Question

In the given figure, AE is the diameter of the circle. Write down the numerical value of ∠ABC + ∠CDE. Give reasons for your answer.

Answer 1

Join OA, OB, OC, OD.

In ΔOAB,

OA = OB    ...(Radii of the same circle)

∠1 = ∠2

Similarly we can prove that

∠3 = ∠4,

∠5 = ∠6,

∠7 = ∠8

In ΔOAB,

∠1 + ∠2 + ∠a = 180°    ...(Angles of a triangle)

Similarly ∠3 + ∠4 + ∠b = 180°

∠5 + ∠6 + ∠c = 180°

∠7 + ∠8 + ∠d = 180°

Adding we get

∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 + ∠7 + ∠8 + ∠a + ∠b + ∠c + ∠d = 4 × 180° = 720°

${\displaystyle \Rightarrow }$ ∠2 + ∠2 + ∠3 + ∠3 + ∠6 + ∠6 + ∠ 7 + ∠7 + ∠a + ∠b + ∠c + ∠d = 720°

${\displaystyle \Rightarrow }$ 2∠2 + 2∠3 + 2∠6 + 2∠7 + ∠a + ∠b + ∠c + ∠d = 720°

${\displaystyle \Rightarrow }$ 2[∠2 + ∠3] + 2[∠6 + ∠7| + 180° = 720°  ...(∠a + ∠b + ∠c + ∠d = 180°)

${\displaystyle \Rightarrow }$ 2∠ABC + 2∠CDE = 720° – 180° = 540°

${\displaystyle \Rightarrow }$ 2(∠ABC + ∠CDE) = 540°

${\displaystyle \Rightarrow }$ ∠ABC + ∠CDE = 270°