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Angles in Circles

Angles in Circles

When playing a free kick in football, the level of curvature is predetermined by the angle formed between the foot of the player and the circular ball.

In this article, we discuss hereafter angles in circles.

Finding angles in circles

Angles in circles are angles that are formed between either radii, chords, or tangents of a circle.

Angles in circles can be constructed via the radii, tangents, and chords. If we talk about circles, then the common unit we use to measure the angles in a circle is the degrees.

You have 360 degrees in a circle as shown in the below figure. Having a closer look at this figure, we realize that all of the angles formed are a fraction of the complete angle formed by a circle, that happens to be 360°.

Angles in circles, angle measurement, StudySmarter

Figure 1. Angles formed by rays in a circle are a fraction of the complete angle, StudySmarter Originals

For example, if you take the ray that is at and another ray that goes straight up as shown in figure 2, this makes up one-fourth of the circumference of the circle, so the angle formed is also going to be one-fourth of the total angle. The angle formed by a ray that goes straight up with the other ray which is either left or right is denoted as a perpendicular (right) angle.

Angles in Circles, angle measurement, StudySmarter

Figure 2. 90 degrees formed is one-fourth of the total angle formed by a circle, StudySmarter Originals

Angles in circle rules

This is otherwise referred to as the circle theorem and is various rules upon which problems regarding angles in a circle are being solved. These rules would be discussed in several sections hereafter.

Types of angles in a circle

There are two types of angles that we need to be aware of when dealing with angles in a circle.

Central angles

The angle at the vertex where the vertex is at the center of the circle forms a central angle.

When two radii form an angle whose vertex is located at the center of the circle, we talk about a central angle.

Angles in circles, central angle, StudySmarterFigure 3. The central angle is formed with two radii extended from the center of the circle, StudySmarter Originals

Inscribed angles

For the inscribed angles, the vertex is at the circumference of the circle.

When two chords form an angle at the circumference of the circle where both chords have a common endpoint, we talk about an inscribed angle.

Angles in circles, inscribed angle, StudySmarterFigure 4. An inscribed angle where the vertex is at the circumference of the circle, StudySmarter Originals`

Angle relationships in circles

Basically, the angle relationship which exists in circles is the relationship between a central angle and an inscribed angle.

Relationship between a central angle and an inscribed angle

Have a look at the below figure in which a central angle and an inscribed angle are drawn together.

The relationship between a central angle and an inscribed angle is that an inscribed angle is half of the central angle subtended at the center of the circle. In other words, a central angle is twice the inscribed angle.

Angles in circles, Figure 5. A central angle is twice the inscribed angle, StudySmarterFigure 5. A central angle is twice the inscribed angle, StudySmarter Originals

Have a look at the figure below and write down the central angle, inscribed angle, and an equation highlighting the relation between the two angles.

Angles in Circles, An example of a central angle and an inscribed angle, StudySmarterAn example of a central angle and an inscribed angle, StudySmarter Originals

Solution:

As we know that a central angle is formed by two radii having a vertex at the center of a circle, the central angle for the above figure becomes,

For an inscribed angle, the two chords having a common vertex at the circumference will be considered. So, for the inscribed angle,

An inscribed angle is half of the central angle, so for the above figure the equation can be written as,

Intersecting angles in a circle

The intersecting angles in a circle are also known as the chord-chord angle. This angle is formed with the intersection of two chords. The below figure illustrates two chords AE and CD that intersect at point B. The angle and are congruent as they are vertical angles.

For the figure below, the angle ABC is the average of the sum of the arc AC and DE.

Angles in circles, chord-chord angle, StudySmarter Figure 6. Two intersecting chords, StudySmarter Originals

Find the angles x and y from the figure below. All the readings given are in degrees.

Angles in Circles, Example on two intersecting chords, StudySmarterExample on two intersecting chords, StudySmarter Originals

Solution:

We know that the average sum of the arcs DE and AC constitute Y. Hence,

Angle B also happens to be 82.5° as it is a vertical angle. Notice that the angles form linear pairs as Y + X is 180° . So,

Hereon, some terms would be used which you need to be conversant with.

A tangent - is a line outside a circle that touches the circumference of a circle at only one point. This line is perpendicular to the radius of a circle.

Angles in Circles, Illustrating the tangent of a circle, StudySmarter Illustrating the tangent of a circle, StudySmarter Originals

A secant - is a line that cuts through a circle touching the circumference at two points.

Angles in Circles, Illustrating the secant of a circle, StudySmarter Illustrating the secant of a circle, StudySmarter Originals

A vertex - is the point where either two secants, two tangents or a secant and tangent meets. An angle is formed at the vertex.

Angles in Circles, Illustrating a vertex formed by a secant and tangent line, StudySmarterIllustrating a vertex formed by a secant and tangent line, StudySmarter Originals

Inner arcs and outer arcs - inner arcs are arcs that bound either or both the tangents and secants inwardly. Meanwhile, outer arcs bound either or both tangents and secants outwardly.

