Geometry Secants, Tangents, & Angle Measures

Welcome back to Educator.com.0000

For the next lesson, we are going to go over secants, tangents, and angle measures.0002

A secant is a line that crosses through a circle, intersecting it at two points.0010

If we have a line that intersects a circle at one point, that is a tangent; but if it intersects a circle at two points, it is a secant.0020

And then, we know that, if it is just a segment that intersects a circle at two points, that is a chord.0029

So, the difference between a secant and a chord is that a secant is a line that crosses through the circle,0037

whereas a chord is just a segment within a circle, where the endpoints are on the circle.0044

Here, we have a secant, and we have a tangent; now, when the secant and the tangent meet,0052

at the point of tangency, then the angles that are formed (we have two angles: we have this angle here,0061

and we have this angle here) are going to be half the measure of the intercepted arc.0070

This angle measure right here is going to be half the measure of the arc right here, this arc--the measure of the arc (not the length--the measure).0080

If this arc measure is, let's say, 100 degrees, then this angle measure is going to be 50 degrees--it is going to be half.0093

Now, the same thing happens with the other side: this angle is a secant and a tangent meeting at that point.0109

This arc measure is going to be 360 - 100; so that is going to be 260; then this angle measure is going to be half, so that is going to be 130.0119

That is this angle measure here; and be careful that the angle measure that is formed is with this and this--it is not with the arc.0138

It is with this and this; and that makes sense, because these two angles have to make a linear pair.0147

It has to be like this; this is 130; this is 50; together, they have to add up to 180, because it is a linear pair.0155

If a tangent forms an angle with a secant, then it is half the measure of the intercepted arc.0169

Interior angles: now, when we have two secants that intersect to form angles within a circle, this is different than central angles and inscribed angles.0178

If I say that the center is right here, that is not the center; that is just a point where they intersect.0196

If this is the center (the center is P), and these two secants intersect just anywhere in the circle (not on the center,0201

and not on the circle), then they are just interior angles.0212

So, be very careful; we went over central angles, inscribed angles...we are going to go over, right now, interior angles;0216

and then, we are going to go over angles on the outside of the circle (that is exterior angles).0233

There are four different types of angles: central angles, when the vertex is on the center (we know that they are0239

the same measure as the intercepted arc); inscribed angles, when the vertex is on the circle0246

(the angle measure of the inscribed angle is half the measure of the intercepted arc); and then,0255

for interior angles, angles that are anywhere inside the circle, but not on the center, and not on the circle--0260

just anywhere inside--then what you are going to do: let's say that we are going to look for this angle right here;0268

let's say the measure of angle 1...then you are going to take this arc right here, the measure of that arc;0275

and then, this arc on the opposite side, because this is the same angle.0289

If this is the measure of angle 1, then this is also the measure of angle 1, because they are vertical angles.0293

You are going to take these two arcs, and you are going to add them up: the measure of this one and the measure of that one.0300

You are going to add them up, and then divide by 2; so it is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.0306

If this is 80 degrees, and this is also 80 degrees--or let's say this is 90 degrees--then you are going to add them up.0318

The measure of angle 1 is going to be 1/2, or I can say 80 degrees, plus the 90 degrees; and then, 1/2 just means divided by 2.0338

80 + 90 is going to be 170, divided by 2 is going to be 85.0349

Here, the measure of angle 1 is 85; and then, this angle right here is going to be 85.0365

Now, the same thing works for this side: if we are looking for the measure of angle 2,0372

well, you are going to take this arc, the measure of that arc, the measure of this arc...0378

you add them up and divide by 2, and that is the measure of angle 2.0386

Now, given these two angle measures, since we know that the measure of angle 1 is 85,0390

I know that the measure of angle 1 and the measure of angle 2 are going to be supplementary;0398

it is a linear pair; this angle and this angle have to be 180; so you can just take 180...0402

this is the measure of angle 2...equals 180 - 85; that is going to be 95 degrees, so this is the measure of angle 2, right there.0411

And that is interior angles; again, not for central, not for inscribed, but for interior,0426

you are going to add up the two intercepted arcs and then divide by 2 to find the angle measure.0432

For exterior angles (this is the fourth type of angle), it is when the angle formed is on the outside of the circle.0443

There are three ways that exterior angles are formed: by two secants (here is the angle, right there);0452

by a secant and a tangent, which forms that angle right there; or two tangents (and there is that angle).0461

There are three different ways that exterior angles are formed; it doesn't matter how the angle is formed.0473

