Question

Are ΔABC and ΔPQR congruent?
Statement 1: The triangles are right-angled triangles
Statement 2: The base and heights of two triangles are equal

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question.

Answer 1

The Correct Option is C

To determine if ΔABC and ΔPQR are congruent, we need to assess the statements given: 

1. Statement 1: The triangles are right-angled triangles.
   • This statement alone indicates only the type of triangles being compared (right-angled), but does not provide any specific measurements or angle information to establish congruence.
   • Right-angled triangles can be congruent only if we have additional specific information such as side lengths or angles that correspond. Therefore, Statement 1 alone is insufficient.
2. Statement 2: The base and heights of two triangles are equal.
   • This tells us that the two triangles have the same area since the formula for the area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\).
   • However, equal areas do not directly imply that two triangles are congruent, as their shapes or other side lengths and angles may differ. Hence, this statement alone is also insufficient.

Conclusion: Both statements provide important information about the triangles, but neither is sufficient on its own. When combined, they imply that both triangles are right-angled, have equal bases, and equal heights. In right-angled triangles, this matches the conditions for congruence under the RHS (Right angle, Hypotenuse, Side) or other congruence criteria if additional information is implied.

Thus, both statements together are needed to conclusively determine that ΔABC and ΔPQR are congruent. Therefore, the correct answer is: both the statements together are needed to answer the question.