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Lesson Video: Angles of Tangency Mathematics • 11th Grade

In this video, we will learn how to identify the angle of tangency in a circle and find its measure using the measure of its subtended arc, inscribed angle, or central angle subtended by the same arc.

18:05

Video Transcript

In this video, we will learn how to identify the angle of tangency in a circle and find its measure using the measure of its subtended angle, inscribed angle, or central angle subtended by the same arc. We will begin by recalling that a tangent line to a circle is a line that intersects the circle at just one point. We also note that the line segment from the point of intersection 𝐴 to the center of the circle 𝑀 is a radius of the circle. And this radius is perpendicular to the tangent line.

In this video, we want to discuss angles of tangency. If we consider a tangent to a circle that meets with a chord of the circle at the point 𝐴 as shown, then the angle πœƒ between the chord and the tangent is known as the angle of tangency. Calculating this angle can be done with the help of several theorems and angle properties. During this video, we will also need to keep in mind some of the properties of triangles, for example, isosceles and equilateral triangles. Specifically, because the radii of a circle all have the same length, two radii can form the sides of an isosceles triangle as shown. This can be useful as it tells us that the measures of angles 𝑀𝐴𝐡 and 𝑀𝐡𝐴 are equal.

Additionally, we recall the inscribed angle theorem or central angle theorem. We recall that two points on a circle 𝐴 and 𝐡 divide a circle into two arcs, a major arc and a minor arc, noting that when the arcs are the same length, they divide the circle into two semicircular arcs. We can then form an inscribed angle with any point 𝐢 on the major arc as shown. This leads us to the inscribed angle theorem, which states let 𝐴 and 𝐡 be two points on a circle, 𝑀 be the center of the circle, and 𝐢 be any point on the major arc. Then, the measure of the central angle 𝐴𝑀𝐡 is double the measure of the inscribed angle 𝐴𝐢𝐡. Another way of phrasing this is that the measure of the central angle subtended by two points on a circle is twice the measure of the inscribed angle subtended by those points.

Having recalled this theorem, let’s now learn a new theorem that deals with angles of tangency in a circle. The alternate segment theorem states the following. Let 𝐴 and 𝐡 be two points on a circle and 𝐢 be the point where a tangent passing through 𝐸, 𝐢, and 𝐷 intersects the circle. Then, the angles of tangency 𝐴𝐢𝐷 and 𝐡𝐢𝐸 are equal to the angles in the alternate segments 𝐴𝐡𝐢 and 𝐡𝐴𝐢, respectively.

This theorem can be proved as follows. We recall that a tangent of the circle at point 𝐢 forms a right angle with the radius 𝑀𝐢 at the point of intersection. Labeling one of the angles of tangency 𝐴𝐢𝐷 as πœƒ, then the measure of angle 𝐴𝐢𝑀 must be 90 degrees minus πœƒ. And since we have an isosceles triangle, the measure of angle 𝐢𝐴𝑀 must also be equal to 90 degrees minus πœƒ. Since angles in a triangle sum to 180 degrees, the measure of angle 𝐴𝑀𝐢 plus 90 degrees minus πœƒ plus 90 degrees minus πœƒ must equal 180 degrees. The left-hand side of this equation simplifies as shown. We can then subtract 180 degrees and add two πœƒ to both sides such that the measure of angle 𝐴𝑀𝐢 is two πœƒ. Since the measure of the central angle is two πœƒ, using the inscribed angle theorem, we can conclude that the measure of angle 𝐴𝐡𝐢 is half the central angle two πœƒ and is therefore equal to πœƒ. This confirms that angle 𝐴𝐢𝐷 is equal to angle 𝐴𝐡𝐢.

Let’s now look at an example where we can use this theorem.

Given that line 𝐡𝐢 is a tangent to the circle, find the measure of angle 𝐴𝐡𝐢.

