Question

D is the midpoint of AC, ∠AED ≅ ∠CFD and ∠EDA ≅ ∠FDC. Prove ΔAED ≅ ΔCFD

Answer 1

To prove that ΔAED ≅ ΔCFD, we will use the two given angle equalities and the fact that D is the midpoint of AC:

Given: D is the midpoint of AC, ∠AED ≅ ∠CFD, and ∠EDA ≅ ∠FDC

To prove: ΔAED ≅ ΔCFD

Proof:

1. Since D is the midpoint of AC, we know that AD = DC and CF = FA.
2. Since ∠AED ≅ ∠CFD and ∠EDA ≅ ∠FDC, we have two pairs of corresponding angles that are equal.
3. Therefore, by the Angle-Angle (AA) similarity postulate, we can conclude that ΔAED ≅ ΔCFD.
4. Additionally, using the fact that AD = DC and CF = FA, we can conclude that ΔAED is congruent to ΔFAC by the Side-Angle-Side (SAS) similarity postulate.
5. Thus, we have ΔAED ≅ ΔCFD and ΔAED ≅ ΔFAC.
6. By the Transitive Property of Congruence, we can conclude that ΔCFD ≅ ΔFAC.
7. Finally, using the fact that CF = FA, we can conclude that ΔCFD is congruent to ΔFAC by the Side-Side-Side (SSS) congruence postulate.
8. Therefore, we have ΔAED ≅ ΔCFD ≅ ΔFAC.

Thus, we have proved that ΔAED ≅ ΔCFD.