Question

1. ABCD is a quadrilateral in which angle A = angle C and angle B = angle D

Prove that, if the opposite angles of a quadrilateral are equal, then the quadrilateral is a
parallelogram.

Answer 1

Quadrilateral ABCD is a parallelogram, quadrilateral with opposite sides parallel

Explanation:

Statement, Reason

∠A = ∠C, Given

∠B = ∠D, Given

Therefore, we have

∠A + ∠B + ∠C + ∠D = 360°, Sum of interior angles of a quadrilateral

2×∠A + 2×∠B = 360°, Substitution property of equality

∴ ∠A + ∠B = 360°/2 = 180°

∠A + ∠B, are supplementary angles, Angles that sum up to 180°

Similarly,

∠C + ∠D, are supplementary angles, Angles that sum up to 180°

Therefore, segment DA is parallel to segment BC, (Lines with supplementary angles on the same side of the transversal)

Similarly, given ∠A = ∠C and ∠B = ∠D, we have;

∠A + ∠D and ∠C + ∠B are supplementary angles, Angles that sum up to 180°

Therefore, segment AB is parallel to segment DC, (Lines with supplementary angles on the same side of the transversal)

Therefore;

Quadrilateral ABCD is a parallelogram, quadrilateral with parallel opposite sides.