WEBVTT mathematics/geometry/pyo
00:00:00.000 --> 00:00:03.300
Welcome back to Educator.com.
00:00:03.300 --> 00:00:07.200
This lesson, we are going to prove triangles congruent.
00:00:07.200 --> 00:00:13.200
In the previous lesson, remember, we talked about the definition of congruent triangles,
00:00:13.200 --> 00:00:19.000
and how, if we have two triangles, their corresponding parts are going to be congruent.
00:00:19.000 --> 00:00:27.800
We are going to take a closer look at some of the methods we can use to prove that two triangles are congruent.
00:00:27.800 --> 00:00:32.900
The first method to prove that two triangles are congruent...
00:00:32.900 --> 00:00:46.800
Now, according to the definition of congruent triangles, if all of the corresponding parts of the two triangles are congruent, then the two triangles are congruent.
00:00:46.800 --> 00:00:51.700
But that is kind of a lot to do; that is a lot of work, because then, you would have to prove
00:00:51.700 --> 00:00:58.000
that six parts are going to be congruent in order to prove that the triangles are congruent.
00:00:58.000 --> 00:01:05.500
Instead, we have postulates and theorems that make it easier.
00:01:05.500 --> 00:01:17.300
The first one is the SSS Postulate, or you could just call it SSS, which stands for Side-Side-Side Postulate.
00:01:17.300 --> 00:01:26.400
And that is just saying that, if all of the sides of one triangle are congruent to the sides of the second triangle, then the triangles are congruent.
00:01:26.400 --> 00:01:34.900
This is one method: instead of having to show that all six corresponding parts are congruent,
00:01:34.900 --> 00:01:40.400
in order to prove that the triangles are congruent, you only have to do the three.
00:01:40.400 --> 00:01:49.900
So, if AC is congruent to DF, AB is congruent to DE, and BC is congruent to EF,
00:01:49.900 --> 00:01:55.100
once you have shown that those three sides are congruent to the three sides of the other triangle,
00:01:55.100 --> 00:01:58.400
then you have proven that the two triangles are congruent.
00:01:58.400 --> 00:02:03.400
So, the Side-Side-Side Postulate is that, if three sides of one triangle are congruent
00:02:03.400 --> 00:02:25.700
to three sides of the other triangle, then triangle ABC is congruent to triangle DEF.
00:02:25.700 --> 00:02:38.700
The next one: now, there are going to be a few different methods--the next method is SAS, which is Side-Angle-Side.
00:02:38.700 --> 00:02:46.900
Again, as long as you can prove that a side is congruent to a side of the other triangle,
00:02:46.900 --> 00:02:57.300
an angle is congruent to the angle, and another side; then you can prove that those two triangles are congruent.
00:02:57.300 --> 00:03:08.100
But there is a condition: the angle, this angle right here, has to be an included angle.
00:03:08.100 --> 00:03:18.400
"Included" means that the angle has to be in between the two sides--
00:03:18.400 --> 00:03:28.600
a side, and then the angle, and then the side; side, side, side, angle, angle, and then side, side.
00:03:28.600 --> 00:03:39.700
It can't be an angle that is one of the two outside angles; it has to be the included angle, between the two sides that you are showing are congruent.
00:03:39.700 --> 00:03:44.700
That is side-angle-side--very important.
00:03:44.700 --> 00:03:59.000
If two sides of one triangle are congruent to two sides of the other triangle, and the included angle, then the triangles are congruent.
00:03:59.000 --> 00:04:05.300
Let's do one proof on the SAS Postulate.
00:04:05.300 --> 00:04:18.900
Now, this is not actually a proof to prove this postulate; it is just a proof to show that the two triangles are congruent, using the SAS Postulate.
00:04:18.900 --> 00:04:33.200
Given that E, this point right here, is the midpoint of BD, and then E is the midpoint of AC,
00:04:33.200 --> 00:04:39.400
I know that if E is the midpoint, then that means that the two segments are congruent.
00:04:39.400 --> 00:04:48.400
And then, I want to prove that triangle AEB is congruent to triangle CED.
00:04:48.400 --> 00:04:59.500
I want to prove that this triangle is congruent to this triangle.
