In this video, we will learn how to use the properties of tangents to find the measure of an angle of tangency or an inscribed angle subtended by the same arc. In geometry, elements are tangent when they meet at only one point. And we can call this the point of tangency. When we look at our second image, we can see that lines can be tangent to more than one circle. And like our first image shows, circles can be tangent to other circles.
Something else we need to remember before we start working with tangents and circles is the shortest distance from a point to a line. If we have line 𝑙 and point 𝐴, the shortest distance between point 𝐴 and line 𝑙 is the line that forms a right angle or a perpendicular angle between the point and the line. If we look at this distance from point 𝐴 to line 𝑙, it does not form a right angle and is very clearly a longer line than the distance 𝑑 we’ve already labeled. And if we overlay a circle onto this image, we illustrate an important point. That is that a line tangent to a circle creates a right angle between the radius and the tangent line. This is one of the first properties of tangents we’ll need to know to solve for missing angles in circles.
We’ll now look at three more fundamental properties of tangents. Let’s look first at two tangents and an external point. If two tangent segments are drawn from the same external point, then the two segments are equal. We can say that this is external point 𝑃. These two lines are tangent segments, which means they form right angles with the radii they intersect with. And based on this property, we can say that these two line segments are equal in length. We can also say that the line between the center of a circle drawn to an external point that is the intersection of two segments bisects the angle between the two tangents. On this diagram, that would mean this angle is equal to this angle.
The final property we’re going to look at is called the alternate segment theorem. And it says that the angle between a tangent and a chord is equal to the angle in the alternate segment. In our example, here is an angle between a tangent and a chord. And it will be equal in measure to this angle in the alternate segment. The same thing is true for the blue angle. This is an angle between a tangent and a chord. And it’s equal to the angle in the alternate segment. We’re nearly ready to try some examples.
Often when we’re dealing with tangents of circles, we will encounter triangles. So we need to remember that in an isosceles triangle, the two angles opposite the equal sides will be equal. And in an equilateral triangle, all three angles will be equal. And in fact they will all be equal to 60 degrees. So let’s look at an example.
Find the measure of angle 𝐵𝐴𝐶.
When we look at this diagram, we see that the measure of angle 𝐵𝐴𝐶 is located here. And what we can do first is list out the given information based on the diagram. If the point 𝑀 is the center of the circle, then we can say line segment 𝑀𝐵 is a radius of circle 𝑀. We can also say that 𝑀𝐴 is a radius of circle 𝑀. This is because the points 𝐴 and 𝐵 lie on the outside of the circle. We can also say that the line 𝐴𝐶 is tangent to circle 𝑀 at the point 𝐴.
What we wanna do now is take these three pieces of information and draw some conclusions. Therefore, we can say that 𝑀𝐵 and 𝑀𝐴 are going to be equal in length because we know what the definition of a radius is. And no matter where you draw a line from the outside of the circle to the center, all of those lengths will be equal. And that means that triangle 𝐵𝑀𝐴 is an isosceles triangle because an isosceles triangle has two sides that are equal in length. From there, we can say that the measure of angle 𝑀𝐵𝐴 is equal to the measure of angle 𝑀𝐴𝐵, which is a property of isosceles triangles. And we’ll let the measure of angle 𝑀𝐴𝐵 be equal to 𝑥 degrees.
So we can say that 90 degrees plus 𝑥 degrees plus 𝑥 degrees must equal 180 degrees because of the sum of the interior angles in any triangle. We can simplify that to an equation that says 90 plus two 𝑥 equals 180. To solve for 𝑥, we subtract 90 from both sides. Two 𝑥 equals 90. And then we divide both sides of the equation by two to see that 𝑥 equals 45. We can add that to our diagram.
We can also say that the measure of angle 𝑀𝐴𝐶 is 90 degrees, which is a property of a tangent line to a circle. And our perpendicular angle 𝑀𝐴𝐶 is made up of two smaller angles, angle 𝑀𝐴𝐵 and angle 𝐵𝐴𝐶. The larger angle is 90 degrees. Angle 𝑀𝐴𝐵 is 45 degrees. And because we know that 45 plus 45 equals 90, we have to say that the measure of angle 𝐵𝐴𝐶 is equal to 45 degrees.
