Question

Find the value of x in the following figure, where O is the centre of the circle: 

Answer 1

Given:

∠BAD = 40°

∠ABD = 35°

We must find ∠ADC = x.

We must find ∠ADC = x.

Observe that A, B, C, and D are all points on the circle.

Step 1: Use triangle ABD

Triangle ABD is inside the circle.

∠BAD = 40°
∠ABD = 35°

Find ∠ADB (the angle at D inside triangle ABD):

Sum of angles of a triangle:

∠ADB = 180° − (40° + 35°)
∠ADB = 180° − 75°
∠ADB = 105°

So the angle at D between DA and DB is:

∠ADB = 105°.

Step 2: ∠ADB and ∠ACD subtend the same arc AB

Look at the circle:

∠ADB is an inscribed angle that subtends arc AB.

∠ACD (or the angle x region) is also an inscribed angle that subtends the same arc AB.

Therefore:

x = ∠ADB

x = 105°