WEBVTT mathematics/geometry/pyo
00:00:00.000 --> 00:00:02.400
Welcome back to Educator.com.
00:00:02.400 --> 00:00:09.900
For this next lesson, we are going to be taking a look at isosceles and equilateral triangles.
00:00:09.900 --> 00:00:22.000
OK, the isosceles triangle theorem: now, just to review, an isosceles triangle is a triangle with two or more congruent sides.
00:00:22.000 --> 00:00:27.100
Here we have an isosceles triangle, because AC is congruent to BC.
00:00:27.100 --> 00:00:37.100
So, a triangle where two sides are congruent (or three) is known as an isosceles triangle.
00:00:37.100 --> 00:00:49.300
And the isosceles triangle theorem says that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent,
00:00:49.300 --> 00:00:59.800
meaning...remember: we learned that these two sides that are congruent are called legs;
00:00:59.800 --> 00:01:09.200
this is called the base--the non-congruent side is called the base; this is called the vertex,
00:01:09.200 --> 00:01:33.800
and these angles are called the base angles; so if the two sides are congruent, then the angles opposite those congruent sides--
00:01:33.800 --> 00:01:41.700
that means if this side is congruent, then the angle opposite is this angle right here,
00:01:41.700 --> 00:01:51.900
and this angle right here--these angles are also congruent.
00:01:51.900 --> 00:01:56.600
That means that we will have this angle being congruent to this angle.
00:01:56.600 --> 00:02:16.300
This isosceles triangle theorem is also called the base angles theorem, because it is saying that for an isosceles triangle, base angles are congruent.
00:02:16.300 --> 00:02:28.100
See how the angles opposite the sides are actually the base angles; so, this is also called the base angles theorem.
00:02:28.100 --> 00:02:38.400
OK, here we have an isosceles triangle; here are the legs; that means that this angle right here and this angle right here
00:02:38.400 --> 00:02:47.700
are the base angles, and they are congruent; so if I want to find x, then I can just make them equal to each other.
00:02:47.700 --> 00:03:13.400
5x + 20 = 8x - 10; if I subtract the 8x over, I am going to get -3x; if I subtract the +20 over, I am going to get -30; divide the -3; x is 10.
00:03:13.400 --> 00:03:19.000
So, in this problem, they just want us to find x; that is my answer.
00:03:19.000 --> 00:03:31.100
Again, when we have an isosceles triangle, the theorem says that the base angles are congruent.
00:03:31.100 --> 00:03:40.200
Now, the converse of the isosceles triangle theorem, we know, is the opposite.
00:03:40.200 --> 00:03:48.700
The original isosceles triangle theorem says that, if two sides of a triangle are congruent, then the angles opposite them are congruent.
00:03:48.700 --> 00:03:58.700
The converse says that if the two angles of a triangle are congruent, then the sides opposite are congruent.
00:03:58.700 --> 00:04:09.000
Here are the angles that are congruent; those are the opposite sides.
00:04:09.000 --> 00:04:15.800
That means that those sides are congruent.
00:04:15.800 --> 00:04:22.200
The isosceles triangle theorem works both ways: you can say that, if these sides are congruent, then the base angles are congruent;
00:04:22.200 --> 00:04:31.900
or if the base angles are congruent, then the sides opposite are congruent.
00:04:31.900 --> 00:04:39.500
Some corollaries: again, corollaries are kind of like theorems, where they are supposed to be proved;
00:04:39.500 --> 00:04:56.500
but they are a little more on the common-sense side; it is like they are important, but not as important as theorems.
00:04:56.500 --> 00:04:59.900
And you can prove these by using theorems.
00:04:59.900 --> 00:05:04.100
A triangle is equilateral if and only if it is equiangular.
00:05:04.100 --> 00:05:13.800
Remember "if and only if": this means that this conditional statement and its converse are both true.
00:05:13.800 --> 00:05:22.700
So, I can say, "If a triangle is equilateral, then it is equiangular," but the "if and only if" says that the converse can also be true;
00:05:22.700 --> 00:05:31.800
so, "If it is equiangular, then the triangle is equilateral."
00:05:31.800 --> 00:05:48.800
That means that, if I have this, then I have this; or the other way around--if I have this, then I have this.
