The missing information in the paragraph proof is the Inscribed Angle Theorem.
Let's break down the proof step by step:
1. Given Data:
- Let the measure of the arc [tex]\( \overline{BCD} = a^\circ \)[/tex].
- Since [tex]\( \overline{BCD} \)[/tex] and [tex]\( \overline{BAD} \)[/tex] form a complete circle, the measure of the arc [tex]\( \overline{BAD} \)[/tex] is [tex]\( 360^\circ - a^\circ \)[/tex].
2. Applying the Inscribed Angle Theorem:
- According to the Inscribed Angle Theorem, the measure of an inscribed angle is half of the measure of its intercepted arc.
- Therefore, the measure of angle [tex]\( \angle A \)[/tex] is [tex]\( \frac{a}{2} \)[/tex] degrees, since it intercepts the arc [tex]\( \overline{BCD} \)[/tex].
- Similarly, the measure of angle [tex]\( \angle C \)[/tex] is [tex]\( \frac{360 - a}{2} \)[/tex] degrees, since it intercepts the arc [tex]\( \overline{BAD} \)[/tex].
3. Summing Angles A and C:
- The sum of the measures of angles [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] is:
[tex]\[
m\angle A + m\angle C = \frac{a}{2} + \frac{360 - a}{2}
\][/tex]
- Simplifying this, we have:
[tex]\[
\frac{a}{2} + \frac{360 - a}{2} = \frac{a + 360 - a}{2} = \frac{360}{2} = 180^\circ
\][/tex]
- Therefore, angles [tex]\( \angle A \)[/tex] and [tex]\( \angle C \)[/tex] are supplementary because their measures add up to [tex]\( 180^\circ \)[/tex].
4. Angles in a Quadrilateral:
- The sum of the measures 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]
- We know [tex]\( m\angle A + m\angle C = 180^\circ \)[/tex], so:
[tex]\[
180^\circ + m\angle B + m\angle D = 360^\circ
\][/tex]
- By subtracting 180° from both sides, we get:
[tex]\[
m\angle B + m\angle D = 180^\circ
\][/tex]
- Therefore, angles [tex]\( \angle B \)[/tex] and [tex]\( \angle D \)[/tex] are also supplementary because their measures add up to [tex]\( 180^\circ \)[/tex].
To conclude, the missing information in the proof is the Inscribed Angle Theorem, which states that the measure of an inscribed angle is half the measure of its intercepted arc.
