles.
- Option (B) has mismatched term inversion.
- Option (C) incorrectly uses side $PQ$.
- Option (D) exactly matches our rearranged proportional equation: $\frac{AC}{BC} = \frac{PR}{QR}$.
Step 4: Final Answer:
The statement that was proved true is $\frac{AC}{BC} = \frac{PR}{QR}$.
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