Mathematics
In the figure (i) given below, chord ED is parallel to the diameter AC of the circle. Given ∠CBE = 65°, calculate ∠DEC.
Circles
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Answer
Consider △AEC and △EBC,
∠EAC = ∠EBC = 65° (∵ angles in same segment are equal.)
In △AEC,
∠AEC = 90° (∵ angle in semicircle is 90°.)
We know that sum of angles of a triangle is 180°.
⇒ ∠AEC + ∠EAC + ∠ACE = 180°.
⇒ 90° + 65° + ∠ACE = 180°
⇒ ∠ACE + 155° = 180°
⇒ ∠ACE = 180° - 155°
⇒ ∠ACE = 25°.
∠DEC = ∠ACE (∵ Alternate angles)
∴∠DEC = 25°
Hence, the value of ∠DEC = 25°.
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