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, ∠ ACB is the inscribed angle and AB is the intercepted angle of ∠ ACB. Next, because an intercepted arc and its central angle have the same measure, we consider the central angle of AB.
We can find the measure of AB by measuring ∠ AOB. To do so we will use a protractor.
The measure of ∠ AOB is 120^(∘), so the measure of AB is also 120^(∘). Now we will measure ∠ ACB.
We can see that the measure of ∠ ACB is 60^(∘).
Inscribed Angle & Intercepted Arc
m∠ ACB=60 & mAB=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 115 ^(∘). 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
