ABCD is a quadrilateral such that diagonal AC bisects the angles A and C. Prove that AB = AD and CB = CD.

Solution:

Given, ABCD is a quadrilateral

The diagonal AC bisects the angles A and C.

We have to prove that AB = AD and CB = CD.

Considering triangles ADC and ABC,

AC is the bisector of angle A

So, ∠DAC = ∠BAC

AC is the bisector of angle C

So, ∠DCA = ∠BCA

Common side = AC

SAS criterion states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are congruent

By SAS criterion, the triangles ADC and ABC are congruent

By CPCTC,

AD = AB

CD = CB

Therefore, it is proven that AD = AB and CD = CB.

✦ Try This: In figure, the diagonal AC of a quadrilateral ABCD bisects the angle A and C. Prove that AB=AD and CB=CD.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 7

NCERT Exemplar Class 9 Maths Exercise 7.4 Problem 17

ABCD is a quadrilateral such that diagonal AC bisects the angles A and C. Prove that AB = AD and CB = CD

Summary:

ABCD is a quadrilateral such that diagonal AC bisects the angles A and C. It is proven that AB = AD and CB = CD by CPCTC

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