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 reason
abcd is inscribed given
__ = 2×m∠a; = 2×m∠c __
__+ = 360° __
__+__ = 360° substitution
m∠a + m∠c = 180° division
m\\(widehat{bad}\\) m\\(widehat{bcd}\\) m∠a 2×m∠a m∠c 2×m∠c inscribed angle theorem arc addition postulate

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. So, $m\overparen{BCD}=2\times m\angle A$ and $m\overparen{BAD}=2\times m\angle C$ (Inscribed Angle Theorem).

Step2: Use arc - addition postulate

The sum of the measures of the arcs of a circle is $360^{\circ}$. So, $m\overparen{BAD}+m\overparen{BCD}=360^{\circ}$ (Arc Addition Postulate).

Step3: Substitute

Substitute $m\overparen{BCD}=2\times m\angle A$ and $m\overparen{BAD}=2\times m\angle C$ into the equation $m\overparen{BAD}+m\overparen{BCD}=360^{\circ}$, we get $2\times m\angle C + 2\times m\angle A=360^{\circ}$.

Step4: Divide by 2

Divide both sides of the equation $2\times m\angle C + 2\times m\angle A=360^{\circ}$ by 2. So, $m\angle A + m\angle C=180^{\circ}$.

Answer:

Statement	Reason
$m\overparen{BAD}+m\overparen{BCD}=360^{\circ}$	Arc Addition Postulate
$2\times m\angle C+2\times m\angle A = 360^{\circ}$	Substitution
$m\angle A + m\angle C=180^{\circ}$	Division