Question 2 of 10

Mathematics

2 answers:

Reptile [31]3 years ago

7 0

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 $\mathrm{\hat{A}}$ 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 $\alpha$ degrees. By the inscribed angle theorem for circles, The green angle will measure $\rm 2\hat{A} = 2\alpha$ .

Similarly, the arc that line CD and CB intercept will equal to twice the measure of $\rm \hat{C}$ . The angle of that arc is the red angle at the origin. The value of that arc will equal to $360\textdegree{} - 2\alpha$ .

Angle $\hat{C}$ is half the measure of that arc. That is:

$\rm \displaystyle \hat{C} = \frac{1}{2} (360 \textdegree - 2\alpha) = 180 - \alpha$ .

Note that $\rm \hat{A} + \hat{C} = \alpha + (180\textdegree - \alpha) = 180\textdegree$ . 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.