Option E is not a valid congruence criterion and thus is not true. Therefore, option E is correct
Given the additional information that side BA is equal to side DE, side BC is equal to side EF, and angle C is a 90-degree angle, let's evaluate each statement:
A. If m∠D + m∠E = 90, then m∠F = 90. The triangles are congruent by HL.**
- If m∠D + m∠E = 90 , then m∠F , being the third angle in a triangle, would indeed be 90 degrees because the sum of angles in a triangle is 180 degrees. Given that both triangles are right-angled and have two sides congruent, they would be congruent by the HL (Hypotenuse-Leg) congruence theorem, where the hypotenuse is the side opposite the right angle and the leg is one of the other sides.
B. If m∠D = 37 , then m∠A = 37 . The triangles are congruent by AAS.
- If m∠D = 37 , then m∠A is also 37 degrees because of the given that side BA is equal to side DE and side BC is equal to side EF, implying that triangles ABC and DEF are congruent by the Angle-Side-Angle (ASA) postulate. However, the AAS (Angle-Angle-Side) congruence theorem would also apply here, as two angles and the non-included side in one triangle are congruent to two angles and the non-included side in another triangle.
C. If ∠E ≅ ∠B , then the triangles are congruent by SAS.
- If ∠E is congruent to ∠B, and we are given that side BA is equal to side DE and side BC is equal to side EF, then the triangles are congruent by Side-Angle-Side (SAS), where two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle.
D. **If ∠F is a right angle, then the triangles are congruent by HL.
- If ∠F is a right angle, then triangle DEF is a right triangle. Given that side BA (hypotenuse of triangle ABC) is equal to side DE and side BC is equal to side EF (legs of the right triangles), then triangles ABC and DEF would be congruent by the HL (Hypotenuse-Leg) congruence theorem for right triangles.
E. If m∠D + m∠E = 90, then m∠F = 90. The triangles are congruent by SSA.
- While the angle statement is correct, SSA (Side-Side-Angle) is not a valid congruence criterion because it can produce two different triangles or no triangle at all. However, because we are dealing with right triangles, the correct theorem is HL, not SSA.
Option E is not a valid congruence criterion and thus is not true.