Lesson Video: Angles of Intersecting Lines in a Circle Mathematics

Video Transcript

Angles of Intersecting Lines in a Circle

In this video, we will learn how to find the measures of angles resulting from the intersection of two chords, two secants, two tangents, or a tangent and a secant in a circle. To do this, let’s start with the case of two chords intersecting inside of a circle. Here, we’ve drawn the chord 𝐴𝐵 intersecting the chord 𝐶𝐷. We’ll call the point of intersection 𝐸.

We want to find an expression for one of the angles between these two chords. Let’s try and find an expression for the measure of the angle 𝐵𝐸𝐶. To do this, we start by joining the points 𝐴 and 𝐶, giving us a triangle 𝐴𝐸𝐶. The first thing we note is angles 𝐴𝐸𝐶 and 𝐵𝐸𝐶 make up a straight line. So the sum of their measures is 180 degrees.

Next, we also have that 𝐴𝐸𝐶 is a triangle. So the sum of the measures of the internal angles of this triangle will be 180 degrees. Both of the expressions on the left-hand side of the equation are equal to 180 degrees. So the left-hand side of both equations must be equal. And if we remove the measure of angle 𝐴𝐸𝐶 from both sides of this equation, we have the measure of angle 𝐵𝐸𝐶 must be equal to the measure of angle 𝐴𝐶𝐸 plus the measure of angle 𝐸𝐴𝐶.

Another way of thinking about this is both sides of the equation add to the measure of angle 𝐴𝐸𝐶 to make 180 degrees. We can rewrite this equation even further. First, let’s take a look at angle 𝐴𝐶𝐸. We can see that angle 𝐴𝐶𝐸 is exactly the same as angle 𝐴𝐶𝐷. It’s on the circumference of our circle. In particular, this angle is subtended by the minor arc from 𝐴 to 𝐷. And we recall, whenever this happens, this means the measure of the angle will be one-half the measure of the arc. The measure of angle 𝐴𝐶𝐸 is one-half the measure of the arc 𝐴𝐷. We can then do exactly the same for our other angle, angle 𝐸𝐴𝐶. This time, the minor arc from 𝐵 to 𝐶 subtends this angle. Therefore, the measure of angle 𝐸𝐴𝐶 is equal to one-half the measure of the arc 𝐵𝐶.

We can substitute both of these expressions for the angles into our equation. This gives us the measure of angle 𝐵𝐸𝐶 is one-half the measure of arc 𝐴𝐷 plus one-half the measure of arc 𝐵𝐶. We can then take out a factor of one-half to get the following equation. The measure of angle 𝐵𝐸𝐶 is one-half the sum of the measure of arc 𝐴𝐷 and the measure of arc 𝐵𝐶.

Another way of thinking about this is we’re taking the average of the measures of the two arcs opposite our angle in the circle. And in exactly the same way, we can find an expression for one of the other angles at point 𝐸. In exactly the same way, the measure of the angle will be one-half the sum of the measures of the arcs opposite the angle. The measure of angle 𝐴𝐸𝐶 is equal to one-half times the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷.

We can write this result formally as follows. If two chords 𝐴𝐵 and 𝐶𝐷 in a circle meet at a point 𝐸, then the measure of either angle between the two chords is half the sum of the measures of the arcs opposite the angle, giving us the following two formulas. The measure of angle 𝐵𝐸𝐶 is one-half the measure of arc 𝐴𝐷 plus the measure of arc 𝐵𝐶. And the measure of angle 𝐴𝐸𝐶 is one-half the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷. Let’s see an example of applying this result to find the measure of an angle between two chords in a circle.

Find 𝑥.

In this question, we’re asked to find the value of 𝑥. And we can see that 𝑥 is the angle between two chords in our circle. That’s the chord 𝐴𝐵 and the chord 𝐶𝐷. We can recall the following fact. The angle between two chords in a circle is one-half the sum of the measures of the arcs opposite the angle. And in our diagram, we’re given the measures of both of the arcs opposite our angle 𝑥. That’s the arc 𝐴𝐶, which has measure 73 degrees, and the arc 𝐷𝐵, which has measure 133 degrees. So by applying this result, we must have that 𝑥 is equal to one-half times 73 degrees plus 133 degrees. We can then evaluate this expression. 73 plus 133 is 206, and then one-half of this is 103 degrees.

Therefore, we were able to show that if 𝑥 is the angle shown in the diagram, 𝑥 is equal to 103 degrees.