00:05:48.800 --> 00:05:56.500
Equilateral and equiangular go hand-in-hand; if you have one, then you have both.
00:05:56.500 --> 00:06:01.500
The next one: Each angle of an equilateral triangle measures 60 degrees.
00:06:01.500 --> 00:06:08.500
Well, if I have an equilateral triangle, I know that I have an equiangular triangle, also.
00:06:08.500 --> 00:06:17.500
And the angle sum theorem says that all of the angles of a triangle have to add up to 180;
00:06:17.500 --> 00:06:30.900
so if it is equiangular, then if that is x, then this has to be x, and this has to be x, because they are all the same--equiangular.
00:06:30.900 --> 00:06:46.000
So then, all three angles are going to add up to be 180; so x + x + x = 180; 3x = 180; divide by 3; x = 60 degrees.
00:06:46.000 --> 00:06:56.700
That means that, if I have an equilateral triangle, or an equiangular triangle, then each angle measure is going to be 60 degrees.
00:06:56.700 --> 00:07:03.700
It must, must, must be, because they have to be the same measure, and then they all have to add up to 180.
00:07:03.700 --> 00:07:09.900
So, it is just 180 divided by 3.
00:07:09.900 --> 00:07:15.500
So, let's go over our examples: Find the value of x.
00:07:15.500 --> 00:07:31.100
The first one: let's look at this: we have an isosceles right triangle, meaning that our two legs are congruent, and the hypotenuse is our base.
00:07:31.100 --> 00:07:40.600
Remember: I think two lessons ago, we talked about the angle sum theorem.
00:07:40.600 --> 00:07:52.000
If we have a right triangle, since all three angles have to add up to 180, and one of the angles is a right angle (that is 90 degrees--
00:07:52.000 --> 00:08:00.100
this is 90)--automatically, we know that the other two angles, the remaining two angles, have to add up to 90,
00:08:00.100 --> 00:08:08.500
because since all three add up to 180, this angle already used up half of that; half of 180 is 90.
00:08:08.500 --> 00:08:14.600
That means that the other two, the remaining two, are going to have to add up to the second half, which is 90.
00:08:14.600 --> 00:08:31.500
So, this angle, angle A...the measure of angle A, plus the measure of angle B, is going to equal 90 degrees.
00:08:31.500 --> 00:08:36.500
We know that the measure of angle A is x; what is the measure of angle B?
00:08:36.500 --> 00:08:43.900
Well, look: it is an isosceles triangle still, so if these sides are congruent, then the base angles have to be congruent.
00:08:43.900 --> 00:08:57.900
That means that this has to be congruent to this; so if this is x, then this has to be x, also; x + x = 90; 2x = 90; so x = 45.
00:08:57.900 --> 00:09:05.500
If this is 90, then this has to be 45 and 45.
00:09:05.500 --> 00:09:16.900
The next one: we have the measure of angle A being 71, the measure of angle B being 71, AC as x, and BC as 22.
00:09:16.900 --> 00:09:27.200
I don't have any markings to show any congruence, so I have to just look at this.
00:09:27.200 --> 00:09:39.500
Angle A and angle B have the same measure of 71, so I know that they are congruent.
00:09:39.500 --> 00:09:44.400
If these are congruent, then the converse of the base angles theorem, or the isosceles triangle theorem,
00:09:44.400 --> 00:09:54.400
says that the sides opposite them (this side and this side) must also be congruent.
00:09:54.400 --> 00:10:03.700
So then, in this case, if this is 22, then x has to be 22.
00:10:03.700 --> 00:10:14.600
OK, for this next problem, I have a triangle here with base angles congruent, which means that this is congruent to this side.
00:10:14.600 --> 00:10:25.600
And then, I have another triangle here with base angles congruent, which makes this side congruent to this side.
00:10:25.600 --> 00:10:35.600
So then, if I look at this top triangle again, this measures 60, and these two are the same.
00:10:35.600 --> 00:10:41.100
So, if I make this y, this has to also be y, because they are congruent.
00:10:41.100 --> 00:10:52.600
And the angle sum theorem says that all three angles have to add up to 180.
00:10:52.600 --> 00:11:02.800
That is 60 + 2y = 180; so 2y = 120, and then y = 60.
00:11:02.800 --> 00:11:09.200
That means that each of these angles measures 60 degrees.