00:04:59.500 --> 00:05:05.800
Whenever you get a proof where you have to prove triangles congruent, you can use any of the methods,
00:05:05.800 --> 00:05:11.900
depending on what information you have--what is congruent to what--what you can show to be congruent.
00:05:11.900 --> 00:05:18.200
But this one is going to be SAS, because we are just going to practice using this one.
00:05:18.200 --> 00:05:29.600
And so far, I have a side of one of the triangles congruent to a side of the other triangle; so I have an S.
00:05:29.600 --> 00:05:37.400
Then, I have another side congruent to the side of another triangle; so I have the other S.
00:05:37.400 --> 00:05:45.300
Now, I need an A; but the A, the angle, has to be the included angle, so it has to be in between the two sides.
00:05:45.300 --> 00:05:52.900
That means that that angle is going to be this angle right here--this angle, and then this angle.
00:05:52.900 --> 00:05:56.700
But do I have a reason--can I just say that they are congruent?
00:05:56.700 --> 00:06:03.100
No, the reason would be that they are vertical, and we know that vertical angles are congruent.
00:06:03.100 --> 00:06:10.600
So, now I have all three parts, and I can go ahead and write out my proof.
00:06:10.600 --> 00:06:18.100
Let's do a two-column proof, where I have statements and reasons.
00:06:18.100 --> 00:06:23.600
Statements/reasons: I am just going to write it out like this.
00:06:23.600 --> 00:06:31.400
#1: The given is always number 1.
00:06:31.400 --> 00:06:37.500
Now, sometimes, if you look at examples of proofs in your book, the first one might not always be the given.
00:06:37.500 --> 00:06:43.500
You might have the given as #1, and then maybe #3, or later on in the proof.
00:06:43.500 --> 00:06:51.300
Sometimes, if you have two different given statements, you don't have to write both for step 1.
00:06:51.300 --> 00:06:56.900
You can write one of them that you are going to use at a time, that you are going to use first.
00:06:56.900 --> 00:07:00.100
And then, if you don't need the other given statement until later on in the proof,
00:07:00.100 --> 00:07:08.400
then you can just wait until that step where you need it and then write it in; and your reason will still be "Given."
00:07:08.400 --> 00:07:14.200
Or you can just write the whole thing under step 1.
00:07:14.200 --> 00:07:36.900
E is the midpoint of BD, and E is the midpoint of AC; my reason is "Given."
00:07:36.900 --> 00:07:46.200
OK, so then, from here, I have to prove each of these; and then I can prove that the triangles are congruent,
00:07:46.200 --> 00:07:52.800
because the postulate says that if the side with the corresponding side and the angle with the corresponding angle
00:07:52.800 --> 00:07:56.800
and the side with the side are congruent, then the triangles are congruent.
00:07:56.800 --> 00:08:02.600
So then, I have to prove all of these first.
00:08:02.600 --> 00:08:14.200
I can say that, since E is the midpoint of BD, BE is congruent to DE.
00:08:14.200 --> 00:08:22.300
Make sure that, if you are going to say BE, then you have to say DE, because it has to be corresponding--the way you write it in order.
00:08:22.300 --> 00:08:30.500
So, if I decide to write it as EB, that is fine; then you would have to write it as ED next.
00:08:30.500 --> 00:08:35.300
So, BE is congruent to DE; what is the reason?
00:08:35.300 --> 00:08:47.500
It is "definition of midpoint," because the definition of midpoint says that, if you have a midpoint,
00:08:47.500 --> 00:08:56.800
then the midpoint will cut the segment into two equal parts.
00:08:56.800 --> 00:09:04.000
So then, that is why these parts will be congruent--the definition of midpoint.
00:09:04.000 --> 00:09:25.200
And then, I can say that angle AEB (I am looking at this angle right here for the second one) is congruent to angle CED.
00:09:25.200 --> 00:09:33.300
And that is because they are vertical, and we know that vertical angles are congruent.
00:09:33.300 --> 00:09:39.700
So, any time you have vertical angles, you can say that they are congruent; the reason is that vertical angles are congruent.
00:09:39.700 --> 00:09:45.400
Now, my next step: I have to say that the other sides, these sides now, are congruent.
00:09:45.400 --> 00:10:03.600
AE, this side of one of the triangles, is congruent to side CE; that is also because of the definition of midpoint.