We can follow a very similar method to our last proof to help us find an angle between two secant lines which intersect outside of a circle, where we remember a secant line is the line extension of a chord. For example, let’s consider the following diagram, which has the secant line 𝐴𝐵 and the secant line 𝐶𝐷, which intersect at point 𝐸. We want to find an expression for the measure of angle 𝐵𝐸𝐷. To do this, we’ll once again create a triangle. This time, we’ll create a triangle by connecting the points 𝐴 and 𝐷 with a line.

We can follow the exact same method we did in the last proof to find an expression for the measure of angle 𝐵𝐸𝐷. First, angle 𝐶𝐷𝐴 and angle 𝐴𝐷𝐸 make up a straight line, so their sum is 180 degrees. And we also know the sum of the measures of the internal angles of the triangle 𝐴𝐷𝐸 will also add to 180 degrees. So the measure of angle 𝐴𝐸𝐷 plus the measure of angle 𝐷𝐴𝐸 plus the measure of angle 𝐴𝐷𝐸 is equal to 180 degrees. Both of these expressions are equal to 180 degrees. So we can equate the left-hand side of both equations. And we can also note that the left-hand side of the equation both have a term measure of angle 𝐴𝐷𝐸, so we can remove this. This gives us the measure of angle 𝐴𝐸𝐷 plus the measure of angle 𝐷𝐴𝐸 is equal to the measure of angle 𝐶𝐷𝐴.

One way of seeing this is that both sides of our equation add to the measure of angle 𝐴𝐷𝐸 to give us a value of 180 degrees. We want to find an expression for the measure of angle 𝐴𝐸𝐷. So we’ll subtract the measure of angle 𝐷𝐴𝐸 from both sides of the equation. This gives us that the measure of angle 𝐴𝐸𝐷 is the measure of angle 𝐶𝐷𝐴 minus the measure of angle 𝐷𝐴𝐸.

Finally, we can find expressions for both of these angles since both of these angles are subtended by arcs in our circle. First, angle 𝐷𝐴𝐸 is subtended by the arc from 𝐵 to 𝐷. Second, angle 𝐶𝐷𝐴 is subtended by the arc from 𝐴 to 𝐶. And we recall that inscribed angles are one-half the measure of the arcs that they are subtended by. Therefore, the measure of angle 𝐶𝐷𝐴 is one-half the measure of the arc from 𝐴 to 𝐶 and the measure of angle 𝐷𝐴𝐸 is one-half the measure of the arc from 𝐵 to 𝐷. And the difference between these two values is the measure of angle 𝐴𝐸𝐷.

Finally, we can take out the factor of one-half to get the measure of angle 𝐴𝐸𝐷 is one-half the measure of arc 𝐴𝐶 minus the measure of arc 𝐵𝐷. We can write this more formally as follows. If 𝐴𝐵 and 𝐶𝐷 are secants which intersect at a point 𝐸 outside of our circle, then the measure of the angle between the two secants is one-half the positive difference between the measures of both arcs intercepted by the sides of the angle. In other words, the measure of angle 𝐴𝐸𝐷 is one-half the positive difference of the measure of arc 𝐴𝐶 and the measure of arc 𝐵𝐷. Let’s see an example of how we can use this property to determine the angle between two secants which intersect outside of a circle.

Find 𝑥.

In this question, we’re asked to find the value of 𝑥. And we can see in our diagram that 𝑥 is the angle between two secant lines which intersect outside of our circle. And we can find the measure of 𝑥 by recalling the following fact. The angle between two secant lines in a circle which intersect outside of a circle is one-half the positive difference of the measures of the arcs intercepted by the sides of the angle.

To apply this property, let’s do this step by step. First, let’s mark the sides of the angle of 𝑥. We can see that 𝑥 is the angle between the lines 𝐴𝐶 and 𝐴𝐸. So the two sides of our angle are the line segment 𝐴𝐶 and the line segment 𝐴𝐸. Next, we need to find the measures of the arcs intercepted by the two sides of our angle. The first side of our angle intersects the circle at the point 𝐵, and the second side of our angle intersects the circle at the point 𝐷. So one of the arcs we’re going to use is the arc from 𝐵 to 𝐷. Similarly, the first side of our angle intercepts the circle at the point 𝐶, and the second side of our angle intercepts the circle at the point 𝐸. So the other arc we’re interested in is the arc from 𝐶 to 𝐸.

Finally, the measure of our angle will be one-half the positive difference between the measures of these two arcs. And since the arc from 𝐶 to 𝐸 is bigger than the arc from 𝐵 to 𝐷, this gives us the following result. 𝑥 will be equal to one-half multiplied by the measure of arc 𝐶𝐸 minus the measure of arc 𝐵𝐷. And we’re given both of these values in the diagram. The measure of arc 𝐶𝐸 is 132 degrees, and the measure of arc 𝐵𝐷 is 36 degrees. So we substitute these values into our formula. We get that 𝑥 is equal to one-half multiplied by 132 degrees minus 36 degrees. And we can then evaluate this expression. 132 minus 36 is equal to 96. And if we multiply this by one-half, we get 48. Therefore, 𝑥 is equal to 48 degrees.

