Consider a circle with a radius of units. An angle whose sides are two chords of the circle is formed as shown. Move the points and along the circle so that the angle is a right angle.

Think about the following questions.

On the circle above, construct an angle with a vertex at the center of the circle. The angle being constructed should also cut off the same arc. In other words, construct the corresponding central angle.

Observe the measures of the angle and the arc intercepted by the angle. Start by moving so that and are collinear. Then, move it once again so that and are collinear.

As can be seen, when and are collinear, becomes the diameter of the circle, and the angle cuts off the semicircle. Furthermore, the measure of is twice the measure of an inscribed angle that intercepts it. This statement can be restated as a theorem.

The measure of an inscribed angle is half the measure of its intercepted arc.

In this figure, the measure of is half the measure of

Let the measures of and be and respectively.

Because the radii of a circle are all congruent, two isosceles triangles can be obtained by drawing and

Therefore, by the Isosceles Triangle Theorem, the measures of and will also be and respectively.

By the Triangle Exterior Angle Theorem, it is recognized that and

By applying similar logic as the procedure above, Case I and Case III can be proven.

In the diagram, the vertex of is on the circle and the sides of the angle are chords of the circle. Given the measure of , find the measure of the angle.

Write the answer without the degree symbol.

The angle shown in the diagram fits the definition of an inscribed angle. For this reason, the measure of the angle can be found using the Inscribed Angle Theorem. The theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.

Multiply

Find the measure of the inscribed angle in the circle.

Similarly, given the measure of an inscribed angle, the measure of its corresponding central angle can be found using the Inscribed Angle Theorem. This can be done because the measure of the central angle is the same as the measure of the arc that the central angle cuts off.

In the circle, measures

Find the measure of the corresponding central angle.

Start by drawing the corresponding central angle.

Recall that a central angle is an angle whose vertex lies at the center of the circle. Additionally, the inscribed angle and its corresponding central angle intercept the same arc for this example. Therefore, the corresponding central angle is

Given the measure of an inscribed angle, find the measure of its corresponding central angle.

Up to now, the relationship between inscribed angles and their corresponding central angles has been discussed. Now the relationship between two inscribed angles that intercept the same arc will be investigated.

As can be observed, the angles are congruent, so long as they intercept the same arc.

If two inscribed angles of a circle intercept the same arc, then they are congruent.

By this theorem, and in the above diagram are congruent angles.

Consider two inscribed angles and that intercept the same arc in a circle.

Mark and Jordan have been asked to find the measure of

Determine which angles intercept the same arc. Use the Inscribed Angles of a Circle Theorem to find .

The inscribed angles and intercept

Inscribed angles, or the central angles, are not the only angles related to circles. In the next part, the angles constructed outside the circles will be examined. To construct an angle outside a circle, tangents can be used.

A line is a tangent to a circle if and only if the line is perpendicular to the endpoint of a radius on the circle’s circumference.

Based on the diagram, the following relation holds true.

Line is tangent to

The theorem will be proven in two parts as it is a biconditional statement. Each will be proven by using an indirect proof.

Assume that line is tangent to the circle centered at and not perpendicular to By the Perpendicular Postulate, there is another segment from that is perpendicular to Let that segment be The goal is to prove that must be that segment. The following diagram shows the mentioned characteristics.

Substitute expressions

Line is tangent to

For the second part, it will be assumed that is perpendicular to the radius at and that line is not tangent to In this case, line intersects at a second point

line is tangent to

Having proven both parts, the proof of the biconditional statement of the theorem is now complete.

Line is tangent to

In the diagram, is tangent to the circle at the point and is a diameter.

It has been given that By the Tangent to Circle Theorem, is perpendicular to In other words,

A circumscribed angle is supplementary to the central angle it cuts off.

The measure of a circumscribed angle is equal to minus the measure of the central angle that intercepts the same arc.

Considering the above diagram, the following relation holds true.

By definition, a circumscribed angle is an angle whose sides are tangents to a circle. Since is a circumscribed angle, and are tangents to at points and respectively. By the Tangent to Circle Theorem, is perpendicular to and is perpendicular to

Find the measure of the central angle.

The following example involving circumscribed angles and inscribed angles could require the use of the previously learned theorems.

Two tangents from to are drawn. The measure of is

Find the measure of the inscribed angle that intercepts the same arc as

This lesson defined three angles related to circles as well as the relationships between these angles. The diagram below shows the definitions and the main theorems of this lesson.