00:10:03.600 --> 00:10:07.000
So then, now can I say that the triangles are congruent?
00:10:07.000 --> 00:10:15.300
This step right here, BE is congruent to DE: see how that is one of the triangle being congruent to one side of the triangle.
00:10:15.300 --> 00:10:22.800
So, I have a side; then my next step was to show that the angles are congruent--I did that;
00:10:22.800 --> 00:10:32.500
angle...and then another side of each of the triangles; did I meet all of the conditions?
00:10:32.500 --> 00:10:42.400
Yes, I showed corresponding sides congruent, corresponding angles congruent, and corresponding sides congruent.
00:10:42.400 --> 00:11:03.200
Now, I can say that the triangles are congruent: so triangle AEB, this triangle right here, is congruent to triangle CED.
00:11:03.200 --> 00:11:14.600
And the reason is the SAS Postulate.
00:11:14.600 --> 00:11:19.600
Again, these are just methods--SAS, SSS, and then we are going to go over a couple more.
00:11:19.600 --> 00:11:25.900
Those are just methods to prove triangles congruent.
00:11:25.900 --> 00:11:36.800
Otherwise, if we didn't have these methods, it would be a lot harder, because you would have to prove that each part, all six corresponding parts, are congruent.
00:11:36.800 --> 00:11:40.700
These actually make it easier; we only have to prove three things congruent, three parts;
00:11:40.700 --> 00:11:47.600
and then we can say that the triangles are congruent.
00:11:47.600 --> 00:12:04.300
The next postulate is Angle-Side-Angle (ASA); here is one angle, angle A, congruent to angle F;
00:12:04.300 --> 00:12:23.100
if angle A is congruent to angle F, and then AC is congruent to (remember, if I am going to say AC, then I have to say) FD
00:12:23.100 --> 00:12:32.700
(because they are corresponding in that order), and angle C is congruent to angle D
00:12:32.700 --> 00:12:42.000
(so then, I have: this is an angle; the corresponding angle is congruent; the corresponding side is congruent;
00:12:42.000 --> 00:12:54.400
and the corresponding angle is congruent), then triangle ABC is congruent to triangle...
00:12:54.400 --> 00:13:04.500
A is corresponding with F, so triangle F...B with E, and then C with D.
00:13:04.500 --> 00:13:09.500
So, to prove that these two triangles are congruent, I can use this postulate.
00:13:09.500 --> 00:13:20.900
Now again, the other thing to mention here is "included side"; if it is ASA, make sure that the side is in between the two angles.
00:13:20.900 --> 00:13:29.200
So then, if I am going to use angle A and angle F and angle C and angle D, the side has to be in between.
00:13:29.200 --> 00:13:40.800
So, if I have, let's say, two triangles, and I did it with the angle here and another angle, the angle here, and side--this side here,
00:13:40.800 --> 00:13:47.100
this is not Angle-Side-Angle, even though I have two angles and I have one side.
00:13:47.100 --> 00:13:57.400
This would not be Angle-Side-Angle, because the side is not included, meaning it is not in between the two angles.
00:13:57.400 --> 00:14:15.000
This is actually going to be Angle-Angle-Side; this is not ASA.
00:14:15.000 --> 00:14:26.700
This one is AAS, Angle-Angle-Side, when we have two angles congruent to the corresponding angles of the second triangle,
00:14:26.700 --> 00:14:35.000
and then a side congruent to the other side, but not included (meaning it is not in between).
00:14:35.000 --> 00:14:42.500
See how this is angle, and then angle, and then side; so it goes in that order: Angle-Angle-Side.
00:14:42.500 --> 00:14:51.200
That is when you can state the Angle-Angle-Side theorem; this is a theorem.
00:14:51.200 --> 00:15:04.400
Make sure that, if you are proving triangles this way, using this theorem, you state that it is Angle-Angle-Side, and not Angle-Side-Angle.
00:15:04.400 --> 00:15:12.900
They both have two angles and one side, but if the side is in between the two angles, then it is Angle-Side-Angle;
00:15:12.900 --> 00:15:22.000
and if the side is not in between--it is not included--then it would be Angle-Angle-Side.
