# Lesson Explainer: Inscribed Angles in a Circle Mathematics

In this explainer, we will learn how to identify theorems of finding the measure of an inscribed angle with respect to its subtended arc or central angle subtended by the same arc and the measures of inscribed angles in a semicircle.

Let us first define an inscribed angle.

### Definition: Inscribed Angle

An inscribed angle is the interior angle between two chords intersecting on a circle’s circumference.

We are now going to prove an important relationship between the measure of an inscribed angle and the measure of the central angle subtended by the same arc. Note that the measure of the central angle subtended by the same arc is, per definition, the same as the measure of the subtended arc of the inscribed angle.

Let us first consider the case where the center of the circle, , belongs to one of the sides of the inscribed angle.

Inscribed angle and central angle are subtended by the same arc, . It is worth noting that the arc that is said to subtend angle is the arc that does not contain point (it is the arc in red in the diagram).

As two sides of the triangle are two radii of the circle, it is an isosceles triangle. This means that, in triangle ,

Hence, as the angles in a triangle add to , we have

 𝑚∠𝐶𝑀𝐴=180−2𝑚∠𝐴𝐶𝐵.∘ (1)

Angles and are on a straight line; hence, we have

 𝑚∠𝐴𝑀𝐵+𝑚∠𝐶𝑀𝐴=180𝑚∠𝐶𝑀𝐴=180−𝑚∠𝐴𝑀𝐵.∘∘ (2)

Equations (1) and (2) lead to that is,

We now consider another situation for an inscribed angle and a central angle subtended by the same arc, namely, when the circle center, , is a point inside the inscribed angle.

We can use our previous result for an inscribed angle that has a side containing the circle center by splitting the inscribed angle into two inscribed angles, and , that have a side containing the circle center (as is a diameter of the circle).

We have and

Since and , we find that

So,

Finally, let us consider the third situation, that is, when the circle center, , is outside the inscribed angle.

As for the previous situation, we consider the two inscribed angles that have a side containing the circle center, and , where is a diameter of the circle.

Since and , we find that

So,

We found the same result in the three possible positions for the circle center, , with respect to the inscribed angle: (i) on one side of the inscribed angle, (ii) inside the inscribed angle, and (iii) outside the inscribed angle.

Remember that the measure of a central angle subtended by an arc is the same as the measure of this arc.

### Theorem: Inscribed Angle Theorem

The measure of an inscribed angle subtended by an arc is half the measure of this arc, that is, half the measure of the central angle subtended by this arc.

Let us now see with our first example how to use this theorem to find the measure of an inscribed angle.

Find .

### Answer

Let us call the center of the circle. It is the intersection point of and .

Angle is an inscribed angle because points , , and are on the circle. and are vertically opposite angles; therefore, they have the same measure, . is the central angle subtended by the same arc as . The inscribed angle theorem states that the measure of an inscribed angle subtended by an arc is half the measure of the central angle subtended by this arc.

Hence, we have

Let us look at an example involving solving linear equations.

### Example 2: Finding the Measure of an Inscribed Angle given Its Arc’s Measure by Solving Two Linear Equations

From the figure, what is ?

### Answer

In the circle of center , is an inscribed angle because points , , and are on the circle. The central angle subtended by the same arc (major arc ) has a measure of . The inscribed angle theorem states that the measure of an inscribed angle subtended by an arc is half the measure of the central angle subtended by this arc.

Hence, we have

Let us look now at an example involving the measure of an arc and solving a linear equation.

### Example 3: Solving Equations Using the Measure of an Inscribed Angle given Its Arc’s Measure

Given that , find .

### Answer

Angle is an inscribed angle subtended by the arc of measure .

The inscribed angle theorem states that the measure of an inscribed angle subtended by an arc is half the measure of this arc. Therefore, we have

In addition, we are told that ; hence,

In the next example, we are going to solve a multistep problem where we are given the measure of an arc.

Find .

### Answer

Angle is an inscribed angle subtended by the arc . The inscribed angle theorem states that the measure of an inscribed angle subtended by an arc is half the measure of this arc. Therefore, we have

Angle is an inscribed angle subtended by the arc . Hence,

The inscribed angle theorem states that the measure of an inscribed angle subtended by an arc is half the measure of this arc. Therefore, we have

Let us look at a corollary of the inscribed angle theorem, namely, when the inscribed angle is drawn in a semicircle (which means that the inscribed angle is subtended by an arc of measure ) or, in other words, when the central angle is a straight angle (the central angle has a measure of ).

Applying the inscribed angle theorem gives us

### Corollary: Inscribed Angle in a Semicircle

An inscribed angle drawn in a semicircle is a right angle.

Let us now solve a system of linear equations to find the measure of an inscribed angle in a semicircle.

### Example 5: Finding the Measure of an Inscribed Angle in a Semicircle

Given that , find and .

### Answer

The inscribed angle is drawn in a semicircle since is a diameter of the circle. An inscribed angle drawn in a semicircle is a right angle. Therefore, we have

In addition, the sum of the angles in a triangle is , which gives

Substituting the value we have found for into this equation gives

We have found that

In our last example, we solve a problem involving an inscribed angle drawn in a semicircle and solving an equation.

### Example 6: Solving Equations Using the Measure of an Inscribed Angle in a Semicircle

Given that and , find the value of .

### Answer

The inscribed angle is drawn in a semicircle since is a diameter of the circle. An inscribed angle drawn in a semicircle is a right angle. Therefore, we have

As the angles in a triangle add to , we find, considering triangle , that

Let us summarize the key points of this explainer.

### Key Points

• An inscribed angle is an angle whose vertex lies on the circle and whose sides contain two chords of the circle.
• The inscribed angle theorem states that the measure of an inscribed angle subtended by an arc is half the measure of this arc, that is, half the measure of the central angle subtended by this arc.
• A corollary to the inscribed angle theorem is that an inscribed angle drawn in a semicircle is a right angle.

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