00:11:09.200 --> 00:11:19.500
And that just means...if all three are 60 degrees, that means that I have an equilateral triangle, or an equiangular triangle.
00:11:19.500 --> 00:11:29.800
That means...if this is 3x + 2, this is also 3x + 2, and this is 3x + 2.
00:11:29.800 --> 00:11:40.200
And then, this right here is 5x - 6, and this is 5x - 6, because of this triangle here.
00:11:40.200 --> 00:11:48.000
Since this side right here is 3x + 2, and it is also 5x - 6, I can just make them equal to each other.
00:11:48.000 --> 00:12:13.500
So, 3x + 2 = 5x - 6; if I subtract the 5x over, I get -2x; if I subtract the 2 over, I get -8; x = 4.
00:12:13.500 --> 00:12:32.800
For the next one: see how we have two triangles; this is an isosceles triangle, because these sides are congruent.
00:12:32.800 --> 00:12:36.600
That means that these base angles have to be congruent.
00:12:36.600 --> 00:12:45.400
Then, for this triangle, the same thing: these are congruent; that means that these base angles have to be congruent.
00:12:45.400 --> 00:12:49.200
And then, if you look here, we have parallel lines.
00:12:49.200 --> 00:13:01.800
Now, parallel lines, with this transversal, mean that we have some congruent angles.
00:13:01.800 --> 00:13:15.200
Since for parallel lines (parallel line, parallel line, transversal), alternate interior angles are congruent,
00:13:15.200 --> 00:13:28.800
that means that this angle right here is also 7x - 6, which is also 6x.
00:13:28.800 --> 00:13:45.800
I can say 6x = 7x - 6; if I subtract 7x over, I get -x = -6, so x is 6.
00:13:45.800 --> 00:13:55.200
You just have to look at it: I have isosceles triangles; if you have parallel lines, that will definitely help you with angles.
00:13:55.200 --> 00:14:05.100
You will need to see those parallel lines to show that these alternate interior angles are congruent.
00:14:05.100 --> 00:14:14.800
The next example: for examples 3 and 4, we are going to be working on a couple of proofs.
00:14:14.800 --> 00:14:22.600
Let's see what is given to us: AB, this side right here, is congruent to DC;
00:14:22.600 --> 00:14:36.300
angle 1 is congruent to angle 4; and I want to prove that these two angles are congruent.
00:14:36.300 --> 00:14:48.400
In order to do this, let's see: Now, what do I have to work with here?
00:14:48.400 --> 00:14:56.300
Well, I know that I am dealing with triangles; so here, I see a lot of triangles.
00:14:56.300 --> 00:15:06.100
Now, if I make these base angles, these congruent angles, then since these base angles are congruent...
00:15:06.100 --> 00:15:15.400
Now, these base angles are from the big triangle, triangle ABD; so that means that this side...
00:15:15.400 --> 00:15:23.500
Now, you have to ignore these two segments right here, because we are just looking at the big one.
00:15:23.500 --> 00:15:36.000
The big triangle with these base angles...I have to make it twice, since there is already one right here...we know that those two are congruent.
00:15:36.000 --> 00:15:45.200
Now, if I want to prove that these two angles are congruent...these are the base angles of this triangle right here;
00:15:45.200 --> 00:15:52.500
So, in order for me to say that these two base angles are congruent, these two sides have to be congruent.
00:15:52.500 --> 00:15:56.100
How can I show that these two sides are congruent?
00:15:56.100 --> 00:16:19.100
Well, let's see: look at how I have Side-Angle-Side.
00:16:19.100 --> 00:16:29.900
From the previous lesson, we can prove that these two triangles (this one right here and this one right here) are congruent.
00:16:29.900 --> 00:16:38.500
Why?--because of Side-Angle-Side: that is one of the rules, one of the methods to proving triangles congruent.
00:16:38.500 --> 00:16:48.000
Then, I can say that, since this is a side (this is a part of this triangle), and this is a part of this triangle--
00:16:48.000 --> 00:16:52.800
once I have proved that these triangles are congruent, then I can say that corresponding parts are congruent.
00:16:52.800 --> 00:16:57.600
So then, I can just say that this is congruent to this.
00:16:57.600 --> 00:17:05.200
And then, once those sides are congruent, then these base angles will be congruent, because of the isosceles triangle theorem.
