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Inscribed Angles

Inscribed Angles

A circle is unique because it does not have any corners or angles, which makes it different from other figures such as triangles, rectangles, and triangles. But specific properties can be explored in detail by introducing angles inside a circle. For instance, the simplest way to create an angle inside a circle is by drawing two chords such that they start at the same point. This might seem unnecessary at first, but by doing so, we can employ many rules of trigonometry and geometry, thus exploring circle properties in more detail.

What is an Inscribed Angle of a Circle?

Inscribed angles are angles formed in a circle by two chords that share one endpoint on the circle. The common endpoint is also known as the vertex of the angle. This is shown in figure 1, where two chords and form an inscribed angle , where the symbol ‘' is used to describe an inscribed angle.

Inscribed Angles, Inscribed Angles, StudySmarterInscribed Angles, StudySmarter Originals

The other endpoints of the two chords form an arc on the circle, which is the arc AC shown below. There are two kinds of arcs that are formed by an inscribed angle.

  • When the measure of the arc is less than a semicircle or , then the arc is defined as a minor arc which is shown in figure 2a.

  • When the measure of the arc is greater than a semicircle or , then the arc is defined as a major arc which is shown in figure 2b.

But how do we create such an arc? By drawing two cords, as we discussed above. But what exactly is a chord? Take any two points on a circle and join them to make a line segment:

A chord is a line segment that joins two points on a circle.

Inscribed Angles, Major arc and minor arc of a circle, StudySmarterMajor arc and Minor arc of a circle, StudySmarter Originals

Now that a chord has been defined, what can one build around a chord? Let‘s start with an arc, and as obvious as it sounds, it is a simple part of the circle defined below:

An arc of a circle is a curve formed by two points in a circle. The length of the arc is the distance between those two points.

  • An arc of a circle that has two endpoints on the diameter, then the arc is equal to a semicircle.
  • The measure of the arc in degrees is the same as the central angle that intercepts that arc.

The length of an arc can be measured using the central angle in both degrees or radians and the radius as shown in the formula below, where θ is the central angle, and π is the mathematical constant. At the same time, r is the radius of the circle.

Inscribed Angles Formula

Several types of inscribed angles are modeled by various formulas based on the number of angles and their shape. Thus a generic formula cannot be created, but such angles can be classified into certain groups.

Inscribed Angle Theorems

Let's look at the various Inscribed Angle Theorems.

Inscribed angle

The inscribed angle theorem relates the measure of the inscribed angle and its intercepted arc.

It states that the measure of the inscribed angle in degrees is equal to half the measure of the intercepted arc, where the measure of the arc is also the measure of the central angle.

Inscribed Angles, Inscribed Angle Theorem,  StudySmarterInscribed Angle Theorem, StudySmarter Originals

Inscribed angles in the same arc

When two inscribed angles intercept the same arc, then the angles are congruent. Congruent angles have the same degree measure. An example is shown in figure 4, where and m<ABC are equal as they intercept the same arc AC:

Congruent Inscribed Angles, Inscribed Angles, StudySmarterCongruent Inscribed Angles, StudySmarter Originals

Inscribed angle in a Semicircle

When an inscribed angle intercepts an arc that is a semicircle, the inscribed angle is a right angle equal to . This is shown below in the figure, where arc is a semicircle with a measure of and its inscribed angle is a right angle with a measure of .

Inscribed Angles, Inscribed Angle in a semicircle, StudySmarterInscribed Angle in a Semicircle, StudySmarter Originals

Inscribed Quadrilateral

If a quadrilateral is inscribed in a circle, which means that the quadrilateral is formed in a circle by chords, then its opposite angles are supplementary. For example, the following diagram shows an inscribed quadrilateral, where is supplementary to and is supplementary to :

Inscribed Angles, Inscribed Quadrilateral, StudySmarterInscribed Quadrilateral, StudySmarter Originals

Inscribed Angles Examples

Find angles and if the central angle shown below is .

Inscribed Angles, Inscribed Angles Example , StudySmarterInscribed angles example, StudySmarter Originals

Solution:

Since angles and intercept the same arc , then they are congruent.

Using the inscribed angle theorem, we know that the central angle is twice the inscribed angle that intercepts the same arc.

Hence the angle is .

What is the measure of angle in the circle shown below if is ?

Inscribed Angles, Congruent Inscribed Angles , StudySmarterCongruent Inscribed Angles, StudySmarter Originals

Solution:

As angles and intercept the same arc , then they are equal . Hence, if is then must also be .

Method for Solving Inscribed Angle Problems

To solve any example of inscribed angles, write down all the angles given. Recognize the angles given by drawing a diagram if not given. Let’s look at some examples.

Find if its intercepted arc has a measure of .

Solution:

Using the inscribed angle theorem, we derive that the inscribed angle equals half of the central angle.

Find and in the inscribed quadrilateral shown below.

Inscribed Angles, Inscribed Quadrilateral Example, StudySmarterInscribed quadrilateral Example, StudySmarter Originals

Solution:

As the quadrilateral shown is inscribed in a circle, its opposite angles are complementary.

Then we substitute the given angles into the equations, and we re-arrange the equations to make the unknown angle the subject.

Find , , and in the diagram below.

Inscribed Angles, An Inscribed quadrilateral, StudySmarterAn inscribed quadrilateral, StudySmarter Originals

Solution:

Inscribed angles and intercept the same arc . Hence they are equal, therefore

Angle is inscribed in a semicircle. Hence <c must be a right angle.

As quadrilateral is inscribed in a circle, its opposite angles must be supplementary.

Inscribed Angles - Key takeaways

  • An inscribed angle is an angle formed in a circle by two chords with a common end point that lies on the circle.
  • Inscribed angle theorem states that the inscribed angle is half the measure of the central angle.
  • Inscribed angles that intercept the same arc are congruent.
  • Inscribed angles in a semicircle are right angles.
  • If a quadrilateral is inscribed in a circle, its opposite angles are supplementary.

Frequently Asked Questions about Inscribed Angles

An inscribed angle is an angle that is  formed in a circle by two chords that have a common end point that lies on the circle.

A central angle is formed by two line segments that are equal to the radius of the circle and inscribed angles are formed by two chords, which are line segments that intersect the circle in two points.

Inscribed angles can be solved using the various inscribed angles theorem, depending on the angle, number of angles and the polygons formed in the circle.

There is not a general formula for calculating inscribed angles. Inscribed angles can be solved using the various inscribed angles theorem, depending on the angle, number of angles and the polygons formed in the circle.

A typical example would be a quadrilateral inscribed in a circle where the angles formed at the corners are inscribed angles.

Final Inscribed Angles Quiz

Question

Find the length of an arc if the central angle is 100  ͦ  and the radius is 5cm. 

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Answer

8.726 cm

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Question

Find the length of an arc if the central angle is 2.53 radians and the radius is 7cm.

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Answer

17.71 cm

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