Question

# Find the value of ∠ABD if AB = AC and DB = DC

- 20°
- 45°
- 30°
- 15°

Hint:

### From figure, in ΔDBC, DB = DC ⇒ ∠DBC = ∠DCB (Angles opposite to equal sides are equal)

Also, ∠DBC + ∠DCB + ∠BDC = 180° (Angle sum property of triangle)

In ΔABC, AB = AC ⇒ ∠ABC = ∠ACB (Angles opposite to equal sides are equal)

Also, ∠ABC + ∠ACB + ∠BAC = 180° (Angle sum property of triangle)

∠ABD = ∠ABC – ∠DBC

## The correct answer is: 15°

From figure, in ΔDBC, DB = DC ⇒ ∠DBC = ∠DCB (Angles opposite to equal sides are equal)

Also, ∠DBC + ∠DCB + ∠BDC = 180° (Angle sum property of triangle)

⇒ 80° + 2∠DBC = 180°

⇒ ∠DBC = 50° ————– (i)

In ΔABC, AB = AC ⇒ ∠ABC = ∠ACB (Angles opposite to equal sides are equal)

Also, ∠ABC + ∠ACB + ∠BAC = 180° (Angle sum property of triangle)

⇒ 50° + 2∠ABC = 180°

⇒ ∠ABC = 65° ————– (i)

Now, ∠ABD = ∠ABC – ∠DBC = 65° – 50° = 15°.

Also, ∠DBC + ∠DCB + ∠BDC = 180° (Angle sum property of triangle)

⇒ 80° + 2∠DBC = 180°

⇒ ∠DBC = 50° ————– (i)

In ΔABC, AB = AC ⇒ ∠ABC = ∠ACB (Angles opposite to equal sides are equal)

Also, ∠ABC + ∠ACB + ∠BAC = 180° (Angle sum property of triangle)

⇒ 50° + 2∠ABC = 180°

⇒ ∠ABC = 65° ————– (i)

Now, ∠ABD = ∠ABC – ∠DBC = 65° – 50° = 15°.

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