In this exercise we will try to find answers for the following questions.

Question I: How are inscribed angles related to their intercepted arcs?
Question II: How are the angles of an inscribed quadrilateral related to each other?

Let's begin with Question I!

Question I

To answer this question we should first recall the definitions of an inscribed angle and an intercepted arc.

An inscribed angle is the angle created when two chords or secants intersect on the circle. An arc lying between two lines, rays, or segments is called an intercepted arc.

We can visualize these terms as follows.

As we can see, $∠ A C B$ is the inscribed angle and $A B$ is the intercepted angle of $∠ A C B .$ Next, because an intercepted arc and its central angle have the same measure, we consider the central angle of $A B .$

We can find the measure of $A B$ by measuring $∠ A O B .$ To do so we will use a protractor.

The measure of $∠ A O B$ is $12 0^{\circ},$ so the measure of $A B$ is also $12 0^{\circ} .$ Now we will measure $∠ A C B .$

We can see that the measure of

∠ A C B

is

6 0^{\circ} .

\underline{Inscribed Angle} m ∠ A C B = 60 \underline{Intercepted Arc} m A B = 120

Therefore, we can conclude that the measure of an inscribed angle is half the measure of its intercepted arc.

Question II

For this question we will begin by constructing a quadrilateral whose vertices are on a circle.

Now, to understand the relation between its angles we will measure each angle using a protractor. Let's begin with $∠ A! .$

The measure of $∠ A$ is $11 5^{\circ} .$ Proceeding in the same way, we can find the measure of the other angles as follows.

From here, looking at the opposite angles of the quadrilateral we can conclude that the opposite angles of an inscribed quadrilateral are supplementary.

m ∠ A + m ∠ C = 180 m ∠ B + m ∠ D = 180

Exercise