Angles in Circles, Illustrating inner and outer arcs, StudySmarterIllustrating inner and outer arcs, StudySmarter Originals

Secant-Secant Angle

Let's assume that two secant lines intersect at point A, the below illustrates the situation. Points B, C, D, and E are the intersecting points on the circle such that two arcs are formed, an inner arc, and an outer arc. If we are to calculate the angle , the equation is half of the difference of the arcs and .

Angles in circles, secant-secant angle, StudySmarter. Figure 7. To calculate the angle at the vertex of the secant lines, the major arc and the minor arc are subtracted and then halved, StudySmarter Originals

Find in the figure below:

Angles in Circles, Example on secant-secant angles, StudySmarterExample on secant-secant angles, StudySmarter Originals

Solution:

From the above, you should note that is a secant-secant angle. The angle of the outer arc is , while that of the inner arc is . Therefore is:

Thus

Secant-Tangent Angle

The calculation of the secant-tangent angle is very similar to the secant-secant angle. In Figure 8, the tangent and the secant line intersect at point B (the vertex). To calculate angle B, you would have to find the difference between the outer arc and the inner arc , and then divide by 2. So,

Angles in circles, secant-tangent angle, StudySmarter. Figure 8. A secant-tangent angle with vertex at point B, StudySmarter Originals

From the figure below, find :

Angles in Circles, Example of the secant-tangent rule, StudySmarterExample of the secant-tangent rule, StudySmarter Originals

Solution:

From the above, you should note that is a secant-tangent angle. The angle of the outer arc is , while that of the inner arc is . Therefore is:

Thus

Tangent-Tangent Angle

For two tangents, in figure 9, the equation to calculate the angle P would become,

Angles in circles, tangent-tangent angle, StudySmarter. Figure 9. Tangent-Tangent Angle, StudySmarter Originals

Calculate the angle P if the major arc is 240° in the figure below.

Angles in Circles, Example on tangent-tangent angles, StudySmarterExample on tangent-tangent angles, StudySmarter Originals

Solution:

A full circle makes an 360° angle and the arc is 240° thus,

Using the equation above to calculate the angle P yields,

Angles in Circles - Key takeaways

  • A complete circle is constituted of 360 degrees.
  • When two radii from an angle where the vertex is at the center of the circle, it is a central angle.
  • Two chords that form an angle at the circumference of the circle where both chords have a common endpoint is called an inscribed angle.
  • An inscribed angle is half of the central angle subtended at the center of the circle.
  • For the chord-chord angle, the angle at the vertex is calculated by the average of the sum of the opposing arcs.
  • To calculate the vertex angle for the secant-tangent, secant-secant, and tangent-tangent angles, the major arc is subtracted from the minor arc and then halved.

Frequently Asked Questions about Angles in Circles

You can find the angles in a circle by using the properties of angles in a circle. 

There are eight 45 degree angles in a circle as 360/45 = 8. 

If we divide a circle using a big plus sign, then a circle has 4 right angles. Also, 360/90 = 4. 

You measure the angles in a circle by applying the angle in circle theorems.

The central angle is that angle formed by two radii, such that the vertex of both radii form an angle at the center of the circle.

Final Angles in Circles Quiz

Question

What are angle relationships?

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Answer

Angle relationships tell the connection and relationship between angles. 

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Question

How do you identify angle relationships?

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Answer

Angle relationships can be identified by first knowing the different types of relationships that can exist and examining the figure to know which applies.

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Question

Where can lines intersect in a circle?

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Answer

Lines can intersect on the circumference of a circle, in the circle and outside the circle.

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Question

What is the theorem for lines intersecting inside a circle?

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Answer

When two chords intersect inside a circle, the measure of each angle is half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

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Question

What is the theorem for lines intersecting outside a circle?

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Answer

If a tangent and a secant, two tangents or two secants intersect outside a circle, the measure of the angle formed is half the difference between the measure of the intersected arcs. 

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Question

What is the theorem for lines intersecting on the circumference of a circle?

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Answer

The measure of the angles formed in a circle when the chord and the tangent intersect is half the measure of the intercepted arc.

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Question

What are supplementary angles?

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Answer

Supplementary angles are angles that have a sum of 180 degrees.

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Question

What are Linear pairs?

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Answer

Linear pair is a pair of adjacent angles that have a common vertex and are formed at the point of intersection on a line. Linear pairs are supplementary because they add up to 180 degrees.

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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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Question

What are angles in circles?

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Answer

Angles in circles are angles that are formed between either radii, chords, or tangents of a circle.

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Question

How many types of angles are used in angles in a circle?

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Answer

2

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Question

What are the types of angles in a circle

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Answer

Central and inscribed angles

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Question

When is a central angle formed?

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Answer

When two radii form an angle whose vertex is located at the center of the circle

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Question

When is an inscribed angle formed?

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Answer

When two chords form an angle at the circumference of the circle where both chords have a common endpoint

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Question

What rule exists between an inscribed and central angle in a circle?

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Answer

An inscribed angle is half of the central angle subtended at the center of the circle.

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Question

What other name is used to refer to intersecting angles in a circle?

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Answer

chord-chord angles

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