In ΔABC, AB = AC, O is a point on BC such that BO = CO and OD is perpendicular to AB and OE is perpendicular to AC. If ∠BOD = 60°, then measure of ∠AOE is:
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Correct answer: C
Explanation
The correct answer is Option 3. i.e. 30°.
In ΔABC, AB = AC, O is a point on BC such that BO = CO and OD is perpendicular to AB and OE is perpendicular to AC.
∠BOD = 60°
In ΔABC, ∠A + ∠B + ∠C = 180°
In ΔABC, ∠OBD is a right-angled triangle at D and ∠BOD = 60°
So, ∠OBD = 90° - 60° = 30°
Since in ΔABC, AB = AC, ∠OBD = ∠ACB = ∠ABC = 30°
∠BAC = 180° - 2 × 30° = 120
Since BO = CO, AO is the median
Then, ∠BAO = ∠OAC = ∠OAE = 120°/2 = 60°
Since OE is perpendicular to AC, ΔOEA is a right-angled triangle at E.
Therefore, ∠AOE = 180° - 90° - 60° = 30°