00:17:05.200 --> 00:17:07.200
So, that is how I am going to go about my proof.
00:17:07.200 --> 00:17:15.400
So again, step 1 is to prove that these two triangles are congruent.
00:17:15.400 --> 00:17:23.400
Then, I am going to say that these two sides are congruent, because of CPCTC.
00:17:23.400 --> 00:17:32.200
And then, I can say that angles 2 and 3 are congruent, because of the isosceles triangle theorem.
00:17:32.200 --> 00:17:45.600
So now, I just have to write everything out: statements and reasons.
00:17:45.600 --> 00:18:01.900
Statements: #1: AB is congruent to DC, and angle 1 is congruent to angle 4; what is the reason?--"Given."
00:18:01.900 --> 00:18:08.300
Again, just to explain what I am doing here: I am going to prove that these red triangles are congruent,
00:18:08.300 --> 00:18:21.900
so that these sides will be congruent, so that these angles will be congruent.
00:18:21.900 --> 00:18:25.700
My first focus is to prove that those two triangles are congruent.
00:18:25.700 --> 00:18:30.100
And I am going to do that by one of the methods that we went over in the previous lessons.
00:18:30.100 --> 00:18:54.200
So, I have AB--I have a side; there is a side; I have an angle; I need one more thing, which would be AE being congruent to DE.
00:18:54.200 --> 00:19:02.800
What is the reason for that? Well, remember how we said that these sides are congruent, because the base angles are congruent.
00:19:02.800 --> 00:19:17.800
Then, we can say "isosceles triangle theorem--converse"; "converse" is because it was given to us that the angles are congruent,
00:19:17.800 --> 00:19:21.900
and then from there we concluded that the sides were congruent.
00:19:21.900 --> 00:19:32.400
And then, now I have all of the parts that I need to prove that the triangles are congruent.
00:19:32.400 --> 00:19:40.700
I can say now that triangle ABE (it doesn't matter how I label it for the first triangle,
00:19:40.700 --> 00:19:54.200
and then the second triangle depends on that)...so what is corresponding with A? D; with B, the C, and then E and E;
00:19:54.200 --> 00:20:03.000
so again, I don't want to just say Side-Angle-Side in this order until I look at it and make sure.
00:20:03.000 --> 00:20:08.000
It is Side-Angle-Side, because the angle is included; it is in between the two sides.
00:20:08.000 --> 00:20:18.000
So then, this is the Side-Angle-Side postulate.
00:20:18.000 --> 00:20:27.900
I prove that the triangles are congruent; once the triangles are congruent, I can now say that any of the parts are congruent.
00:20:27.900 --> 00:20:37.200
Now, I can't automatically just write that angles 2 and 3 are congruent, because they are not angles of the triangles that we proved.
00:20:37.200 --> 00:20:39.800
So, we can't say that these angles are congruent.
00:20:39.800 --> 00:20:58.700
Instead, we can say that EB is congruent to EC, because they are parts of the triangles that we just proved congruent.
00:20:58.700 --> 00:21:07.000
So, they are corresponding parts of congruent triangles that are congruent.
00:21:07.000 --> 00:21:20.900
So again, if you want to use this as a reason for corresponding parts to be congruent, this has to first be stated--two triangles being congruent.
00:21:20.900 --> 00:21:32.000
And then, I can say that angle 2 is congruent to angle 3, my final step, because,
00:21:32.000 --> 00:21:43.500
now that this is congruent because of CPCTC, I can say that the angles opposite, the base angles, are congruent now.
00:21:43.500 --> 00:22:00.000
So then, here it is going to be "isosceles triangle theorem," and that wouldn't be the converse; that would just be the regular theorem.
00:22:00.000 --> 00:22:12.500
That is it for this one; I know this was a little bit more difficult, but sometimes you have to work backwards.
00:22:12.500 --> 00:22:16.400
I looked at what I had; that was the first thing I did--I looked at my given.
00:22:16.400 --> 00:22:20.700
I made little markings if it is not already there.
00:22:20.700 --> 00:22:26.900
And then, I have to see what I have to prove--what is going to be my last step.
00:22:26.900 --> 00:22:34.600
So, from there, I can say, "OK, well, I can use this theorem if I prove that, and then I can prove that by proving something else."