Therefore, we were able to find the value of 𝑥 in the given diagram. It was one-half the difference between the measure of arc 𝐶𝐸 and the measure of arc 𝐵𝐷, which was 48 degrees.

Let’s now consider how we would find the angle between two tangent lines to a circle which intersect at a point outside of the circle. For example, let’s consider the following tangent lines which intersect at the point 𝐶. We want to determine the measure of angle 𝐴𝐶𝐵. If we call the center of our circle 𝑀, then 𝑀𝐴𝐶𝐵 is a quadrilateral. And the internal angles of a quadrilateral add to 360 degrees.

So the measure of angle 𝑀 plus the measure of angle 𝐴 plus the measure of angle 𝐶 plus the measure of angle 𝐵 is equal to 360 degrees. Since 𝐴 and 𝐵 are the points of tangency and 𝑀 is the center of our circle, the angle at 𝐴 and the angle at 𝐵 will be right angles. So these are both equal to 90 degrees. So we can subtract 180 degrees from both sides of the equation to get the measure of angle 𝑀 plus the measure of angle 𝐶 is equal to 180 degrees. And we can see in our diagram that angle 𝑀 is the central angle of a circle and is subtended by the arc from 𝐴 to 𝐶. And we know the measure of an arc is equal to the measure of its central angle. So in this case, the measure of angle 𝑀 is equal to the measure of the arc from 𝐴 to 𝐵.

We can then substitute this into our equation and then rearrange to find an expression for the measure of angle 𝐶. We get that the measure of angle 𝐶 is equal to 180 degrees minus the measure of the arc from 𝐴 to 𝐵. And we can also write the result we’ve just proven formally as follows. If two tangents to a circle at points 𝐴 and 𝐵 intersect at a point 𝐶, then the measure of the angle between the tangents is 180 degrees minus the measure of the arc between the two points of tangency. The measure of angle 𝐶 is equal to 180 degrees minus the measure of the arc from 𝐴 to 𝐵.

Let’s now see an example where we use this property to determine the angle between two tangent lines of a circle which intersect at a point outside of the circle.

Find 𝑥.

In this question, we’re asked to find the value of 𝑥. And we can see that 𝑥 is the angle between two tangent lines to our circle. That’s the line from 𝐴 to 𝐶 and the line from 𝐴 to 𝐵. They just touch the circle at a single point, so these are tangent lines. And we can find the value of 𝑥 by recalling the following property for the angle between two tangent lines which intersect at a point outside of our circle.

We recall the angle between two tangent lines which intersect at a point is 180 degrees minus the measure of the arc between the two points of tangency. In our diagram, the points of tangency are the points 𝐵 and 𝐶. And the arc between 𝐵 and 𝐶 will be the minor arc shown. And we know the measure of this arc; its measure is 151 degrees. Then, our property tells us that the value of 𝑥 is equal to 180 degrees minus the measure of arc 𝐵𝐶. So we can substitute the measure of arc 𝐵𝐶, being 151 degrees, to get 𝑥 is equal to 180 degrees minus 151 degrees, which we can calculate is 29 degrees.

Therefore, by using the fact that the angle between two tangent lines which intersect at a point outside of a circle is 180 degrees minus the measure of the arc between the two points of tangency, we were able to show that 𝑥 is equal to 29 degrees.

Finally, let’s try and find the angle between a tangent line and a secant line which intersect outside of a circle. In this diagram, the tangent line is 𝐴𝐷 and the secant line is 𝐶𝐵. And we want to find the measure of the angle 𝐴𝐷𝐵. We’ll do this by using a very similar method to the last three proofs. We’ll start by connecting 𝐴 and 𝐵 to construct a triangle 𝐴𝐵𝐷. We see that angle 𝐶𝐵𝐴 and angle 𝐴𝐵𝐷 are on a straight line, so their measures add to 180 degrees. So we have the measure of angle 𝐶𝐵𝐴 plus the measure of angle 𝐴𝐵𝐷 is equal to 180 degrees.

We then also have that the sum of the measures of the internal angles in a triangle add to 180 degrees. So we have the measure of angle 𝐵𝐷𝐴 plus the measure of angle 𝐵𝐴𝐷 plus the measure of angle 𝐴𝐵𝐷 is equal to 180 degrees. And now we have two different expressions which when added to the measure of angle 𝐴𝐵𝐷 is equal to 180 degrees. So these two expressions must be equal. The measure of angle 𝐶𝐵𝐴 is equal to the measure of angle 𝐵𝐷𝐴 added to the measure of angle 𝐵𝐴𝐷.

