Question

ABCD is a square and EF is parallel to diagonal BD and EM=FM. prove that DF=BE AM bisects angle BAD

Answer 1

Solution:-
Since diagonal of a square bisects the vertex BD is the diagonal of square ABCD.
∴ CBD = CDB = 90°/2 = 45°
Given - EF is parallel to BD
⇒ CEF = CBD = 45° and CEF = CDB = 45° (Corresponding angles)
⇒ CEF = CFE
⇒ CE = CF (Sides opposite to equal angles are equal) ....(1)
Now, BC = CD (Sides of square) ....(2)
Subtracting (1) from (2), we get
⇒ BC CE = CD CF
⇒ BE = DF  Proved
Δ ABE ≡ Δ ADF (SAS congruency criterion)
⇒ BAE = DAF and AE = AF ....(3)
and, Δ AEM ≡ Δ AFM (By SSS criterion)
⇒ EAM = FAM .....(4)
Now, adding (3) and (4), we get
⇒ BAE + EAM = DAF + FAM
⇒ BAM = DAM
i.e. AM bisects angle BAD
Hence proved.