Question

given: a circle with inscribed quadrilateral abcd. prove: ∠a and ∠c are supplementary. 1. let m∠dcb = a°. then m∠dab = 360 - a. 2. by the inscribed - angle theorem, m∠a = a/2. 3. also by the inscribed - angle theorem, m∠c=(360 - a)/2. 4. therefore, m∠a + m∠c = a/2+(360 - a)/2, which simplifies to 180. 5. therefore, m∠a and m∠c are by the definition of supplementary an.

Explanation:

Step1: Recall angle - sum property of a circle

The sum of the measures of the arcs of a circle is 360°. If we let the measure of arc DCB be \(a^{\circ}\), then the measure of arc DAB is \(360 - a^{\circ}\).

Step2: Apply inscribed - angle theorem for \(\angle A\)

The measure of an inscribed angle is half the measure of its intercepted arc. For \(\angle A\), which intercepts arc DCB, \(m\angle A=\frac{a}{2}\) by the inscribed - angle theorem.

Step3: Apply inscribed - angle theorem for \(\angle C\)

For \(\angle C\), which intercepts arc DAB, \(m\angle C = \frac{360 - a}{2}\) by the inscribed - angle theorem.

Step4: Calculate \(m\angle A+m\angle C\)

\[

$$\begin{align*} m\angle A+m\angle C&=\frac{a}{2}+\frac{360 - a}{2}\\ &=\frac{a + 360 - a}{2}\\ &=\frac{360}{2}\\ &=180 \end{align*}$$

\]

Step5: Determine the relationship between \(\angle A\) and \(\angle C\)

Since \(m\angle A + m\angle C=180^{\circ}\), by the definition of supplementary angles (two angles whose sum is 180°), \(\angle A\) and \(\angle C\) are supplementary.

Answer:

supplementary