We will begin by labeling angle 𝐴𝐡𝐢 as πœƒ. This angle is an angle of tangency, as πœƒ lies between a chord and tangent. Since 𝐡𝐢 is a tangent to the circle and 𝐴, 𝐡, and 𝐷 are three points on the circle, we can use the alternate segment theorem. This states that an angle of tangency is equal to the angle in the alternate segment. In this case, the measure of angle 𝐴𝐡𝐢 is equal to the measure of angle 𝐴𝐷𝐡. We are told that this is equal to 78 degrees. And we can therefore conclude that the measure of angle 𝐴𝐡𝐢 is equal to 78 degrees via the alternate segment theorem.

Let’s now consider how angles of tangency relate to central angles. When proving the alternate segment theorem, we saw that the angle of tangency is half the central angle subtended by the same arc. This can be more formally stated as follows. Let 𝐴 be a point on a circle of center 𝑀 and 𝐡 be the point where a tangent passing through 𝐡 and 𝐢 intersects the circle. Then, the angle of tangency 𝐴𝐡𝐢 is half of the central angle 𝐴𝑀𝐡. Let’s now look at an example where we can use this.

Find the measure of angle 𝐡𝐴𝐢.

Let’s begin by marking the angle we are trying to find on the diagram. The angle we are trying to find is an angle of tangency, as it lies between a tangent and a chord. We are told that angle 𝐴𝑀𝐡 is a right angle and is therefore equal to 90 degrees. This is also the central angle subtended by the same arc 𝐴𝐡 as the angle of tangency. And we recall that the angle of tangency is equal to a half the central angle. This means that the measure of angle 𝐡𝐴𝐢 is equal to a half of the measure of angle 𝐴𝑀𝐡. The angle of tangency is therefore equal to a half of 90 degrees, which is equal to 45 degrees.

An alternative method here would be to recognize that triangle 𝐴𝑀𝐡 is isosceles. This means that angles 𝐡𝐴𝑀 and 𝐴𝐡𝑀 are equal in measure. And since the angles in a triangle sum to 180 degrees, they must be equal to 45 degrees. We can then use the fact that a tangent is perpendicular to the radius at the point of contact, which means that πœƒ plus 45 degrees equals 90 degrees. And subtracting 45 degrees from both sides, we have πœƒ is equal to 45 degrees. This confirms that the measure of angle 𝐡𝐴𝐢 is 45 degrees.

So far in this video, we have seen how to calculate an angle of tangency using a central angle and using an inscribed angle. We can also calculate angles of tangency using the measure of an arc. We recall that the measure of an arc is the angle that the arc makes at the center of the circle. For example, in the diagram drawn, the measure of the major arc of the circle created by the two points 𝐴 and 𝐡 is equivalent to the measure of the angle between the radii formed by 𝐴 and 𝐡 in the inside of the circle.

Recalling that the central angle of a circle is double the angle of tangency, then we can extend this to apply to the arc of a circle as shown. Since both the measure of the central angle and arc measure are two πœƒ, they are both twice the angle of tangency. More formally, if we let 𝐴 be a point on a circle and 𝐡 be the point where a tangent passing through 𝐡 and 𝐢 intersects the circle, then the angle of tangency 𝐴𝐡𝐢 is half of the measure of the arc 𝐴𝐡 formed on the same side. It is important to note that this holds for both the minor and major arcs as shown.

Let’s now look at an example where we can use this theorem together with the fact that the measures of the two arcs of a circle sum to 360 degrees.

Given that line 𝐡𝐢 is a tangent to the circle below, find the measure of angle 𝐴𝐡𝐢.

We will begin by marking the angle we need to find on the diagram. We note that this is an angle of tangency, as it lies between a tangent and a chord, in this case the tangent 𝐡𝐢 and the chord 𝐴𝐡. We know that the angle of tangency is half of the measure of the arc formed on the same side. This means that the measure of the angle 𝐴𝐡𝐢 is half the measure of the arc 𝐴𝐡 formed on the same side as shown. The measure of the arc we have been given, 190 degrees, is not on the same side. However, we can find the correct arc measure by using the fact that the measures of two arcs of a circle sum to 360 degrees. The correct arc measure labeled two πœƒ is equal to 360 degrees minus 190 degrees, which is equal to 170 degrees. Adding this to our diagram, we can now calculate the angle of tangency. It is equal to a half of 170 degrees, which is equal to 85 degrees. This is the measure of angle 𝐴𝐡𝐢.

