Question

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

Answer 1

Error in the question :- "such that AD = AB", not "show that AB = AD"

Correct question :- Triangle ABC is an isosceles triangle in which AB = AC . side BA is produced to D such that AD = AB. Show that angle BCD is a right angle.

Solution :-

Since, AB = AC

$\angle ABC = \angle ACB$

Also, AB = AD

→ AC = AD (since, AB = AC)

$Again, \angle ACD = \angle ADC$

Now, by exterior angle property,

$\angle DAC = \angle ABC + \angle ACB \\ \\ \angle DAC = 2\angle ACB \\ (since \ \angle ABC = \angle ACB)$

Now, by angle sum property of triangle,

$\angle DAC + \angle ADC + \angle ACD = 180^{\circ} \\ \\ 2\angle ACB + 2\angle ACD = 180^{\circ} \\ (since, \angle DAC = 2\angle ACB) \\ (and \angle ADC = \angle ACD)$

$2(\angle ACB + \angle ACD) = 180^{\circ} \\ \\ \implies \angle ACB + \angle ACD = 90^{\circ} \\ \\ Now, \angle ACB + \angle ACD = \angle BCD \\ \\ \implies, \angle BCD = 90^{\circ}$

Hence, angle BCD is a right angle.

Answer 2

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