Question

In a rectangle ABCD, X and Y are points on the sides AD and BC respectively such that AY = BX. Prove that: BY = AX and ∠BAY = ∠ABX.

Answer 1

Given:

• ABCD is a rectangle.
• X is on AD and Y is on BC.
• AY = BX.

To Prove:

• BY = AX.
• ∠BAY = ∠ABX.

Proof [Step-wise]:

1. In a rectangle AB ⟂ AD and AD || BC, AB || CD (definition/properties of a rectangle).

Reason: Opposite sides of a rectangle are parallel and adjacent sides are perpendicular.

2. AB ⟂ BY because BY lies on BC and BC || AD (so BY is perpendicular to AB).

Hence, ∠ABY = 90°, so triangle AYB is a right triangle with right angle at B.

Reason: BC is perpendicular to AB (From step 1).

3. AB ⟂ AX because AX lies on AD and AD ⟂ AB.

Hence, ∠BAX = 90°, so triangle ABX or XBA is a right triangle with right angle at A.

Reason: AD is perpendicular to AB (From step 1).

4. Consider right triangles ΔAYB and ΔXBA:

Hypotenuse AY in ΔAYB equals hypotenuse BX in ΔXBA (Given).

AB is a side common to both triangles.

Therefore, the hypotenuse and one leg of one right triangle equal the hypotenuse and the corresponding leg of the other right triangle. By the RHS (right-angle–hypotenuse–side) congruence criterion, ΔAYB ≅ ΔXBA.

5. From the congruence (CPCT, corresponding parts of congruent triangles):

BY = AX

∠BAY = ∠ABX

BY = AX and ∠BAY = ∠ABX, as required.