Question

In Fig. AB is a diameter of a circle, with centre O. If ∠ABC = 70°, ∠CAD = 30° and ∠BAE = 60°, find ∠BAC, ∠ACD and ∠AВЕ.

Answer 1

Given

• AB is a diameter → ∠ACB = 90°
• ∠ABC = 70°
• ∠CAD = 30°
• ∠BAE = 60°

Must find ∠BAC, ∠ACD, and ∠ABE.

Find ∠BAC

In ΔABC, AB is a diameter ⇒ ∠ACB = 90°.

So,

∠BAC + ∠ABC + ∠ACB = 180°

∠BAC + 70° + 90° = 180°

∠BAC = 20°

Find ∠ABE

∠BAE = 60°

∠BAC = 20°

∠CAE = 60° − 20° = 40°

Now look at quadrilateral A–C–E–B on the circle.

Angles ∠CAE and ∠CBE stand on the same arc CE, so:

∠CBE = ∠CAE = 40°

In triangle ABC:

∠ABC = 70°

∠ABE + ∠CBE = 70°

∠ABE = 70° − 40° = 30°

Find ∠ACD

∠CBE = 40°

Angles ∠CBE and ∠ACD subtend the same arc CD. 

∠ACD = ∠CBE = 40°