A circle is circumscribed on a quadrilateral ABCD. If ∠DAB = 100°, ∠ADB = 35° and ∠CDB = 40°, then find the measure of ∠DBC.

60°

We know that,

In the case of a cyclic quadrilateral, the sum of opposite angles of the quadrilateral is 180°.

The sum of all internal angles of a triangle is 180°.

We have,

∠DAB = 100°,

∠ADB = 35° and ∠CDB = 40°

Considering ΔDAB, 

∠DAB + ∠ADB + ∠DBA = 180°

= 100° + 35° + ∠DBA = 180°

= ∠DBA = 180° - 135° = 45°

= ∠ABD = 45°

Because the circle is circumscribed on a quadrilateral ABCD, ABCD is a cyclic quadrilateral.

So, ∠ADC + ∠ABC = 180°

= (∠ADB + ∠CDB) + (∠ABD + ∠DBC) = 180°

= 35° + 40° + 45° + ∠DBC = 180°

= ∠DBC = 180° - 120° 

= ∠DBC = 60°