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
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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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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