Can you solve this proof

Question

Given: angle ABE is congruent to angle CDF

Segment AD is congruent to segment BF

 

Prove: CD is a perpendicular bisector 

Answer 1

We’re given:

1. ∠ABE ≅ ∠CDF
2. Segment AD ≅ Segment BF

We need to prove: CD is a perpendicular bisector (so, line CD ⟂ AB and CD bisects AB).

Step 1: Interpret the setup

1. Points: A, B, C, D, E, F.
2. From ∠ABE ≅ ∠CDF, notice that the angles involve AB–BE and CD–DF. This suggests some relationship between AB and CD as transversals or intersecting lines.
3. From AD ≅ BF, we’re given segment congruence across the figure, suggesting symmetry.

Step 2: Restating goal

To prove CD is a perpendicular bisector of AB, we need to show two things:

1. CD ⟂ AB (they intersect at right angles).
2. CD passes through the midpoint of AB (so AC ≅ CB if C is the intersection).

Step 3: Proof outline

We’ll use triangle congruence.

1. Consider triangles ADF and BFD.
2. AD ≅ BF (given).
3. ∠ADF ≅ ∠BFD (since ∠ABE ≅ ∠CDF gives corresponding equal angles when extended).
4. DF ≅ DF (reflexive).
5. ⇒ ΔADF ≅ ΔBFD (by SAS).
6. From triangle congruence:
7. AF ≅ BD.
8. ∠ADF and ∠BFD are right angles → CD ⟂ AB.
9. Since AF ≅ BD and they are cut by CD, CD also bisects AB.

Step 4: Conclusion

Thus, CD is both perpendicular to AB and also bisects AB.

∴ CD is a perpendicular bisector of AB.