Question

In the quadrilateral ABCD, AB = AD, BC = CD and BC > AB. Prove that ∠x > ∠y. [Hint: Join AC.]

Answer 1

Given:

Quadrilateral ABCD where AB = AD, BC = CD and BC > AB.

To Prove: ∠x > ∠y

Proof:

Join AC in the quadrilateral ABCD.

Since AB = AD and BC = CD, triangles ABC and ADC have two pairs of equal sides AB = AD and BC = CD.

Consider triangles ABC and ADC. We have AB = AD given, BC = CD given and AC is common.

Now observe the two triangles ABC and ADC: They are isosceles triangles with sides AB = AD and BC = CD respectively.

Given BC > AB. This implies in triangle ABC, side BC > side AB.

In an isosceles triangle, the larger side lies opposite the larger angle.

Since BC > AB. 

Therefore ∠x angle opposite side BC in triangle ABC is greater than ∠y angle opposite side CD in triangle ADC.

Hence, ∠x > ∠y.

In the quadrilateral ABCD where AB = AD, BC = CD and BC > AB, it is proven that ∠x > ∠y by considering the triangles formed by joining AC and applying properties of isosceles triangles and the relation between sides and opposite angles.