Answer:
Choice B. False. Rather, the opposite angles of a circumscribed quadrilateral are always supplementary with a sum of 180 degrees.
Step-by-step explanation:
Refer to the sketch attached.
Consider the quadrilateral ABCD inscribed in the circle O. Angle [tex]\mathrm{\hat{A}}[/tex] is the angle that line BA and DA inscribe. The green angle at O will be the arc that the two lines intercepts. Let angle A equals [tex]\alpha[/tex] degrees. By the inscribed angle theorem for circles, The green angle will measure [tex]\rm 2\hat{A} = 2\alpha[/tex].
Similarly, the arc that line CD and CB intercept will equal to twice the measure of [tex]\rm \hat{C}[/tex]. The angle of that arc is the red angle at the origin. The value of that arc will equal to [tex]360\textdegree{} - 2\alpha[/tex].
Angle [tex]\hat{C}[/tex] is half the measure of that arc. That is:
[tex]\rm \displaystyle \hat{C} = \frac{1}{2} (360 \textdegree - 2\alpha) = 180 - \alpha[/tex].
Note that [tex]\rm \hat{A} + \hat{C} = \alpha + (180\textdegree - \alpha) = 180\textdegree[/tex]. The sum of A and angle C is 180 degrees. In other words, the two angles are supplementary. The claim that the two angles are complementary (which describes two angles with a sum of 90 degrees) will thus be false.
