Question

quadrilateral abcd is inscribed in a circle. move measures and reasons into the blanks to complete the proof that shows ∠a and ∠c are supplementary.
statement\treason
abcd is inscribed\tgiven
m\\(stackrel{\frown}{bad}\\)=2×m∠a; m\\(stackrel{\frown}{bcd}\\)=2×m∠c\tinscribed angle theorem
__ + __ = 360°\tarc addition postulate
__ + __ = 360°\tsubstitution
m∠a + m∠c = 180°\tdivision
m\\(stackrel{\frown}{bad}\\)\tm\\(stackrel{\frown}{bcd}\\)\tm∠a\t2×m∠a\tm∠c\t2×m∠c
inscribed angle theorem\tarc addition postulate

Explanation:

Step1: Recall arc - related facts

By the Arc Addition Postulate, the sum of the measures of the arcs of a circle is 360°. For a cyclic quadrilateral \(ABCD\) inscribed in a circle, the sum of the measures of arc \(\widehat{BAD}\) and arc \(\widehat{BCD}\) is 360°, so the blanks in the third - row statement are \(\text{m}\widehat{BAD}\) and \(\text{m}\widehat{BCD}\).

Step2: Substitute arc - angle relationships

We know that \(\text{m}\widehat{BAD}=2\times\text{m}\angle A\) and \(\text{m}\widehat{BCD}=2\times\text{m}\angle C\). Substituting these into the equation \(\text{m}\widehat{BAD}+\text{m}\widehat{BCD} = 360^{\circ}\), we get \(2\times\text{m}\angle A+2\times\text{m}\angle C=360^{\circ}\). So the blanks in the fourth - row statement are \(2\times\text{m}\angle A\) and \(2\times\text{m}\angle C\).

Answer:

Third - row statement: \(\text{m}\widehat{BAD}+\text{m}\widehat{BCD}=360^{\circ}\)
Fourth - row statement: \(2\times\text{m}\angle A + 2\times\text{m}\angle C=360^{\circ}\)