Are triangles ΔABC and ΔPQR congruent? I. Areas of ΔABC and ΔPQR are equal. II. ΔABC and ΔPQR are right triangles.

Introduction / Context:Triangle congruence typically requires side-angle-side (SAS), side-side-side (SSS), angle-side-angle (ASA), or right-angle-hypotenuse-side (RHS) information. Equal area and both being right-angled do not automatically imply congruence.

Given Data / Assumptions:

• Statement I: area(ΔABC) = area(ΔPQR).
• Statement II: both triangles are right triangles.

Concept / Approach:Equal area does not fix shape or side lengths; many non-congruent triangles can share the same area. Knowing both are right triangles still leaves infinitely many non-congruent possibilities with the same area (different legs producing the same 1/2 * leg1 * leg2).

Step-by-Step Solution:With I alone: triangles can have equal areas yet different side sets ⇒ not sufficient.With II alone: both right-angled does not determine equality of corresponding sides ⇒ not sufficient.With I and II together: right triangles with area A can be formed by infinitely many integer or real leg pairs (x, 2A/x), leading to different hypotenuse lengths. Hence still not sufficient to conclude congruence.

Verification / Alternative check:Example: Right triangles with legs (3,8) and (4,6) both have area 12 but are not congruent.

Why Other Options Are Wrong:Neither statement alone reaches a congruence criterion; together they also fall short of SAS/SSS/ASA/RHS with matching corresponding measures.

Common Pitfalls:Confusing equal area with congruence; congruence requires stricter conditions.

Final Answer:Statements I and II together are not sufficient to assert congruence.