ABC is an isosceles triangie with AB=AC,P and Q are points on AB and AC respectively such that AP=AQ

i) Is Triangle ABQ Congruent Triangle ACP

ii) Is Triangle BPC CongruentTriangle CQB ?

Give reasons in support of your answer

ΔABC is an isosceles triangle with AB = AC. P and Q are point on AB and AC respectively such that AP = AQ.

In ΔABC,

AB = AC (Given)

∴ ∠C = ∠B (Equal sides have equal angles opposite to them)

AB = AC (Given)

AP = AQ (Given)

∴ AB – AP = AC – AQ

⇒ BP = CQ

In ΔABQ and ΔACP,

AQ = AP (Given)

∠A = ∠A (Common)

AB = AC (Given)

∴ ΔABQ ΔACP (SAS congruence criterion)

In ΔBPC and ΔCQB,

BP = CQ (Proved)

∠B = ∠C (Proved)

BC = BC (Common)

∴ ΔBPC ΔCQB (SAS congruence criterion)

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