Question

In the given figure, if ray BA || ray DE, ∠C = 50^° and ∠D = 100^°. Find the measure of ∠ABC.

(Hint: Draw a line passing through point C and parallel to line AB.)

Answer 1

Draw a line XY passing through point C and parallel to AB.

We have, AB || DE and AB || XY.

It is known that if two lines in a plane are parallel to a third line in the plane, then those two lines are parallel to each other.

∴ DE || XY

Since DE || XY and DC is a transversal intersecting them at D and C, then

∠EDC + ∠YCD = 180^∘    ...(Pairs of interior angles on the same side of transversal are supplementary)

⇒ 100^∘ + ∠YCD = 180^∘ 

∠YCD = 180^∘ − 100^∘ = 80^∘

Since, sum of all the angles on a straight line at a point is 180^∘, then

∠XCB + ∠BCD + ∠YCD = 180^∘

⇒ ∠XCB + 50^∘ + 80^∘ = 180^∘

⇒ ∠XCB + 130^∘ = 180^∘

⇒ ∠XCB = 180^∘ − 130^∘ 

⇒ ∠XCB = 50^∘

Since, AB || XY and BC is a transversal intersecting them at B and C, then

∠ABC + ∠XCB = 180^∘    ...(Pairs of interior angles on the same side of transversal are supplementary)

⇒ ∠ABC + 50^∘ = 180^∘

⇒ ∠ABC = 180^∘ − 50^∘ 

⇒ ∠ABC = 130^∘