ΔABC is an isosceles triangle in which AB = AC. Side BA is produced t

1.	ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see the given figure). Show that ∠BCD is a right angle.
Answer» In ΔABC, AB = AC (Given) ∴ ∠ACB = ∠ABC (Angles opposite to equal sides of a triangle are also equal) In ΔACD, AC = AD ∴ ∠ADC = ∠ACD (Angles opposite to equal sides of a triangle are also equal) In ΔBCD, ∠ABC + ∠BCD + ∠ADC = 180º (Angle sum property of a triangle) ∴ ∠ACB + ∠ACB +∠ACD + ∠ACD = 180º ∴ 2(∠ACB + ∠ACD) = 180º ∴ 2(∠BCD) = 180º ∴ ∠BCD = 90º

ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see the given figure). Show that ∠BCD is a right angle.

Answer»

In ΔABC,

AB = AC (Given)

∴ ∠ACB = ∠ABC (Angles opposite to equal sides of a triangle are also equal)

In ΔACD,

AC = AD

∴ ∠ADC = ∠ACD (Angles opposite to equal sides of a triangle are also equal) In ΔBCD,

∠ABC + ∠BCD + ∠ADC = 180º (Angle sum property of a triangle)

∴ ∠ACB + ∠ACB +∠ACD + ∠ACD = 180º

∴ 2(∠ACB + ∠ACD) = 180º

∴ 2(∠BCD) = 180º

∴ ∠BCD = 90º