Here’s another example.
Given that the measure of angle 𝑀𝐴𝐶 equals 36 degrees, determine the measure of angle 𝐵𝐴𝑀 and the measure of angle 𝐴𝑀𝐶.
We want to know the measure of angle 𝐵𝐴𝑀 and the measure of angle 𝐴𝑀𝐶. Let’s start by identifying what we know. We know that the measure of angle 𝑀𝐴𝐶 is 36 degrees, so we can add that to the diagram. We’ll also wanna list other information we can identify from the diagram. We know that line segment 𝐴𝐶 and the line segment 𝐴𝐵 are tangent to the circle 𝑀 since each segment intersects the circle at exactly one point. Line segment 𝐴𝐶 and line segment 𝐴𝐵 intersect at point 𝐴. From there, we say that line segment 𝑀𝐶 and the line segment 𝑀𝐵 are radii of circle 𝑀. We can take these four pieces of information and draw some conclusions.
We can say that line segment 𝐴𝐶 is equal to line segment 𝐴𝐵 in length, since this is one of the properties of tangents when they intersect at an external point. We can also say, based on circle properties, that the line segment 𝑀𝐴 bisects the angle 𝐵𝐴𝐶. This is another one of our circle theorems. And since that’s true, angle 𝐵𝐴𝑀 has to be equal in measure to angle 𝑀𝐴𝐶. And angle 𝑀𝐴𝐶 is 36 degrees, which makes angle 𝐵𝐴𝑀 36 degrees. That’s the first part of the question.
Now we’re moving on to find the angle of 𝐴𝑀𝐶. We recognize that the points 𝐴, 𝐶, and 𝑀 form a triangle. And since the line segment 𝐴𝐶 is tangent to this circle at point 𝐶, there’s a right angle here. The measure of angle 𝑀𝐶𝐴 is 90 degrees. And once we see that, we can say that 90 degrees plus 36 degrees plus our missing angle must equal 180 degrees because we know the sum of interior angles in triangles must be 180. 90 plus 36 is 126. And when we subtract 126 degrees from both sides, we see that the measure of angle 𝐴𝑀𝐶 is 54 degrees.
Here’s another example, this time with an equilateral triangle in the middle.
Find the measure of angle 𝐶𝐴𝐵.
First, we wanna list out the information we’re given in this diagram. We can say that line segment 𝐷𝐶 is equal to line segment 𝐴𝐶, which is equal to line segment 𝐷𝐴. Then, the line 𝐴𝐵 is tangent to the circle at point 𝐴. Using this information, we want to know the measure of angle 𝐶𝐴𝐵. From the first two given statements, there are some conclusions we can draw. First, we can say that triangle 𝐴𝐶𝐷 is equilateral. Since we know that this is an equilateral triangle, we can say that all three interior angles will measure 60 degrees. But to go any further, we need to remember the alternate segment theorem.
The alternate segment theorem tells us that the angle between a tangent and a chord is equal to the angle in the alternate segment. Our angle, angle 𝐶𝐴𝐵, is the angle between a tangent and a chord. And it will be equal to this angle in the alternate segment. And so we would say that the measure of angle 𝐶𝐴𝐵 equals 60 degrees by the alternate segment theorem.
Our next example might initially seem more complicated, but we’ll follow the same process to find some missing angles.
Given that the measure of angle 𝐵𝐸𝐶 is 31 degrees, find the measure of angle 𝐶 and the measure of angle 𝐵𝐷𝐴.
We’ve been given that angle 𝐵𝐸𝐶 is 31 degrees. That’s our first piece of information. We wanna go ahead and list out the other things we know based on the diagram. We can say that line segment 𝑁𝐹, line segment 𝑁𝐴, line segment 𝑁𝐸, and line segment 𝑁𝐵 are all radii of the circle 𝑁. And we can say that ray 𝐶𝐵 is tangent to this circle at the point 𝐵. We want to know the measure of angle 𝐶, which is here, and the measure of angle 𝐵𝐷𝐴, which is here.
