Question

Find the measures of

Answer 1

Answer:

Step-by-step explanation:

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

From the picture attached,

m(∠A) = $\frac{1}{2}m(\text{arc BD})$

           = $\frac{1}{2}[m(\text{BC})+m(\text{CD}]$

           = $\frac{1}{2}[55^{\circ}+145^{\circ}]$

           = 100°

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

           = $\frac{1}{2}(360^{\circ}-200^{\circ})$

           = 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°