Question

triangle ABC is an isosceles triangle in which AB= AC side BA is produced to D such that AD = AB show that AD =AB . Show that angle BCA is a right angle

Answer 1

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ANS.( GIVEN ) = AB = AC
= AD = AB
= AC = AB = AD

(TO PROOF) = ∠BCD = 90'

( PROOF) = ANGLE ABC
= AB = AC
=∠ACB = ABC

Angle opposites to equal side are equal.

In ANGLE ACB.....

AC = AD
∠ADC = ACD

Angle opposites to equal side are equal.

IN ANGLE BCD......

∠ ABC + ∠BCD + ∠BDC =180'

∠ACB + ∠BCD + ∠ACD = 180'

[∠ACB. + ∠ACD] + ∠BCD = 180'

[ ∠BCD] + ∠BCD = 180'

2∠BCD = 180'

∠BCD = 180'/2
∠BCD = 90' Proved.....


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Answer 2

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$\bold\pink{Solution:}$

AB=AC         (Given)

It means that ∠DBC=∠ACB           (In triangle, angles opposite to equal sides are equal)     

Let ∠DBC=∠ACB=x         .......(1)

AC=AD          (Given)

It means that ∠ACD=∠BDC         (In triangle, angles opposite to equal sides are equal)     

Let ∠ACD=∠BDC=y           ......(2)

In ∆BDC, we have

∠BDC+∠BCD+∠DBC=180°     (Angle sum property of triangle)

⇒∠BDC+∠ACB+∠ACD+∠DBC=180°

Putting (1) and (2) in the above equation, we get

y+x+y+x=180°

⇒2x+2y=180°

⇒2(x+y)=180°

⇒(x+y)=180/2=90°

Therefore, ∠BCD=90°

hope, this will help you.☺

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