Question

6. ABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB. Show that BCD is a right angle (See figure).


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Answer 1

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△ABC is an isosceles triangle in which AB=AC. Sides BA is produced to D such that AD=AB. Show that ∠BCD is a right angle.

463839

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Solution To Question ID 463839

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Answer

In △ABC, we have

AB=AC ∣ given

∠ACB=∠ABC ... (1) ∣ Since angles opp. to equal sides are equal

Now, AB=AD ∣ Given

∴AD=AC ∣ Since AB=AC

Thus , in △ADC, we have

AD=AC

⇒∠ACD=∠ADC ... (2) ∣ Since angles opp. to equal sides are equal

Adding (1) and (2) , we get

∠ACB+∠ACD=∠ABC+∠ADC

⇒∠BCD=∠ABC+∠BDC ∣ Since∠ADC=∠BDC

⇒∠BCD+∠BCD=∠ABC+∠BDC+∠BCD ∣ Adding ∠BCD on both sides

⇒2∠BCD=180

∣ Angle sum property

⇒∠BCD=90

Hence, ∠BCD is a right angle.

Answer 2

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Step-by-step explanation:

In A ABC, we have

AB=AC

AB=AC (given)

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

Now,

AB=AD (Given)

AD=AC (Since AB=AC)

Thus, in ∆ABC, we have

AD=AC

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

Adding (1) and (2), we get

∠ACB+∠ACD=∠ABC+∠ADC

→∠BCD=∠ABC+∠BDC (Since∠ADC=∠BDC)

→∠BCD+∠BCD=∠ABC+∠BDC+∠BCD

Adding ∠BCD on both sides,

→ 2∠BCD=180

Angle sum property

→ ∠BCD=90

Hence, ∠BCD is a right angle.