In the figure (2) given below, AB || DC and ∠C = ∠D. Prove that

(i) AD = BC

(ii) AC = BD.

(i) Draw AE ⊥ CD, BF ⊥ CD.

Considering △ADE and △BCF we get,

∠ADE = ∠BCF (Given)

∠AED = ∠BFC = 90°

AE = BF (Distance between parallel lines are equal)

Hence, △ADE ≅ △BCF by AAS axiom.

We know that corresponding parts of congruent triangles are equal.

∴ AD = BC.

Hence, proved that AD = BC.

(ii) Join AC, BD.

Considering △ACD and △BDC we get,

∠ADC = ∠BCD (Given)

AD = BC (Proved)

DC = DC (Common)

Hence, △ACD ≅ △BDC by SAS axiom.

We know that corresponding parts of congruent triangles are equal.

∴ AC = BD.

Hence, proved that AC = BD.