ABD is a triangle such that ∠ADB = 20° and C is a point on BD such that AB = AC and CD = CA. The measure of ∠ABC :

1. 40°

2. 50°

3. 55°

4. 60°

In △ADC,

CD = CA

∠ADC = ∠CAD = 20° (Angles opposite to equal sides in a triangle are equal)

In △ACD,

By angle sum property of triangle,

⇒ ∠ACD + ∠ADC + ∠CAD = 180°

⇒ ∠ACD + 20° + 20° = 180°

⇒ ∠ACD + 40° = 180°

⇒ ∠ACD = 180° - 40°

⇒ ∠ACD = 140°

From figure,

∠ACB + ∠ACD = 180° (Linear pair)

⇒ ∠ACB + 140° = 180°

⇒ ∠ACB = 180° - 140°

⇒ ∠ACB = 40°

In △ABC,

AB = AC

∠ABC = ∠ACB = 40° (Angles opposite to equal sides in a triangle are equal)

Hence, option 1 is the correct option.