Question

In \(\triangle ABC\) & \(\triangle PQR\), if \(AB = QR\), \(BC = PR\) and \(CA = PQ\), then :

• A. \(\triangle ABC \cong \triangle PQR\)
• B. \(\triangle CBA \cong \triangle PRQ\)
• C. \(\triangle BAC \cong \triangle RPQ\)
• D. \(\triangle PQR \cong \triangle BCA\)

Hint:

To quickly verify congruence options, check the corresponding segment pairs.
For option (B) \(\triangle CBA \cong \triangle PRQ\), the first two letters give segment \(CB\) corresponding to \(PR\).
The given data says \(BC = PR\), which matches.
The last two letters give \(BA\) corresponding to \(RQ\) (\(AB = QR\)), which also matches.
This systematic check helps avoid mistakes with vertex ordering.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question is from Triangle Geometry, specifically focusing on the congruence of triangles under the SSS (Side-Side-Side) criterion and the correct order of matching vertices.

Step 2: Key Formula or Approach:
If three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent (SSS Congruence Rule).
When writing the congruence statement, the vertices must be written in the exact corresponding order:
If vertex \(X\) corresponds to vertex \(Y\), then they must occupy the same relative position in the names of the congruent triangles.

Step 3: Detailed Explanation:
We are given:
1. \[ AB = QR \]
2. \[ BC = PR \]
3. \[ CA = PQ \]
Let us find the vertex-to-vertex correspondence:
- Look at \(AB = QR\) and \(CA = PQ\). Vertex \(A\) is common to \(AB\) and \(CA\), and vertex \(Q\) is common to \(QR\) and \(PQ\). Thus, vertex \(A\) corresponds to vertex \(Q\) (\(A \leftrightarrow Q\)).
- Look at \(AB = QR\) and \(BC = PR\). Vertex \(B\) is common to \(AB\) and \(BC\), and vertex \(R\) is common to \(QR\) and \(PR\). Thus, vertex \(B\) corresponds to vertex \(R\) (\(B \leftrightarrow R\)).
- Look at \(BC = PR\) and \(CA = PQ\). Vertex \(C\) is common to \(BC\) and \(CA\), and vertex \(P\) is common to \(PR\) and \(PQ\). Thus, vertex \(C\) corresponds to vertex \(P\) (\(C \leftrightarrow P\)).
Now, write the congruence relation matching the vertices in order:
If we write the first triangle as \(\triangle CBA\):
- The first letter \(C\) corresponds to \(P\).
- The second letter \(B\) corresponds to \(R\).
- The third letter \(A\) corresponds to \(Q\).
Therefore, the correct congruence statement is:
\[ \triangle CBA \cong \triangle PRQ \]
Let us verify the sides from this statement:
- \(CB = PR\) (Given as \(BC = PR\))
- \(BA = RQ\) (Given as \(AB = QR\))
- \(CA = PQ\) (Given as \(CA = PQ\))
This confirms that the statement is correct.

Step 4: Final Answer:
The correct congruence statement is \(\triangle CBA \cong \triangle PRQ\).