Solve this question please!

Dear Student!

(i) Since diagonal of a square bisects the vertex and BD is the diagonal of square ABCD

Given : EF || BD

⇒ ∠CEF = ∠CBD = 45°  and ∠CEF = ∠CDB = 45°  (corresponding angles)

⇒ ∠CEF = ∠CFE

⇒ CE = CF    (sides opposite of equal angles are equal)    .......(1)

Now  BC = CD  (Sides of square)    ........(2)

Subtracting (1) from (2) we get

⇒ BC – CE = CD – CF

⇒ BE = DF

(ii) ∆ABE ≅ ∆ADF  (by SAS congruency criterion)

⇒ ∠BAE = ∠DAF and AE = AF  ........(3)

and ∆AER ≅ ∆AFR    (by SSS congruency criterion)

⇒ ∠EAR = ∠FAR    .........(4)

Now Adding (3) and (4) we get

⇒ ∠BAE + ∠EAR = ∠DAF + ∠FAR

⇒ ∠BAR = ∠DAR

i.e AR bisects ∠BAD

(iii) Since AR bisects ∠BAD and we know that in a square its diagonal bisects the angle of vertex.

∴ AR is along the diagonal AC.

Hence,AR produced will pass through C.

Regards