We can prove the triangle congruence that the question asks in three steps. In each step, we will prove the congruence of two triangles.
Triangles △ MFC and △ MFE
Let's start with triangles △ MFC and △ MFE.
Segment MF is a common side and it is given that the angles at M and at F are congruent in the two triangles. According to the Angle-Side-Angle (ASA) Congruence Postulate, this means that the two triangles are congruent.
△ MFC≅ △ MFE
Corresponding sides of congruent triangles are congruent, so the sides opposite to M and the sides opposite to F are congruent in the two triangles.
CF&≅EF
CM&≅EM
Let's mark these congruent segments on the diagram, and let's shift our focus on triangles △ DFC and △ DFE.
Triangles △ DFC and △ DFE
Let's summarize the properties of these two triangles.
Congruence
Justification
∠ CFD≅∠ EFD
Given
CF≅EF
Proven above
We also know that segment DF is a common side in the two triangles. According to the Side-Angle-Side (SAS) Congruence Postulate, this means that the two triangles are congruent.
△ DFC≅ △ DFE
Corresponding sides of congruent triangles are congruent, so the sides opposite to F are congruent.
CD&≅ED
Let's mark these congruent segments on the diagram, and let's shift our focus to triangles △ DMC and △ DME.
Triangles △ DMC and △ DME
Let's summarize the properties of these two triangles.
Congruence
Justification
CD≅ED
Proven above
CM≅EM
Proven above
We also know that segment DM is a common side in the two triangles. According to the Side-Side-Side (SSS) Congruence Postulate, this means that the two triangles are congruent.
△ DMC≅ △ DME
We can summarize the steps above in a two-column proof.
