Question

Abcd is a parallelogram the circle through a b and c intersects cd produced if necessary at e prove that a is equals to 80

Answer 1

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ajr11

Secondary SchoolMath 5+3 pts

ABCD IS A PARALLELOGRAM.THE CIRCLE THROUGH A,B,C INTERSECT CD AT E.PROVE THAT AD=AE

Report by Vinayakgarg 28.02.2017

Answers

joe30

Joe30 Expert

Given: ABCD is a parallelogram. A circle through A, B, C intersects CD produced at E.

To prove: AE = AD

Proof:

∠AED + ∠ABC = 180° ... (1) (Sum of opposite angles of cyclic quadrilaterals is 180°)

∠ADE + ∠ADC = 180° ... (2) (linear pair)

∠ABC = ∠ADC ... (3) (opposite angles of parallelogram are equal)

From (1) and (2)

∠AED + ∠ABC = ∠ADE + ∠ADC

⇒∠AED = ∠ADE (using (3))

In ∆AED,

∠AED = ∠ADE

⇒AD = AE (equal sides have equal angles opposite to them)

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