Two line segments AC and BD bisect each other at P. Draw the diagram and prove that

(i) AB = CD

(ii) ∠BAC = ∠DCA

(i) Given, AC and BD bisect each other at P.

Join CD, BC, AB and AD.

In △BPA and △CPD,

As, P bisects AC and BD

⇒ PA = PC (P bisects AC and BD)

⇒ PB = PD (P bisects AC and BD)

⇒ ∠BPA = ∠CPD (Vertically opposite angles are equal).

∴ △BPA ≅ △CPD by SAS axiom.

We know that corresponding sides of congruent triangles are equal.

∴ AB = CD (By C.P.C.T.C.)

Hence, proved that AB = CD.

(ii) As proved in part (i),

△BPA ≅ △CPD by SAS axiom.

We know that corresponding angles of congruent triangles are equal.

∠DCP = ∠PAB ……………………(1)

From figure we get,

∠DCP = ∠DCA and ∠PAB = ∠BAC.

Substituting above values in equation (1) we get,

∠DCA = ∠BAC.

Hence, proved that ∠BAC = ∠DCA.