Question

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the third corollary to the inscribed angle theorem
given: a circle with inscribed quadrilateral abcd
prove: ∠a and ∠c are supplementary.

1. let m∠dcb = a°

then m∠dab = 360 - a

1. by the inscribed angle theorem, m∠a = a/2
2. also by the inscribed angle theorem, m∠c = (360 - a)/2
3. therefore, m∠a + m∠c = a/2+(360 - a)/2, which simplifies to 180
4. therefore, m∠a and m∠c are dropdown with options: complementary, supplementary by the definition of supplementary ang

check

Explanation:

Step1: Set angle measure

Let $m\angle DCB = a^{\circ}$. The arc $DAB$ is the arc that is not the arc $DCB$. Since the sum of the measures of the arcs of a circle is $360^{\circ}$, then $m\overset{\frown}{DAB}=360 - a$.

Step2: Apply inscribed - angle theorem for $\angle A$

The measure of an inscribed angle is half the measure of its intercepted arc. $\angle A$ intercepts arc $DCB$. So, $m\angle A=\frac{a}{2}$ by the inscribed - angle theorem.

Step3: Apply inscribed - angle theorem for $\angle C$

$\angle C$ intercepts arc $DAB$. So, $m\angle C=\frac{360 - a}{2}$ by the inscribed - angle theorem.

Step4: Calculate the sum of $\angle A$ and $\angle C$

$m\angle A + m\angle C=\frac{a}{2}+\frac{360 - a}{2}=\frac{a + 360 - a}{2}=\frac{360}{2}=180$.

Answer:

supplementary