We wanna use our given statements and then draw some conclusions. First of all, we can say that all of the radii will be equal in length to one another because we know the definition of a radius. We can also say that triangle 𝐴𝑁𝐹 and triangle 𝑁𝐸𝐵 are isosceles triangles because they both have two sides of the triangle that are equal in length. We can also say that the measure of angle 𝐴𝑁𝐹 will be equal to the measure of angle 𝐵𝑁𝐸 because they’re vertical angles. And that means we can say that triangle 𝐵𝑁𝐸 is congruent to triangle 𝐴𝑁𝐹 because they have a side, an angle, and a side that are congruent. And since these triangles are congruent, all of these angles will be equal to one another. They will be congruent angles, all measuring in at 31 degrees.
Something else we can now identify is that the measure of angle 𝐴𝐵𝐶 is 90 degrees because of tangent line properties. So far, we’ve identified a lot of the angles, but not the ones we need to find. We now wanna focus in on the triangle created from points 𝐸, 𝐶, and 𝐵. The angle 𝐶𝐵𝐸 is created from a 90-degree angle and a 31-degree angle. That means the angle 𝐶𝐵𝐸 is 121 degrees. The measure of angle 𝐶 plus 121 degrees plus 31 degrees must equal 180 degrees. 121 plus 31 equals 152. We subtract that value from both sides of the equation. And we get that the measure of angle 𝐶 is 28 degrees.
To find the measure of angle 𝐵𝐷𝐴, we need to follow a really similar procedure. We’re gonna focus on the triangle 𝐴𝐵𝐷, which has a right angle and a 31-degree angle. In this case, the measure of angle 𝐵𝐷𝐴 plus 90 degrees plus 31 degrees will equal 180 degrees. 90 plus 31 is 121. So we subtract 121 from both sides of the equation, which tells us that the measure of angle 𝐵𝐷𝐴 is 59 degrees. And we found both of these answers by knowing that the sum of the interior angles in a triangle must be equal to 180 degrees.
In our final example, we’re not given a diagram. Instead, we’re just given information about a figure.
A circle with center 𝑀 has a diameter line segment 𝐴𝐵. If line 𝐴𝐶 and line 𝐵𝐷 are two tangents to the circle, what can you say about them?
To solve this, let’s go ahead and sketch a circle. This is the circle 𝑀, with a diameter of 𝐴𝐵. We know that line segment 𝐴𝐵 must pass through the center 𝑀 since it is a diameter. And then we have a line 𝐴𝐶. We know that we name lines by two points that fall on that line. And because 𝐴𝐶 is tangent to this circle, it can only intersect the circle at point 𝐴. We can then sketch a line that looks like this for 𝐴𝐶. The line 𝐴𝐶 meets the diameter at a right angle. A similar thing is true for line segment 𝐵𝐷. We name the lines by points along that line. And because we know it’s tangent, it can only intersect the circle at point 𝐵. And so we have line 𝐵𝐷. This line also forms a right angle with the diameter.
What we’re seeing in this image is that line 𝐴𝐶 and line 𝐵𝐷 are parallel. The way we’ve drawn it, line 𝐴𝐶 and line 𝐵𝐷 are vertical and the diameter 𝐴𝐵 is horizontal. But this is not the only way we could draw the image. Let’s say we have the diameter drawn in this way. 𝐴𝐶 still forms a right angle with the diameter, as does 𝐵𝐷. And again, we’ll see that these two lines are parallel. They will never intersect. If both of these images still don’t convince you, we could do a short kind of proof.
If these lines are not parallel, then at some point in the distance, they will intersect. And we could call that point 𝑃. And if they intersect somewhere really far out in the distance, they would form a triangle. And the triangle would be 𝐴𝑃𝐵. The problem is we know that angles in a triangle must add up to 180 degrees. And since the measure of angle 𝐴 is 90 degrees and the measure of angle 𝐵 is 90 degrees, combined, they already equal 180 degrees. This confirms that these two lines can never intersect. They could not form a triangle and are therefore parallel. Line 𝐴𝐶 and line 𝐵𝐷 are parallel.
Let’s do a quick review to summarize what we’ve learned. A line tangent to a circle creates a right angle between the radius and the tangent line. And from one external point, two tangents drawn to a circle have equal tangent segments, those segments being the segment between the external point and the point of tangency.