We can subtract the measure of angle 𝐵𝐴𝐷 from both sides to find an expression for the measure of angle 𝐵𝐷𝐴. We have the measure of angle 𝐵𝐷𝐴 is equal to the measure of angle 𝐵𝐴𝐷 minus the measure of angle 𝐶𝐵𝐴. We can find an expression for the measure of angle 𝐵𝐴𝐷 by first adding the following two radii to our diagram. And then we’ll use the fact that the measure of the internal angles of quadrilateral 𝑀𝐴𝐷𝐵 add to 360 degrees. Since 𝐴 is a point of tangency for our tangent line, angle 𝑀𝐴𝐷 is a right angle. So the sum of the internal angles of this quadrilateral — the measure of angle 𝐴𝑀𝐵 plus 90 degrees plus the measure of angle 𝐵𝐷𝐴 plus the measure of angle 𝐷𝐵𝑀 — is equal to 360 degrees.

We know the measure of the central angle 𝐴𝑀𝐵 will be equal to the measure of the arc 𝐴𝐵. So we can substitute this into our expression to give us the following. And by considering the internal angles of triangle 𝐴𝐵𝐷, the internal angles sum to 180 degrees. So the measure of angle 𝐴𝐵𝐷 is 180 degrees minus the sum of the other two angles, the measure of angle 𝐵𝐴𝐷 and the measure of angle 𝐵𝐷𝐴. Finally, since 𝑀𝐴 and 𝑀𝐵 are radii, this means 𝑀𝐴𝐵 is an isosceles triangle. Therefore, the measure of angle 𝑀𝐴𝐵 and the measure of angle 𝑀𝐵𝐴 are equal. In particular, since angle 𝑀𝐴𝐷 is a right angle, we have the measure of angle 𝑀𝐴𝐵 is equal to 90 minus the measure of angle 𝐵𝐴𝐷.

Now, all we need to use is the fact that the measure of angle 𝐷𝐵𝑀 is the sum of the measure of angle 𝐴𝐵𝐷 and the measure of angle 𝑀𝐴𝐵. We would substitute these into our expression and then simplify. And we would be able to find the following result. The measure of angle 𝐵𝐴𝐷 is one-half the measure of the arc from 𝐴 to 𝐵. To do this, let’s clear some space and go back to the following equation.

We can find an expression for the measure of angle 𝐶𝐵𝐴 from our diagram. Angle 𝐶𝐵𝐴 is subtended by the major arc from 𝐴 to 𝐶. And the measure of an inscribed angle is one-half the measure of the arc it’s subtended by. So this is one-half the measure of the arc from 𝐴 to 𝐶. We can substitute our expression for the measure of angle 𝐵𝐴𝐷, giving us the following equation, which we can rearrange for the measure of angle 𝐵𝐷𝐴, which gives us the following. The measure of angle 𝐵𝐷𝐴 is one-half the measure of the major arc from 𝐴 to 𝐶 minus the measure of the arc from 𝐴 to 𝐵.

An easy way to remember this is the measure of the angle is one-half the difference of the measures of the two arcs intercepted by the sides of the angle. And of course we take the positive value for this difference.

Before we finish, there’s one more property we can show. We’ve already shown the measure of the angle between two tangents of a circle which meet at a point is 180 degrees minus the measure of the minor arc between the two points of tangency. We can relate this to our other results by considering the measure of the other arc; let’s call this 𝑦. These two arcs make up a full circle, so the sum of their measures is 360 degrees. Subtracting 𝑥 from both sides of the equation gives us 𝑦 is 360 degrees minus 𝑥. And we want to use this to consider one-half the difference between these two arcs. That’s one-half 𝑦 minus 𝑥.

We’ll substitute this expression for 𝑦 into one-half the difference. This gives us one-half 360 minus 𝑥 minus 𝑥, which if we simplify is 180 degrees minus 𝑥, which by using our first result is the measure of angle 𝐴𝐶𝐵. In other words, we can also think of the measure of the angle between two tangents which meet outside of a circle as one-half the difference between the two arcs between the points of tangency.

Let’s go over the key points of this video. First, we saw if two chords intersect at a point in the circle, then the measure of the angle between the two chords is half the sum of the measures of the two arcs opposite the angle. Next, we saw if two secants, two tangents, or a secant and a tangent intersect at a point outside of a circle, then the measure of the angle between them is half the positive difference between the measures of both arcs intercepted by the sides of the angle. Finally, we saw the measure of the angle between two tangents which intersect outside of a circle is 180 degrees minus the measure of the minor arc between the two points of tangency.