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Question

ABC and DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC as in figure.If AD is extended to intersect BC at P,Show that AP bisects A as well as D
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Solution

Given:ABC is isosceles,

AB=AC .....(1)

Also,DBC is isosceles,

DB=DC .....(2)

Proof:In ABD and DBC, we have

AB=AC from (1)

BD=DC from (2)

AD=AD (common)

ABDACD by SSS congruence rule.

So,BAP=PAC by CPCT .........(3)

In ABP and ACP,

AB=AC(given)

BAP=PAC by CPCT (from(3))

AP=AP(common)

ABPACP by SAS congruence rule.

BP=CP ......(4)

In BDP and CDP, we have

BD=CD(Given)

BP=CP(from (4))

DP=DP(common)

So,BDPCDP by SSS congruence rule

BDP=PDC by CPCT

Thus, AP bisects D

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