Complete the flow proof for a Hypotenuse-Angle Theorem. Given AC congruent DF, <C congruent <F. <B and <E are right angles. Prove triangle ABC congruent triangle DEF. Thanks for tutors.

To complete the flow proof for the Hypotenuse-Angle Theorem, we need to use the given information and apply the congruence statements to prove that triangle ABC is congruent to triangle DEF.

Here is the complete flow proof:

Statement | Reason
-------------------------------------------------------
1. AC ≅ DF | Given
2. ∠C ≅ ∠F | Given
3. ∠B and ∠E are right angles. | Given
4. AB ≅ DE | Hypotenuse-Leg Congruence (HL Congruence)
5. ∠A ≅ ∠D | Right Angle Congruence (RA Congruence)
6. BC ≅ EF | Hypotenuse-Leg Congruence (HL Congruence)
--------------------------------------------------------
7. △ABC ≅ △DEF | By using the criteria: SSS Congruence (Side-Side-Side)

Let's explain the reasoning for each statement:

1. AC ≅ DF: Given. It is given that side AC is congruent to side DF.

2. ∠C ≅ ∠F: Given. It is given that angle C is congruent to angle F.

3. ∠B and ∠E are right angles: Given. It is given that angles B and E are right angles, i.e., they measure 90 degrees.

4. AB ≅ DE: Hypotenuse-Leg Congruence (HL Congruence). As both triangles have a pair of congruent sides (AC ≅ DF) and a pair of congruent angles (∠C ≅ ∠F), we can conclude that the remaining sides, AB and DE, are also congruent.

5. ∠A ≅ ∠D: Right Angle Congruence (RA Congruence). Both triangles have a right angle at vertex B and vertex E (since ∠B and ∠E are right angles). Thus, these angles are congruent.

6. BC ≅ EF: Hypotenuse-Leg Congruence (HL Congruence). Since AC ≅ DF and ∠C ≅ ∠F, the remaining sides, BC and EF, are congruent.

7. △ABC ≅ △DEF: By using the criteria SSS Congruence (Side-Side-Side). We have shown that all three corresponding sides of the triangles are congruent (AB ≅ DE, BC ≅ EF), and all three corresponding angles are congruent (∠A ≅ ∠D, ∠C ≅ ∠F). Therefore, the two triangles are congruent.

Thus, we have successfully completed the flow proof for the Hypotenuse-Angle Theorem, proving that triangle ABC is congruent to triangle DEF based on the given information.

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