00:15:22.000 --> 00:15:45.900
If angle A is congruent to angle F, and angle B is congruent to angle E, and BC is congruent to ED, then I can say,
00:15:45.900 --> 00:15:57.400
because again, this is an angle, and the next is an angle, and the next is a side--then I can say that triangle ABC
00:15:57.400 --> 00:16:06.700
is congruent to triangle...what is corresponding with A? F; with B, E; and then D.
00:16:06.700 --> 00:16:17.600
So, once you prove that those three parts are congruent, then you can prove that those two triangles are congruent.
00:16:17.600 --> 00:16:33.900
So, just to go over the methods again: these four, again, are methods to prove that two triangles are congruent.
00:16:33.900 --> 00:16:52.300
SSS means Side-Side-Side; if I have two triangles that I want to prove congruent, then this is one method, one way to do it,
00:16:52.300 --> 00:17:06.900
by showing that all three sides are congruent; so if this, then the triangles are congruent.
00:17:06.900 --> 00:17:28.600
This is Side-Angle-Side; and that is saying that a side with an included angle, and then a side, like that;
00:17:28.600 --> 00:17:39.800
the angle has to be in between the two sides; if it is this angle or this angle, then that is not it.
00:17:39.800 --> 00:17:51.800
And then, if this is true--if I prove that these parts are congruent--then the triangles are congruent.
00:17:51.800 --> 00:18:18.700
This is Angle-Side-Angle; if you can prove that those three parts are congruent, then you can prove that the triangles are congruent.
00:18:18.700 --> 00:18:32.800
Angle-Angle-Side: now, when you use these in your proofs, you don't have to actually write out "Side-Side-Side,"
00:18:32.800 --> 00:18:41.000
"Side-Angle-Side"...you can just write SSS; and these three are postulates, and this one is a theorem.
00:18:41.000 --> 00:18:45.300
The last one, Angle-Angle-Side, is a theorem; you can just write SSS, though; that should be OK.
00:18:45.300 --> 00:18:47.800
Or you can just write these.
00:18:47.800 --> 00:19:01.800
This is Angle-Angle-Side: see how there is a difference between this and this.
00:19:01.800 --> 00:19:06.600
In this one, the side is in between the two angles; and in this one, the side is not in between the two angles.
00:19:06.600 --> 00:19:14.700
So, if you prove that these parts are congruent, then you can say that the triangles are congruent.
00:19:14.700 --> 00:19:30.100
Now, this last one is from the last section, the last lesson: CPCTC is actually not used to prove that triangles are congruent.
00:19:30.100 --> 00:19:49.300
This one stands for Corresponding Parts of Congruent Triangles are Congruent.
00:19:49.300 --> 00:19:53.000
That means that the triangles have to be congruent first.
00:19:53.000 --> 00:20:09.100
So, if I want to prove that...let's say I have this, and I have to prove that maybe this side is congruent to this side;
00:20:09.100 --> 00:20:15.500
now, if there is no simple way to just prove that those two sides are congruent, then what I can do is:
00:20:15.500 --> 00:20:22.200
using one of these methods, first prove that the triangles are congruent.
00:20:22.200 --> 00:20:31.300
So, with just Side-Side-Side, or whatever it is, prove that these two triangles are congruent.
00:20:31.300 --> 00:20:41.400
Once those triangles are congruent, then I can say that any corresponding parts are congruent.
00:20:41.400 --> 00:20:49.700
Let's say it is given to me; if it is given to me, whether it is given or whether you proved it, once you say
00:20:49.700 --> 00:21:08.500
that triangle ABC is congruent to triangle DEF, once this statement is written, then I can say AB is congruent to DE.
00:21:08.500 --> 00:21:14.400
And the reason would be CPCTC.
00:21:14.400 --> 00:21:17.000
Corresponding parts of congruent triangles are congruent.
00:21:17.000 --> 00:21:29.600
These four are used to actually prove that the triangles are congruent; this one is used after the triangles are congruent.
00:21:29.600 --> 00:21:37.800
So, let's go over our examples: Determine which postulate or theorem can be used to prove that the triangles are congruent.
00:21:37.800 --> 00:21:53.400
Now, for this one, I have a side; here is the first triangle, and here is the other triangle.
00:21:53.400 --> 00:22:00.800
Side and side--that is one part that is congruent; and then you have another side.
