Question:easy

Are triangle ABC and triangle PQR congruent?

Statement 1: The triangles are right angled triangles.
Statement 2: The base and heights of the two triangles are equal.

Show Hint

Recall which congruency rule needs two sides plus the included angle.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: Set the congruency rule to apply.
SAS congruency needs two sides and the angle between them to match exactly in both triangles.

Step 2: Test statement 1 alone.
Being right angled fixes only one angle in each triangle, with no length data at all.
Right triangles of many different sizes exist, so congruency cannot be claimed from this fact alone.

Step 3: Test statement 2 alone.
Matching base and height fixes equal areas, since area $ = \frac{1}{2} \times base \times height $ works out the same for both.
Equal area triangles are not always the same shape, so shape and congruency stay unproven.

Step 4: Combine both statements.
With right angles confirmed and the base and height, which are the two legs meeting at that right angle, equal in both triangles, the two legs and the included angle all match.
That is exactly the SAS pattern, so the triangles must be congruent.

Final Answer:
Only the combined data proves congruency between the two triangles. \[ \boxed{\text{Both statements together are needed (option c)}} \]
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