ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.

Solution:

Given, ABCD is a quadrilateral in which AB = BC and AD = CD.

We have to show that BD bisects both the angles ABC and ADC.

Given, AB = BC

We know that the angles opposite to equal sides are equal.

So, ∠BCA = ∠BAC

∠2 = ∠1 ------------------ (1)

Given, AD = CD

We know that the angles opposite to equal sides are equal.

∠DCA = ∠CAD

So, ∠4 = ∠3 ----------- (2)

Adding (1) and (2),

∠2 + ∠4 = ∠1 + ∠3

So, ∠BCD = ∠BAD ------------ (3)

Considering triangles BAD and BCD,

Given, AB = BC

From (3), ∠BAD = ∠BCD

Given, AD = CD

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 BAD and BCD are congruent.

By CPCTC,

∠ABD = ∠CBD

∠ADB = ∠CDB

This implies that BD bisects the angles ABC and ADC

Therefore, BD bisects the angles ABC and ADC.

✦ Try This: In figure, D is the mid-point of side BC and AE⊥BC. If BC=a, AC=b, AB=c, ED=x, AD=p and AE=h, prove that b² = p² + ax + a²/4

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

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NCERT Exemplar Class 9 Maths Exercise 7.4 Problem 5

ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC

Summary:

ABCD is a quadrilateral in which AB = BC and AD = CD. It is shown that BD bisects both the angles ABC and ADC by CPCT which states that when two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements

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