00:22:00.800 --> 00:22:05.500
But then, you only have two; you need three; so I know that vertical angles,
00:22:05.500 --> 00:22:12.700
these angles (even if it is not given to me, just from the fact that they are vertical, I can say that they) are congruent.
00:22:12.700 --> 00:22:27.600
So then, this one...which one is this going to be?--Side-Side-Angle.
00:22:27.600 --> 00:22:37.900
This is not Side-Angle-Side, because the congruent angles are not included; it is not in between the two congruent sides.
00:22:37.900 --> 00:22:41.400
This is not Side-Angle-Side; this would be Side-Side-Angle.
00:22:41.400 --> 00:22:47.900
Now, there is no such thing as Side-Side-Angle; that doesn't exist.
00:22:47.900 --> 00:22:54.600
If you spell this backwards, the other way around, then it spells out a bad word.
00:22:54.600 --> 00:23:02.400
If it spells out a bad word, it doesn't exist; just think of it that way.
00:23:02.400 --> 00:23:06.300
You cannot use this to prove that the triangles are congruent;
00:23:06.300 --> 00:23:12.300
there is no postulate or theorem that says that you can prove these, and then the triangles are congruent.
00:23:12.300 --> 00:23:22.900
In this case, for this problem, we can't prove that they are congruent, because this is not a rule.
00:23:22.900 --> 00:23:32.100
Now, if I said that angle D is congruent to angle B, then you can say Side-Angle-Side, and that is a rule.
00:23:32.100 --> 00:23:37.000
So then, you can prove that the triangles are congruent that way.
00:23:37.000 --> 00:23:46.300
But for this one, this is all that is given--this side with this side, this side with this side, and then vertical angles; so that is not one.
00:23:46.300 --> 00:23:56.400
The next one: Now, again, I only have a side; I am trying to prove that this triangle is congruent to this triangle.
00:23:56.400 --> 00:23:59.700
I have a side congruent to a side; I have a side.
00:23:59.700 --> 00:24:08.100
I have another side congruent to this side; but then I am missing one more thing.
00:24:08.100 --> 00:24:17.700
So, I have to try to see if there is a given--if there is something here that, even if they didn't give it to me, I know automatically that it is congruent.
00:24:17.700 --> 00:24:20.900
And what that is, is this side right here.
00:24:20.900 --> 00:24:32.600
So, if I split up these triangles, it is going to be like this and like that.
00:24:32.600 --> 00:24:40.500
So, I am trying to prove that these two triangles are congruent; this side is congruent, and this side is congruent.
00:24:40.500 --> 00:24:50.400
This right here is the same as this right here; they are sharing that side.
00:24:50.400 --> 00:24:59.400
Since they are sharing that side, it has to be the same; it has to be congruent, automatically.
00:24:59.400 --> 00:25:06.700
And that is the reflexive property, because AC is going to be congruent to itself.
00:25:06.700 --> 00:25:10.600
So, AC is the same for this triangle, and it is the same for this triangle.
00:25:10.600 --> 00:25:19.800
So, in this case, this is Side-Side-Side; even though this side is not given to you, we know that it is still congruent,
00:25:19.800 --> 00:25:25.900
because it is reflexive; it is equaling itself; this is the same side for this, and it is the same side for this.
00:25:25.900 --> 00:25:41.000
Automatically, it is congruent; it would be the Side-Side-Side Postulate.
00:25:41.000 --> 00:25:45.900
OK, we are going to do a few proofs now.
00:25:45.900 --> 00:25:54.700
So then, let's take a look at our given: angle A is congruent to angle E.
00:25:54.700 --> 00:26:10.000
I am just going to do that; C is the midpoint of AE; that means that, if this is the midpoint, then these two parts are congruent.
00:26:10.000 --> 00:26:17.500
That is given; and then I have to prove that this triangle, triangle ABC, is congruent to triangle EDC.
00:26:17.500 --> 00:26:30.000
Since I am proving that two triangles are congruent, I have to use one of the methods: SSS, SAS, ASA, or AAS--one of those four methods.
00:26:30.000 --> 00:26:39.300
Since I only have two parts--I have an angle, and I have a side--I need to look at this and see if anything else is given.
00:26:39.300 --> 00:26:48.300
And I see that this angle right here is congruent to this angle right here, automatically, because they are vertical.
