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 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!
The ratio 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 two triangles, $\Delta ABC$ and $\Delta PQR$.
Devansh proved that these triangles are similar ($\Delta ABC \sim \Delta PQR$) using the Side-Angle-Side (SAS) similarity criterion.
The given corresponding equal angle is $\angle C = \angle R$.
We need to determine the correct ratio of sides that must be proportional to satisfy this SAS similarity criterion.
Step 2: Key Formula or Approach:
The Side-Angle-Side (SAS) 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 two triangles are similar.
For the angle $\angle C$ in $\Delta ABC$, the two containing sides are $AC$ and $BC$.
For the angle $\angle R$ in $\Delta PQR$, the two containing sides are $PR$ and $QR$.
Step 3: Detailed Explanation:
• Identify the sides containing the equal angles:
In $\Delta ABC$, the angle $\angle C$ is formed by the sides $AC$ and $BC$.
In $\Delta PQR$, the angle $\angle R$ is formed by the sides $PR$ and $QR$.
• According to the SAS similarity criterion, the corresponding containing sides must be in the same ratio:
\[ \frac{AC}{PR} = \frac{BC}{QR} \]
• We can rearrange this proportion by cross-multiplying or rearranging the terms:
Multiply both sides by $PR$ and divide both sides by $BC$:
\[ \frac{AC}{BC} = \frac{PR}{QR} \]
• Let us evaluate the given choices to see which one matches this relation:
- Option (A) $\frac{AC}{AB} = \frac{PR}{PQ}$: This involves $AB$ and $PQ$, which do not contain the angles $C$ and $R$.
- Option (B) $\frac{BC}{AC} = \frac{PR}{QR}$: The left side is $\frac{BC}{AC}$, but the right side is $\frac{PR}{QR}$ instead of $\frac{QR}{PR}$.
- Option (C) $\frac{AC}{BC} = \frac{PR}{PQ}$: This incorrectly uses $PQ$.
- Option (D) $\frac{AC}{BC} = \frac{PR}{QR}$: This perfectly matches our derived relation.
Step 4: Final Answer:
Therefore, the statement proved true is $\frac{AC}{BC} = \frac{PR}{QR}$.
We are given two triangles, $\Delta ABC$ and $\Delta PQR$.
Devansh proved that these triangles are similar ($\Delta ABC \sim \Delta PQR$) using the Side-Angle-Side (SAS) similarity criterion.
The given corresponding equal angle is $\angle C = \angle R$.
We need to determine the correct ratio of sides that must be proportional to satisfy this SAS similarity criterion.
Step 2: Key Formula or Approach:
The Side-Angle-Side (SAS) 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 two triangles are similar.
For the angle $\angle C$ in $\Delta ABC$, the two containing sides are $AC$ and $BC$.
For the angle $\angle R$ in $\Delta PQR$, the two containing sides are $PR$ and $QR$.
Step 3: Detailed Explanation:
• Identify the sides containing the equal angles:
In $\Delta ABC$, the angle $\angle C$ is formed by the sides $AC$ and $BC$.
In $\Delta PQR$, the angle $\angle R$ is formed by the sides $PR$ and $QR$.
• According to the SAS similarity criterion, the corresponding containing sides must be in the same ratio:
\[ \frac{AC}{PR} = \frac{BC}{QR} \]
• We can rearrange this proportion by cross-multiplying or rearranging the terms:
Multiply both sides by $PR$ and divide both sides by $BC$:
\[ \frac{AC}{BC} = \frac{PR}{QR} \]
• Let us evaluate the given choices to see which one matches this relation:
- Option (A) $\frac{AC}{AB} = \frac{PR}{PQ}$: This involves $AB$ and $PQ$, which do not contain the angles $C$ and $R$.
- Option (B) $\frac{BC}{AC} = \frac{PR}{QR}$: The left side is $\frac{BC}{AC}$, but the right side is $\frac{PR}{QR}$ instead of $\frac{QR}{PR}$.
- Option (C) $\frac{AC}{BC} = \frac{PR}{PQ}$: This incorrectly uses $PQ$.
- Option (D) $\frac{AC}{BC} = \frac{PR}{QR}$: This perfectly matches our derived relation.
Step 4: Final Answer:
Therefore, the statement proved true is $\frac{AC}{BC} = \frac{PR}{QR}$.
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