Question

AABC is an isosceles triangle in which AB AC
Side BA is produced to D such that AD = AB
(see Fig. 7.34). Show that Z BCD is a right angle.​

Answer 1

Given :

• ∆ ABC in which AB = AC

• Side BA is Produced to D such that AB = AD

To Prove :

• Angle BCD is a right angle.

Proof :

In ∆ ABC we've

• AB = AC

Angles opposite to the equal sides are equal

→ ∠ ACB = ∠ ABC .... i)

Now,

• AB = AD (given)

$\therefore$ AD = AC

In ∆ ADC we've

• AD = AC

Angles opposite to the equal sides are equal

→ ∠ ACD = ∠ ADC .... ii)

Adding Equation i) and ii)

→ ∠ ACB + ∠ ABC + ∠ ACD + ∠ ADC

∠ ADC is equal to the ∠ ABC

→ ∠ BCD = ∠ ABC + ∠ BDC

Adding ∠ BCD on both the sides

→ ∠ BCD + ∠ BCD = ∠ ABC + ∠ BDC + ∠ BCD

Sum of the angles of triangle is 180°

→ 2 ∠ BCD = 180°

→ ∠ BCD = 90°

Hence, proved!

Answer 2

In △ ABC,

→ AB = AC (given)

.°. ∠ACB = ∠ABC

In △ ACD,

→ AD = AB (given)

.°. ∠ADC = ∠ABC

In △ BCD,

→ ∠ABC + ∠BCD + ∠ADC = 180°

→ ∠ABC + ∠ABC + ∠ADC +∠ADC = 180°

→ 2(∠ABC + ∠ADC) = 180°

→ 2∠BCD = 180°

→ ∠BDC = 180° ÷ 2

→ ∠BDC = 90°

Hence,Proved...