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 \(360°\).
Fig. 1. Angles formed by rays in a circle are a fraction of the complete angle.
For example, if you take the ray that is at \(0º\) 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.
Fig. 2. \(90\) degrees formed is one-fourth of the total angle formed by a circle.
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.
Fig. 3. The central angle is formed with two radii extended from the center of the circle.
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.
Fig. 4. An inscribed angle where the vertex is at the circumference of the circle.
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.
Fig. 5. A central angle is twice the inscribed angle.
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.
Fig. 6. An example of a central angle and an inscribed angle.
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,
\[\text{Central Angle}=\angle AOB\]
For an inscribed angle, the two chords having a common vertex at the circumference will be considered. So, for the inscribed angle,
\[\text{Inscribed Angle}=\angle AMB\]
An inscribed angle is half of the central angle, so for the above figure the equation can be written as,
\[\angle AMB=\dfrac{1}{2}\left(\angle AOB\right)\]
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 \(\angle ABC\) and \(\angle DBE\) 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 \(DE\).
\[\angle ABC=\dfrac{1}{2}\left(\widehat{AC}+\widehat{DE}\right)\]
Fig. 7. Two intersecting chords.
Find the angles \(x\) and \(y\) from the figure below. All the readings given are in degrees.
Fig. 8. Example on two intersecting chords.
Solution:
We know that the average sum of the arcs \(DE\) and \(AC\) constitute Y. Hence,
\[Y=\dfrac{1}{2}\left(100º+55º\right)=82.5º\]
Angle \(B\) also happens to be \(82.5°\) as it is a vertical angle. Notice that the angles \(\angle CXE\) and \(\angle DYE\) form linear pairs as \(Y + X\) is \(180°\) . So,
\[\begin{align}180º-Y&=X\\180º-82.5º&=X\\X&=97.5º\end{align}\]
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.
Fig. 9. Illustrating the tangent of a circle.
A secant - is a line that cuts through a circle touching the circumference at two points.
Fig. 10. Illustrating the secant of a circle.
A vertex - is the point where either two secants, two tangents or a secant and tangent meets. An angle is formed at the vertex.
Fig. 11. Illustrating a vertex formed by a secant and tangent line.
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.
Fig. 12. Illustrating inner and outer arcs.
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 \(\widehat{BC}\), and an outer arc\(\widehat{DE}\). If we are to calculate the angle \(\alpha\), the equation is half of the difference of the arcs \(\widehat{DE}\) and \(\widehat{BC}\).
\[\alpha=\dfrac{1}{2}\left(\widehat{DE}-\widehat{BC}\right)\]
Fig. 13. To calculate the angle at the vertex of the secant lines, the major arc and the minor arc are subtracted and then halved.
Find \(\theta\) in the figure below:
Fig. 14. Example on secant-secant angles.
Solution:
From the above, you should note that \(\theta\) is a secant-secant angle. The angle of the outer arc is \(128º\), while that of the inner arc is \(48º\). Therefore \(\theta\) is:
\[\theta=\dfrac{128º-48º}{2}\]
Thus
\[\theta=30º\]
Secant-Tangent Angle
The calculation of the secant-tangent angle is very similar to the secant-secant angle. In Figure 15, 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 \(\widehat{AC}\) and the inner arc \(\widehat{CD}\), and then divide by \(2\). So,
\[X=\dfrac{1}{2}\left(\widehat{AC}-\widehat{CD}\right)\]
Fig. 15. A secant-tangent angle with vertex at point B.
From the figure below, find \(\theta\):
Fig. 16. Example of the secant-tangent rule.
Solution:
From the above, you should note that \(\theta\) is a secant-tangent angle. The angle of the outer arc is \(170º\), while that of the inner arc is \(100º\). Therefore \(\theta\) is:
\[\theta=\dfrac{170º-100º}{2}\]
Thus
\[\theta=35º\]
Tangent-Tangent Angle
For two tangents, in figure 17, the equation to calculate the angle \(P\) would become,
\[\angle P=\dfrac{1}{2}\left(\text{major arc}-\text{minor arc}\right)\]
\[\angle P=\dfrac{1}{2}\left(\widehat{AXB}-\widehat{AB}\right)\]
Fig. 17. Tangent-Tangent Angle.
Calculate the angle \(P\) if the major arc is \(240°\) in the figure below.
Fig. 18. Example on tangent-tangent angles.
Solution:
A full circle makes an \(360°\) angle and the arc \(\widehat{AXB}\) is \(240°\) thus,
\[\widehat{AXB]+\widehat{AB}=360º\]
\[\widehat{AB}=360º-240º\]
\[\widehat{AB}=120º\]
Using the equation above to calculate the angle \(P\) yields,
\[\angle P=\dfrac{1}{2}(240º-120º)\]
\[\angle P=60º\]
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.
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