Question

ABCD is a parallelogram. A circle passes through A,B and C. The circle intersects the line CD produced at E. Prove that AE = AD. [3 MARKS]

Answer 1

Concept: 1 Mark
Application: 2 Marks

Given:ABCD is a parallelogram. A circle through A, B and C passes through E (produced CD).

To prove : AE = AD

Proof: ABCD Is a parallelogram

$\angle ADC=\angle ABC$ (1)

$\angle ADC+\angle ADE={180}^{\circ }$ (2)

$\angle ABC+\angle AED={180}^{\circ }$ (3)

(ABCE is a cyclic quadrilateral)

From (2) and (3) we have

$\angle ADC+\angle ADE=\angle ABC+\angle AED$

$\angle ADC+\angle ADE=\angle ADC+\angle AED$ [Using (1)]

$\angle ADE=\angle AED$

$AE=AD$