Measurements of Angles Involving Tangents, Chords & Secants

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  • 1:20 Two Chords
  • 2:55 Tangent and Chord
  • 3:57 Two Tangent Lines
  • 4:39 Tangent and Secant
  • 5:31 Two Secant Lines
  • 6:10 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

When lines and circles meet, angles are formed. Fortunately, we can determine the measure of these angles, whether they're formed by tangents, secants or chords, just by knowing the measure of the created arcs.

Tangents, Secants, and Chords, Oh My!

Once upon a time, I wanted to be a cartoonist. I read a lot of Calvin and Hobbes, Peanuts and The Far Side, and I was pretty sure being a cartoonist was the best job ever.

I used to get those books on how to draw cartoon characters. I didn't know how to draw, but the books said to start with circles. Then you add lines to the circles and, miraculously, cartoon characters emerged. I could never quite figure out the circle thing well enough to make my own Calvin and Hobbes. In fact, the comics I drew were really quite terrible. But I did learn a thing or two about circles.

The different lines drawn on circles have different names. This one, where the line starts and ends on the circumference, is called a chord. The part of the circle it intercepts is an arc. If a line hits two points on a circle, but keeps going, it's a secant. These also form arcs. But if the arc shrinks because the two points get closer and closer and closer until the line only hits the circle at one point, we have a tangent. Somehow a combination of chords, secants and tangents on a circle can make you a cartoon character. How? I really don't know.

Secant: Tangent: Chord: Arc:
Lines on a circle

Two Chords

This is what it looks like when I try to draw a face. Yeah, not good. But you know what is good? Finding the measure of these angles.

How we do this depends on the kind of lines we're working with. These lines are chords. With two chords, we have vertical angles here and here. The chords also create arcs. And it's the relationship between the arcs and angles that matters. The measure of the angle formed by two chords is 1/2 of the sum of the intercepted arcs. Let's see this in action.

Angles created by two chords
Angles Chord

In my botched drawing, the intercepted arcs are arc AD and arc CB. We want to know the measure of angle AED. We know that arc AD is 175 degrees and arc CB is 85 degrees. So we can say that the measure of angle AED equals 1/2 the measure of arc AD plus arc CB. 175 plus 85 is 260. What's half of 260? 130. So angle AED is 130 degrees.

Finding the measure of an angle
Arcs measured

Now, angle CEB is going to also be 130, since it's a vertical angle. And since angle CEB and angle AEC are supplementary angles, we know angle AEC is 50 degrees, as is angle BED. With just those two arcs, we figured out four angles. Not bad! Though, again, not much of a face.

Tangent and Chord

Sometimes when I draw lines on circles, I miss almost completely, like this. But this is another chance to learn about angles. Line AC is a tangent to our circle. And it forms an angle with chord AB. What's the measure of angle BAC? The measure of the angle formed by a chord and a tangent is 1/2 the measure of the arc the chord creates.

Angle created by a tangent and chord
Tangent Chord

Before we get to my botched drawing, think about a tangent line and a diameter. These lines are perpendicular, so we know the angle is 90 degrees. And what's the measure of this arc? It's half the circle, so it's 180. Did we just prove our formula? I think we did.

As for my weird art, arc AB is 106 degrees. So if angle BAC equals 1/2 the measure of arc AB, then angle BAC is half of 106, which is 53 degrees. That's all there is to it!

Arc measure of a tangent and chord
Chord Tangent Angle

Two Tangent Lines

You know what I can draw? Hats. Check out this awesome party hat. This is not just a hat, it's two tangent lines. And guess what? That means it's time for more math - party hat math.

The angle created when two tangent lines meet is half of the measure of difference in measure of the two arcs. We just subtract the minor, or smaller, arc from the major, or larger arc, then cut that in half.

Angle created by two tangent lines
Tangents Measured Arc

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