Question

`square`ABCD is a cyclic quadrilateral where side AB ≅ side BC, ∠ADC = 110°, AC is the diagonal, then:

1. Draw the figure using given information
2. Find measure of ∠ABC
3. Find measure of ∠BAC
4. Find measure of (arc ABC).

Answer 1

a. 

b. m∠ADC + m∠ABC = 180°    ...[Cyclic quadrilateral theorem]

∴ 110° + m∠ABC = 180°

∴ m∠ABC = 180° − 110°

∴ m∠ABC = 70°

c. ΔABC is an isosceles triangle. (AB = BC)

∴ m∠ABC = m∠BCA    ...[Isosceles triangle theorem]

∴ m∠BAC + m∠ACB + m∠ABC = 180°    ...[Angle sum property of a triangle]

∴ m∠BAC + m∠BAC + 70° = 180°

∴ 2m∠BAC = 180° − 70°

∴ 2m∠BAC = 110°

∴ m∠BAC = 55°

d. m∠ADC = `1/2` m(arc ABC)    ...[Inscribed angle theorem]

∴ 110° = `1/2` m(arc ABC)

∴ m(arc ABC) = 220°