00:22:34.600 --> 00:22:42.400
Sometimes you have to work backwards, and just have to kind of look at it and think about it before you actually begin the proof.
00:22:42.400 --> 00:22:50.700
We are going to do one more for this lesson: again, the first step is to look at my given;
00:22:50.700 --> 00:23:04.400
angle 3 is congruent to angle 4; those angles are congruent; AB is congruent to DC--that is already marked.
00:23:04.400 --> 00:23:14.200
And then, I want to prove that angle 1 is congruent to angle 2.
00:23:14.200 --> 00:23:26.700
Again, if I want to say that these two angles are congruent, there is no way, just by what is given to me;
00:23:26.700 --> 00:23:30.400
I can't just say that these two angles are congruent.
00:23:30.400 --> 00:23:37.900
But then, I know that these two angles are the base angles of this triangle right here.
00:23:37.900 --> 00:23:48.700
So, as long as I can say that the sides opposite, this side and this side, are congruent, then angles 1 and 2 can be congruent.
00:23:48.700 --> 00:23:54.200
Is there any way that I can prove that these sides are congruent, then, instead?
00:23:54.200 --> 00:23:58.300
Since I can't prove that the angles are congruent, can I prove that the sides are congruent?
00:23:58.300 --> 00:24:07.000
Well, let's see: these sides belong to these triangles.
00:24:07.000 --> 00:24:13.100
I can't directly say that these sides are congruent, but if I prove that the triangles are congruent,
00:24:13.100 --> 00:24:22.800
then I can say that these sides that belong to those triangles are going to be congruent, because of CPCTC.
00:24:22.800 --> 00:24:28.500
Then, I can say that these angles are congruent; that is the reasoning behind it.
00:24:28.500 --> 00:24:32.600
Now, am I able to prove that these two triangles are congruent?
00:24:32.600 --> 00:24:39.400
I have a side; I have corresponding angles; and then, I need one more--I only have two.
00:24:39.400 --> 00:24:50.100
So then, I need one more; now, look: I have that angle 5 is congruent to angle 6--that is automatic, because they are vertical angles.
00:24:50.100 --> 00:24:57.400
Now, I have Angle-Angle-Side; is that a valid method?
00:24:57.400 --> 00:25:03.000
Yes, it is; Angle-Angle-Side is valid, so that is what I am going to do.
00:25:03.000 --> 00:25:10.700
I am going to prove that these two triangles are congruent, and then say that these two sides are congruent, and then say that the angles are congruent.
00:25:10.700 --> 00:25:21.500
So, it is kind of similar to the example that we just did.
00:25:21.500 --> 00:25:41.900
Statements and reasons here: Angle 3 is congruent to angle 4, and AB is congruent to DC; "Given."
00:25:41.900 --> 00:25:45.300
What do I have here that pertains to my triangles?
00:25:45.300 --> 00:25:52.300
I have a side, my side; and I have my angles.
00:25:52.300 --> 00:25:59.700
Step 2: I need another side or angle--something; so then, that would be my vertical angles.
00:25:59.700 --> 00:26:12.300
Angle 5 is congruent to angle 6; "Vertical angles are congruent."
00:26:12.300 --> 00:26:21.300
There is my other angle; now that I have everything I need, I have proven that the triangles are congruent.
00:26:21.300 --> 00:26:40.100
Triangle ABE is congruent to triangle...what is corresponding with A?--D; C, because C is corresponding with B; E.
00:26:40.100 --> 00:26:51.800
And is my reason ASA? No, because it is not the right order; it is Angle-Angle-Side, so it is AAS.
00:26:51.800 --> 00:26:57.000
And this is actually the theorem, not the postulate.
00:26:57.000 --> 00:27:13.600
And then, now that I have proven that the triangles are congruent, I can now say that the sides, one of which is AE...that is congruent to DE.
00:27:13.600 --> 00:27:19.500
What is my reason? CPCTC.
00:27:19.500 --> 00:27:31.600
Now that these sides are congruent, I can now say that these angles are congruent, which is my last step.
00:27:31.600 --> 00:27:50.000
And then, the reason is "isosceles triangle theorem," because we just proved that these sides are congruent.
00:27:50.000 --> 00:27:53.000
That is it for this lesson; thank you for watching Educator.com.