Question

proving the third corollary to the inscribed angle theorem
given: a circle with inscribed quadrilateral abcd
prove: ∠a and ∠c are supplementary.

1. let m∠dcb = a°

then m∠dab =
options: 90 - a, 180 - a, 360 - a

Explanation:

Step1: Recall the inscribed - angle theorem related to arcs

The measure of an inscribed angle is half the measure of its intercepted arc. Let \(\angle DCB\) intercept arc \(DAB\) and \(\angle DAB\) intercept arc \(DCB\). The sum of the measures of arcs \(DAB\) and \(DCB\) is \(360^{\circ}\) (since a full - circle has a measure of \(360^{\circ}\)).
If \(m\angle DCB=a^{\circ}\), then the measure of arc \(DAB = 2a^{\circ}\) (by the inscribed - angle theorem: \(m\angle=\frac{1}{2}\text{ measure of intercepted arc}\)). Let \(m\angle DAB = x\). Then the measure of arc \(DCB=2x\).

Step2: Use the property of the sum of arcs in a circle

We know that \(2a + 2x=360\). Divide both sides of the equation \(2a + 2x = 360\) by \(2\): \(a + x=180\).

Answer:

\(180 - a\)