Question

Find the measures of

Answer 1

Explanation:

Measure of an inscribed angle intercepted by an arc is half of the measure of the arc.

From the picture attached,

m(∠A) =
`(1)/(2)m(\text{arc BD})`

=
`(1)/(2)[m(\text{BC})+m(\text{CD}]`

=
`(1)/(2)[55^(\circ)+145^(\circ)]`

= 100°

m(∠C) =
`(1)/(2)[(360^(\circ))-m(\text{arc BCD})]`

=
`(1)/(2)(360^(\circ)-200^(\circ))`

= 80°

m(∠B) + m(∠D) = 180° [ABCD is cyclic quadrilateral]

115° + m(∠D) = 180°

m(∠D) = 65°

m(arc AC) = 2[m(∠D)]

m(arc AB) + m(arc BC) = 2(65°) [Since, m(arc AC) = m(arc AB) + m(arc BC)]

m(arc AB) + 55° = 130°

m(arc AB) = 75°

m(arc ADC) = 2(m∠B)

m(arc AD) + m(arc DC) = 2(115°)

m(arc AD) + 145° = 230°

m(arc AD) = 85°