If you have an exterior angle, then there are two intercepted arcs; there is one right here,0478

and then there is another one right here; there are two of them.0490

See how that angle right there intercepts this arc and that arc.0493

What you are going to do is take the difference of those two arcs; it is going to be the bigger arc, this one, minus the smaller arc.0497

And then, divide it by 2; so it is kind of similar to the interior angles.0508

But the difference is that, for interior angles, you have to add up the two intercepted arcs, and then divide by 2.0514

For exterior angles, you are going to subtract the two arcs, and then divide by 2.0521

Here, if this is, let's say, 100, let's say that this is 40; then to find this angle (let's say that that is the measure of angle 1),0528

you are going to do 100 - 40 (not plus--that is interior angles), divided by 2.0546

60 divided by 2...the measure of angle 1 is 30.0556

Again, there are four angles: central angle, inscribed angle, interior angle, and exterior angle.0566

Central angles have the same measure as the arc; the inscribed angle has half the measure of the arc.0574

The interior angle...you add up the two arcs and divide it by 2; for exterior angles, you subtract the two arcs and divide by 2.0581

For this one, this is the intercepted arc, and this is the intercepted arc.0590

For this one, these are the intercepted arc and intercepted arc.0596

If you have two tangents that form the exterior angle, then you know that it is just dividing the whole circle into two parts,0604

because this is the intercepted arc, and then the other part, the remaining portion, of the circle, is going to be the other intercepted arc.0614

If you remember two tangents...this is just a different theorem, but remember the two tangents:0628

if they meet at a point at a point outside the circle, then these segments would be congruent.0634

We are working with angles now, but that is just a reminder that those two segments would be congruent.0645

Going over examples: the first one: Find the angle measure.0654

This is the angle measure right there that we are looking for; this is 68, and this is 22.0659

And then, remember: this is the exterior angle; always be careful--is it interior, or is it exterior?0669

To find the exterior one, you are going to subtract them; so let's call this the measure of angle 1: 68 - 22, divided by 2;0674

that is going to be 46/2, so the measure of angle 1 is 23 degrees--this is 23.0691

If you added them and divided by 2, that is going to be wrong; you have to make sure you subtract, since it is an exterior angle.0706

Now, this right here is an angle that is formed by a secant and a tangent that are intersecting at the point of tangency.0714

This is a little bit different; it is not any one of those four that we went over--this is just when there is an angle, and it intercepts arcs.0728

Now, if you are trying to find this angle right here, the measure of that angle, we know that this angle is half the measure of this arc.0737

So, if this is 116, do we know what this is?0751

Well, if this is 116, then this has to be 360 - 116; so then, if you subtract this, this is going to be 244; this arc right here is 244.0755

And then, this angle measure is half that arc, because this whole arc is being intercepted by the angle, so the angle is going to be half.0780

This angle...let's say this is angle 1; so the measure of angle 1 is going to be 244 divided by 2.0790

And then, it is going to be 122 degrees.0801

For the next example, we are going to find the value of x; here is x; this is an exterior angle.0812

That means that we have to take intercepted arc #1, subtract it from intercepted arc #2, and then go ahead and divide that by 2.0820

Here is the first intercepted arc; and then, there is the second one.0831

We are going to have to subtract those intercepted arcs and divide by 2; but I have to first find the measure of this arc.0845

Since this is 120 and this is 70, I can add those up and subtract it from 360, and I will find the measure of that arc over there.0851

120 + 70 is going to be 190; I am going to subtract that from 360; it is going to be 170.0862

The measure of this arc here is 170; then, I can go ahead and find x; this arc minus the measure of that arc,0874

divided by 2, is going to be 100/2, which is 50; so the measure of x is 50 degrees.0890

And then, the second one: there is x, and that is an interior angle;0907

that means that we have to add up the two intercepted arcs, and then divide it by 2.0912

For exterior, you are going to subtract; for interior, you are going to add.0917

Now, for this one, we don't have the measures of the intercepted arcs; we don't have that, and we don't have that.0922

So, you can go ahead and solve this one in two ways: the first way is to use this arc and that arc, add them up, and divide by 2.0931

And you are going to find the measure of this angle right here, or this angle; it is the same angle.0945

So, add them; divide by 2; and then, that will give you this arc, because these are the intercepted arcs for these angles.0951

After you find that, you can subtract that from 180, because that, with x, is going to be 180; it is a linear pair.0960

That is the first way you can solve for x; the second way is to use this angle right here.0970

If this is 80 degrees, then this whole thing (because this whole arc right here is the intercepted arc for this angle,0982

and we know that this angle is half the measure of this intercepted arc)--if this is 80, then this arc has to have a measure of 160.0995

