To find the missing information in the paragraph proof, let's take a closer look at the logic and steps presented:
1. Measure of Arcs:
- Given: [tex]$\overline{B C D} = a^{\circ}$[/tex]
- Since [tex]$\overline{B C D}$[/tex] and [tex]$\overline{B A D}$[/tex] form a circle, the measure of [tex]$\overline{B A D}$[/tex] is [tex]\(360 - a^{\circ}\)[/tex].
2. Angles and Theorem:
- Due to the missing theorem, [tex]\(m \angle A = \frac{a}{2}\)[/tex] degrees and [tex]\(m \angle C = \frac{360 - a}{2}\)[/tex] degrees.
Here, we know this must be related to a theorem connecting the arc measures to their respective angles inside the circle. The appropriate theorem is the inscribed angle theorem, which states that an inscribed angle is half the measure of its intercepted arc.
3. Sum of Angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex]:
- The sum of the measures of angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] is [tex]\(\left(\frac{a}{2} + \frac{360 - a}{2}\right)\)[/tex] degrees:
[tex]\[
\frac{a}{2} + \frac{360 - a}{2} = \frac{a + 360 - a}{2} = \frac{360}{2} = 180^{\circ}
\][/tex]
- Hence, angles [tex]\(A\)[/tex] and [tex]\(C\)[/tex] are supplementary, as their measures add up to [tex]\(180^{\circ}\)[/tex].
4. Supplementary Angles [tex]\(B\)[/tex] and [tex]\(D\)[/tex]:
- Since the sum of the measures of the 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]
- Substituting the previous result:
[tex]\[
180^{\circ} + m \angle B + m \angle D = 360^{\circ} \rightarrow m \angle B + m \angle D = 180^{\circ}
\][/tex]
Tracing back through the logic, the theorem that ties the arc measure to the angle within the circle is crucial, and it is the inscribed angle theorem. Thus, the missing information in the paragraph proof is indeed the "inscribed angle" theorem.
