The missing information in the paragraph proof is the inscribed angle theorem.
Here’s a detailed explanation:
1. We start with the measure of arc [tex]\(\overline{BCD}\)[/tex], which is given as [tex]\(a^\circ\)[/tex].
2. Since the arcs [tex]\(\overline{BCD}\)[/tex] and [tex]\(\overline{BAD}\)[/tex] together cover the entire circle, the measure of [tex]\(\overline{BAD}\)[/tex] is [tex]\(360^\circ - a^\circ\)[/tex].
3. According to the inscribed angle theorem:
- The measure of an inscribed angle is half the measure of its intercepted arc.
- Therefore, [tex]\(m \angle A\)[/tex], which intercepts the arc [tex]\(\overline{BCD}\)[/tex], is [tex]\(\frac{a}{2}^\circ\)[/tex].
- Similarly, [tex]\(m \angle C\)[/tex], which intercepts the arc [tex]\(\overline{BAD}\)[/tex], is [tex]\(\frac{360 - a}{2}^\circ\)[/tex].
4. To find the sum of the measures of angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex]:
- Add [tex]\(m \angle A\)[/tex] and [tex]\(m \angle C\)[/tex]:
[tex]\[
m \angle A + m \angle C = \frac{a}{2} + \frac{360 - a}{2}
\][/tex]
- Simplify the expression:
[tex]\[
\frac{a + 360 - a}{2} = \frac{360}{2} = 180^\circ
\][/tex]
- Therefore, [tex]\(\angle A\)[/tex] and [tex]\(\angle C\)[/tex] are supplementary since their measures add up to [tex]\(180^\circ\)[/tex].
5. In a quadrilateral inscribed in a circle (a cyclic quadrilateral), opposite angles are supplementary. Given that [tex]\(\angle A\)[/tex] and [tex]\(\angle C\)[/tex] are supplementary:
- The sum of all angles in a quadrilateral is [tex]\(360^\circ\)[/tex]:
[tex]\[
m \angle A + m \angle C + m \angle B + m \angle D = 360^\circ
\][/tex]
- Substitute the supplementary angles:
[tex]\[
180^\circ + m \angle B + m \angle D = 360^\circ
\][/tex]
- This simplifies to:
[tex]\[
m \angle B + m \angle D = 180^\circ
\][/tex]
- Hence, [tex]\(\angle B\)[/tex] and [tex]\(\angle D\)[/tex] are also supplementary.
Thus, the inscribed angle theorem is the crucial piece of information that was missing to complete the proof.
