Question

given: a circle with inscribed quadrilateral abcd
prove: ∠a and ∠c are supplementary.

1. let ( moverarc{dcb} = a^circ )

then ( moverarc{dab} = 360 - a ).

1. by the inscribed angle theorem, ( mangle a = \frac{a}{2} ).
2. also by the inscribed angle theorem, ( mangle c = \frac{360 - a}{2} ).
3. therefore, ( mangle a + mangle c = \frac{a}{2} + \frac{360 - a}{2} ),

which simplifies to
(options: 90, 180, 360)
check

Explanation:

Step1: Simplify the expression

We have \( m\angle A + m\angle C=\frac{a}{2}+\frac{360 - a}{2} \). First, combine the fractions: \(\frac{a+(360 - a)}{2}\).

Step2: Simplify the numerator

Simplify the numerator \(a+(360 - a)\). The \(a\) and \(-a\) cancel out, leaving \(360\). So we have \(\frac{360}{2}\).

Step3: Calculate the result

\(\frac{360}{2} = 180\).

Answer:

180