Practicing Success
In the given figure, AE is the diameter of the circle. Find $\angle ABC+\angle CDE$: |
360° 180° 540° 270° |
270° |
In △OAB, OA = OB [Radius of same circle] ∠1 = ∠2. In △OBC, OB = OC [Radius of same circle] ∠3 = ∠4. In △OCD, OC = OD [Radius of same circle] ∠5 = ∠6. In △ODE, OD = OE [Radius of same circle] ∠7 = ∠8. In △OAB, ⇒ ∠1 + ∠2 + ∠a = 180° [By angle sum property of triangle] ……….(1) In △OBC, ⇒ ∠3 + ∠4 + ∠b = 180° [By angle sum property of triangle] ……….(2) In △OCD, ⇒ ∠5 + ∠6 + ∠c = 180° [By angle sum property of triangle] ……….(3) In △ODE, ⇒ ∠7 + ∠8 + ∠d = 180° [By angle sum property of triangle] ………(4) Adding (1), (2), (3) and (4) we get, ⇒ ∠1 + ∠2 + ∠a + ∠3 + ∠4 + ∠b + ∠5 + ∠6 + ∠c + ∠7 + ∠8 + ∠d + = 180° + 180° + 180° + 180° ⇒ ∠2 + ∠2 + ∠a + ∠3 + ∠3 + ∠b + ∠6 + ∠6 + ∠c + ∠7 + ∠7 + ∠d + = 720° ⇒ 2∠2 + 2∠3 + 2∠6 + 2∠7 + ∠a + ∠b + ∠c + ∠d = 720° ⇒ 2[∠2 + ∠3] + 2[∠6 + ∠7] + 180° = 720° [As ∠a + ∠b + ∠c + ∠d = 180°] ⇒ 2∠ABC + 2∠CDE = 540° ⇒ ∠ABC + ∠CDE = 270°. Hence, ∠ABC + ∠CDE = 270°. |