In the given figure,BA=FC andDB =EF andLDBC = LAFE,then prove that:!!,.AFE = !!,.CBD

Given: BA = FC, DB = EF, and ∠LDBC = ∠LAFE

To prove: ∠AFE = ∠CBD

Proof:

1. Given that BA = FC and DB = EF, we can conclude that triangle BAC is congruent to triangle FCE by the Side-Angle-Side (SAS) congruence criterion.

2. Therefore, ∠BAC = ∠FCE (corresponding angles of congruent triangles).

3. Given that ∠LDBC = ∠LAFE, we can conclude that triangle LDB is congruent to triangle LFE by the Angle-Side-Angle (ASA) congruence criterion.

4. Therefore, ∠LDB = ∠LFE (corresponding angles of congruent triangles).

5. Since ∠LDB = ∠LFE and DB = EF, we can conclude that triangle LDB is congruent to triangle LFE by the Angle-Side-Angle (ASA) congruence criterion.

6. Therefore, ∠LBD = ∠LEF (corresponding angles of congruent triangles).

7. Now, consider the quadrilateral ABDC. Since ∠BAC = ∠FCE (from step 2) and ∠LBD = ∠LEF (from step 6), we can conclude that ∠CBD = ∠AFE (opposite angles of a quadrilateral).

8. Hence, we have proved that ∠AFE = ∠CBD.

Therefore, in the given figure, if BA = FC, DB = EF, and ∠LDBC = ∠LAFE, then ∠AFE = ∠CBD.