Mathematics
In the given figure, AB = AC = CD and ∠ADC = 38°. Calculate :
(i) Angle ABC
(ii) Angle BEC
Circles
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Answer
Join BE.
(i) AC = CD
∠DAC = ∠ADC = 38° [Angles opposite to equal sides are equal]
In △ACD,
⇒ ∠DAC + ∠ADC + ∠ACD = 180°
⇒ 38° + 38° + ∠ACD = 180°
⇒ 76° + ∠ACD = 180°
⇒ ∠ACD = 180° - 76° = 104°
From figure,
⇒ ∠ACB + ∠ACD = 180° [BCD is a straight line]
⇒ ∠ACB + 104° = 180°
⇒ ∠ACB = 180° - 104° = 76°.
Given,
AB = AC
∴ ∠ABC = ∠ACB = 76°. [As angles opposite to equal sides are equal]
Hence, ∠ABC = 76°.
(ii) In △ABC,
⇒ ∠BAC + ∠ACB + ∠ABC = 180° [Angle sum property of triangle]
⇒ ∠BAC + 76° + 76° = 180°
⇒ ∠BAC + 152° = 180°
⇒ ∠BAC = 180° - 152° = 28°.
We know that,
Angles in same segment are equal.
⇒ ∠BEC = ∠BAC = 28°.
Hence, ∠BEC = 28°.
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