And that is because 160, divided by 2, is going to be 80, which is that angle.1009

What you can do from there is subtract the 90 from 160, because, since all of this is 160,1020

if you subtract it, that is going to give you 70; that means that this has to be 70 degrees.1030

Then, you can take the 160, which is all of that, add it to 144, subtract it from 360 (because it is the whole circle;1035

this is the leftover portion of the circle); then subtract it from 360, and then you can use it to find x...those two arcs.1046

It doesn't matter which way you use; since I already found this, I am going to just go ahead and look for this, and then solve for x that way.1058

160 + 144 is 304; then, I am going to subtract it from 360, and that is going to be 56.1067

Then, here, this is 56 degrees; so to find x, I am going to...this is 56...and then, add the 70 to the 56, and then divide by 2.1085

x equals 126, divided by 2; so x is going to be 63 degrees; this is 63.1117

If you wanted to go ahead and solve it this other way, then you are going to take 90, plus the 1441137

(I'll use a different color): 144 + 90 is going to give you 234; you are going to divide by 2.1143

So then, if you divide this by 2, you are going to get 117, and that is that angle measure.1156

And so, if you add them together, it is going to become...this is 180, which is correct, because this has to be a linear pair.1164

x is 63 degrees; if you found this first, you should have gotten 117 degrees, and then subtracted from 180 to get 63.1179

The third example: Find the measure of angle 1 and the measure of angle 2.1196

It is very similar; we have interior angles--both of these are interior angles; this is 80, and this is 130.1202

I can just use those to find this angle right there: 130 + 80, divided by 2, is going to be 210/2; and then, that is going to be 105.1211

So, right here, it is 105 degrees; that is not our answer, though, because we are looking for the measure of angle 1.1239

Here, since this is a linear pair, I am going to take 180; the measure of angle 1 equals 180 minus 105, and that is going to give us 75 degrees.1249

The measure of angle 2 is vertical to angle 1; so the measure of angle 2 is going to have the same angle measure.1267

And then, for this one here, we can just use these intercepted arcs to find the measure of angle 1: 60 degrees...1284

again, it is an interior angle, so we are going to add up the numbers...plus 42, divided by 2; that is going to be 102/2, which is 51.1296

So, this is 51 degrees; and then, to find the measure of angle 2, it is supplementary; so 180 - 51 is going to be 129 degrees.1312

And the fourth example: we are going to find the measure of arc AB.1350

Here is AB; I want to look for the measure of that arc there.1358

Now, let's see what we are working with here: for this circle, we have two tangents, and this right here is a secant...secant.1364

They intersect, and then here, this arc is 210; this arc is 120; and this is the arc that they want us to find.1388

Since I need this arc and this arc to find this angle measure, the exterior angle, I know that,1402

as long as I can find the exterior angle here, I can find the measure of this arc.1414

In order to help me find the measure of this angle, I can turn to this circle here.1422

And if I find the measure of this angle, then this angle...they are vertical, so then they are the same.1429

And then, I will use that to find the measure of arc AB.1434

So here, if this is 210, then I know that this is going to be 360, subtract 210.1439

360 - 210 is going to be 150; that is this right here.1452

Now that I have both intercepted arc measures, that and that, you can subtract them (remember: with an exterior angle, you subtract).1463

So, it will be 210 - 150, divided by 2; and that is going to be 60/2, which is 30, so this is the measure of angle DCE.1476

The measure of angle DCE is 30, which makes the angle of ACB also 30.1499

Well, if this is 30, I know that this angle is going to be the measure of this arc, minus this arc, divided by 2.1520

The measure of (let me use blue) angle ACB equals the measure of arc FG, minus the measure of arc AB, divided by 2.1534

It is just saying that this angle equals this arc, minus this arc, divided by 2.1566

The measure of angle ACB is 30; that equals...the measure of arc FG is 120, minus...this is what I am looking for...divided by 2.1572

So, it is going to be 60 = 120 minus the measure of arc AB.1588

I just multiplied the 2 to both sides, so that gave me 60; if I subtract the 120 over there, then I get -60 = negative measure of arc AB.1602

Or I can say that positive 60 equals the measure of arc AB; I just multiplied both by the negative.1614

So, it was -60, equal to negative measure of arc AB.1624

Here I found the measure of arc AB, which is 60 degrees; so this has to be 60 in order for this angle measure to be 30,1630

because, if you take 120, subtract 60, and divide it by 2, then you are going to get this angle,1640

which is the measure of angle ACB; and that is 30 degrees...so that equals 60 degrees.1648

And that is it; thank you for watching Educator.com.1664