In the figure below, the lines AB and CD are parallel and M is the midpoint of AB.

i) Computer the angles of ΔAMD, ΔMBC and ΔDCM?

ii) What is special about the quadrilaterals AMCD and MBCD?

Given, lines AB and CD are parallel and M is the midpoint of AB.

As AB and CD are parallel, AD, DM, MC and CB are transversal.

(i) ∠AMD = ∠MBC = 60⁰ (corresponding angles)

∠CMB = ∠DAM = 40⁰ (corresponding angles)

∠CDM = AMD = 60⁰ (alternate interior angles)

∠DCM = ∠CMB = 40⁰ (alternate interior angles)

On straight line AMB,

∠ AMD + ∠DMC + ∠CMB = 180⁰ (angles in a straight line)

60⁰ + ∠DMC + 40⁰ = 180⁰

∠DMC + 100⁰ = 180⁰

∠DMC = 180⁰-100⁰

∠DMC = 80⁰

∠ADM = ∠DMC = 80⁰ (alternate interior angles)

∠MCB = ∠DMC = 80⁰ (alternate interior angles)

Therefore, now we have,

(ii) Quadrilaterals AMCD and MBCD both contain two equal triangles. That is what makes special.

Quad. AMCD consists of ∆AMD and ∆DMC.

Quad. MBCD consists of ∆MBC and ∆DMC.