Question

ΔABC 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.​

Answer 1

As given

1. ∆ABC is an isosceles ∆

2. AB = AC

3. AD = AB

To Prove: ∠BCD is a right angle.

Proof:

In ΔABC,

AB = AC (........................... Given 2. )

⇒ ∠ACB = ∠ABC (Angles opposite to the equal sides are equal.)

In ΔACD,

AD = AB (........................... Given 3. )

⇒ ∠ADC = ∠ACD (Angles opposite to the equal sides are equal.)

Now,

In ΔABC,

∠CAB + ∠ACB + ∠ABC = 180°

⇒ ∠CAB + 2∠ACB = 180°

⇒ ∠CAB = 180° – 2∠ACB .............................. a)

Similarly in ΔADC,

∠CAD = 180° – 2∠ACD ...................................b)

∠CAB + ∠CAD = 180° (BD is a straight line.)

Adding a & b

∠CAB + ∠CAD = 180° – 2∠ACB + 180° – 2∠ACD

⇒ 180° = 360° – 2∠ACB – 2∠ACD

⇒ 2∠ACB + 2∠ACD= 360-180

⇒ 2(∠ACB + ∠ACD) = 180°

⇒ ∠BCD = 90°

Hence angle BCD is a right angle

Answer 2

Hopeit helps you

♡Thank you♡