Question

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.

• A. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

Recall which congruency rule needs two sides plus the included angle.

Answer 1

The Correct Option is C

Step 1: Recall what congruency needs.
Two triangles are congruent only when a full set of matching sides or angles, such as SAS or ASA, is confirmed equal.

Step 2: Check statement 1 alone.
Knowing both triangles are right angled says nothing about their actual side lengths.
Two right triangles can have completely different leg lengths and still both be right angled.
Statement 1 alone is not sufficient.

Step 3: Check statement 2 alone.
Equal base and equal height only guarantee equal area, not equal shape.
Many differently shaped triangles share the same base and height, so this alone cannot prove congruency.
Statement 2 alone is not sufficient.

Step 4: Combine both statements.
If both triangles are right angled and their base and height, which act as the two legs at the right angle, are equal, then two legs of one triangle match two legs of the other.
This satisfies the SAS condition, since the included right angle is also equal, so the triangles must be congruent.

Final Answer:
Both statements together confirm the triangles are congruent. \[ \boxed{\text{Both statements together are needed (option c)}} \]