Question

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

Answer 1

Hello !!✋

Here is your answer.

Given , ∆ABC is an isosceles triangle in which AB = AC.Side BA is produced to D such that AD = AB.

Prof that, < BCD is a right angle.

prof:-

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°.° ABC is a isosceles triangle

.°. < ABC = < ACB -------(1)

°.° AB = AC and AD = AB

.°. AC = AD

.°.In ∆ACD

< CDA = < ACD ( angle opposite to the equal sides are aqual )

=> < CDB = < ACD ----------(2)

From (1) and (2) we get

< ABC + < CDB = < ACB + < ACD

=> < ABC + < CDB = < BCD --------(3)

In ∆BCD,

< BCD + < DBC + < CDB = 180°

=> < BCD + < ABC + < CDB = 180°

=> < BCD + < BCD = 180° [ using (3) ]

=> 2< BCD = 180°

=> < BCD = 180°/ 2 = 90°

.°. < BCD is a right angle.

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Answer 2

construct a triangle an isosceles triangle ABC

extend A to the opposite direction rather than of  b in a  line. draw a line from C to join the extended line drawn from A and name it as point D such that AD=AC.

AB=AC ..... given

AC=AD .......consruction

therefore AB = AD

in triangle ABC and ACD

AB=AC AND AC=AD

ANGLE ABC = ANGLE ACB ..... ISOSCELES TRIANGLE THEOREM...1

AND ANGLE ACD= ANGLE ADC ....ISOSCELES TRIANGLE THEOREM

THEREFORE ANGLE  .....2

IN TRIANGLE BCD

BCD+BDC+DBC=180.... SUM OF MEASURES OF ALL ANGLES OF A TRIANGLE IS 180.

BCD+ACD+ACB=180

BCD+BCD=180... ANGLE ADDITION PROPERTY

2BCD=180

BCD=180/2

THEREFORE BCD=90 IS HENCE PROVED