Let’s now consider one final example that combines several of the theorems.

Given that the measure of angle 𝐸𝐢𝐷 is 54 degrees and the measure of angle 𝐹𝐡𝐷 is 78 degrees, find π‘₯ and 𝑦.

We will begin by adding the information we have been given onto our diagram. We are told that angle 𝐸𝐢𝐷 is equal to 54 degrees and angle 𝐹𝐡𝐷 is equal to 78 degrees. Since both of these angles are angles of tangency, that is, they lie between a tangent and a chord, we will begin by considering the alternate segment theorem. This states that the angles of tangency are equal to the angles in the alternate segments. This means that the measure of angle 𝐢𝐡𝐷 is equal to the measure of angle 𝐸𝐢𝐷. And from the information given, these are both equal to 54 degrees. Likewise, the measure of angle 𝐡𝐢𝐷 is equal to the measure of angle 𝐹𝐡𝐷. And both of these are equal to 78 degrees.

We now have two of the three angles of the inner triangle 𝐡𝐢𝐷. And since angles in a triangle sum to 180 degrees, we have π‘₯ degrees plus 78 degrees plus 54 degrees is equal to 180 degrees. As all our measurements are in degrees, we can rewrite the equation as shown. Subtracting 78 and 54 from both sides, we are left with π‘₯ is equal to 48. The measure of angle 𝐡𝐷𝐢 is 48 degrees.

Having worked out π‘₯, we now need to work out 𝑦. We begin by using the alternate segment theorem once again or noting that angles on a straight line sum to 180 degrees. Using either of these, we find that the measures of angle 𝐴𝐢𝐡 and 𝐴𝐡𝐢 are both equal to 48 degrees. This makes sense, since we recall that two tangent segments meeting at a point have the same length and therefore form an isosceles triangle when connected by a chord. Triangle 𝐴𝐡𝐢 is isosceles, where the two equal angles are 48 degrees. This means that 𝑦 degrees plus 48 degrees plus 48 degrees is equal to 180 degrees. Once again, we can simplify the equation. And subtracting 48 and 48 from both sides gives us 𝑦 is equal to 84. The measure of angle 𝐡𝐴𝐢 is 84 degrees. And we can therefore conclude that the values of π‘₯ and 𝑦 are 48 and 84, respectively.

Before summarizing the key points from this video, we will briefly look at the converse of the alternate segment theorem. This states that if a ray or line segment meets with a chord of a circle on the outside of the circle and the angle it makes with the chord is equal in measure to the angle in an alternate segment of the circle, then the ray or line segment must be a tangent to the circle. If the angles are not equal, then the ray or line segment is not tangent to the circle. This can be demonstrated in the two diagrams as shown. In the first case, the measure of angle 𝐴𝐢𝐷 is equal to the measure of the angle in the alternate segment. This means that line 𝐢𝐷 must be a tangent, whereas in the second case, as the angles are not equal, 𝐢𝐷 cannot be a tangent.

Let’s now recap what we learned about angles of tangency in this video. The alternate segment theorem told us that the angles of tangency 𝐴𝐢𝐷 and 𝐡𝐢𝐸 are equal to the angles in the alternate segments 𝐴𝐡𝐢 and 𝐡𝐴𝐢, respectively. We also saw that for a circle of center 𝑀, the angle of tangency 𝐴𝐡𝐢 is half of the central angle 𝐴𝑀𝐡. Finally, we saw that the angle of tangency 𝐴𝐡𝐢 is half of the measure of the arc 𝐴𝐡 formed on the same side.

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