In triangles ABC and DEF; ∠B = 90°, BC = 8 cm, ∠A = 40°, DE = 8 cm, ∠F = 40°…

Concept: Two angles of a triangle fix the third, since a triangle's angles sum to 180°. Two equal angles together with one equal side then fix a unique congruence rule between two triangles: ASA when the equal side lies between the two given angles (the included side), AAS when it lies opposite one of them (a non-included side), and RHS when, for right triangles specifically, the hypotenuse and one leg are equal.

1. In ΔABC, ∠C = 180° − (∠A + ∠B) = 180° − (40° + 90°) = 50°.

2. In ΔDEF, ∠D = 180° − (∠F + ∠E) = 180° − (40° + 90°) = 50°.

3. Matching equal angles between the two triangles: ∠A = ∠F = 40°, ∠B = ∠E = 90°, ∠C = ∠D = 50° — so the vertex correspondence is A↔F, B↔E, C↔D.

4. The equal side (BC = DE = 8 cm) joins vertices B and C in the first triangle and D and E in the second, so it lies opposite vertex A in the first triangle and opposite vertex F in the second — a side that is NOT included between the two given angles.

5. Two equal angles with a non-included equal side is the AAS pattern, not ASA (the included side would be AB and FE, which were not given).

6. Writing the congruence in the matched vertex order (A↔F, B↔E, C↔D): ΔABC ≅ ΔFED, by AAS.

Cross-check: RHS is ruled out because the equal side (BC, DE) is a leg next to the right angle in each triangle, not the hypotenuse (AC and FD, which are not given); and reversing the correspondence to ΔDFE would require A to pair with D, which contradicts the angle match above — so the AAS correspondence ΔABC ≅ ΔFED is the one that holds.