00:26:48.300 --> 00:27:11.300
So, statements and reasons, right here: the #1 statement is going to be that angle A is congruent to angle E,
00:27:11.300 --> 00:27:24.500
and that C is the midpoint of AE; the reason is "Given."
00:27:24.500 --> 00:27:34.700
Now, since I know that my destination, my last step, my point B, is going to be that these two triangles are congruent,
00:27:34.700 --> 00:27:40.300
do I know already what method I am going to be using to prove that those two triangles are congruent?
00:27:40.300 --> 00:27:48.800
I have an angle, a side, and an angle, so I know that I am going to be using the ASA Postulate.
00:27:48.800 --> 00:27:57.900
Now, I just have to state out each of these; I have to state out the angle; I have to state out the sides; and then I have to state the angles.
00:27:57.900 --> 00:28:07.700
And then, once all three of these are stated, then I can say that these two triangles are congruent.
00:28:07.700 --> 00:28:19.800
The next step: now, see how angle A is congruent to angle E--that angle is already stated, so that here, this is an angle.
00:28:19.800 --> 00:28:28.700
Since C is the midpoint of AE, I am going to say that AC is congruent to EC.
00:28:28.700 --> 00:28:41.900
The reason for that is "definition of midpoint."
00:28:41.900 --> 00:28:56.200
And then, that is my side; and then, I am missing an angle, which is this right here.
00:28:56.200 --> 00:29:01.900
Now, I can't say angle C, because angle C represents so many different angles.
00:29:01.900 --> 00:29:15.100
So, I have to say angle ACB, or BCA; so ACB is congruent to...and if I am going to say angle ACB, then I have to say angle ECD,
00:29:15.100 --> 00:29:21.300
because A and D are corresponding, because they are congruent.
00:29:21.300 --> 00:29:39.900
ECD--see how that is another angle; what is my reason for that?--that vertical angles are congruent.
00:29:39.900 --> 00:29:46.400
Now, see how I stated all of them now: I stated my angle; I stated my sides; and I stated my angle.
00:29:46.400 --> 00:29:59.400
So then, I can say that triangle ABC is congruent to triangle EDC.
00:29:59.400 --> 00:30:13.000
And what is my reason--what method did I use?--Angle-Side-Angle Postulate.
00:30:13.000 --> 00:30:20.500
You are going to do a few of these kinds of proofs, where you are proving the triangles using one of the methods.
00:30:20.500 --> 00:30:23.900
Always look at what is given; look at your diagram.
00:30:23.900 --> 00:30:28.600
Your diagram is going to be very valuable when it comes to proofs, because you want to mark it up
00:30:28.600 --> 00:30:39.000
and see what you have, what you have to work with, what more you have to do, and how you are going to get to this step right here.
00:30:39.000 --> 00:30:43.400
It is like your map.
00:30:43.400 --> 00:30:58.100
BD bisects AE; now, remember "bisector": when something bisects something else, it is cutting it in half.
00:30:58.100 --> 00:31:09.400
So, if BD is bisecting AE, that means that BD is cutting AE in half--think of it that way--it is cutting it in half.
00:31:09.400 --> 00:31:19.600
Don't get confused by what is cut in half, which one is going to be cut in half.
00:31:19.600 --> 00:31:24.600
Is BD cut in half, or is AE cut in half?
00:31:24.600 --> 00:31:28.500
Whichever one is doing the bisecting is the one that is doing the cutting.
00:31:28.500 --> 00:31:46.800
Since BD is bisecting AE, BD is cutting AE in half; if BD is bisecting AE, that means that BD cut AE in half, so AE is cut in half.
00:31:46.800 --> 00:31:59.600
And then, what else is given? Angle B is congruent to angle D.
00:31:59.600 --> 00:32:05.600
I know that angle 1 is congruent to angle 2, because they are vertical angles; I am just going to mark that.
00:32:05.600 --> 00:32:09.400
So, how would I be able to prove that these two triangles are congruent?
00:32:09.400 --> 00:32:19.000
Angle-Angle-Side: it is not Angle-Side-Angle; it is Angle-Angle-Side.
00:32:19.000 --> 00:32:23.500
Now, but then, look at my "prove" statement; it is not asking me to prove that the triangles are congruent.
00:32:23.500 --> 00:32:30.900
It is asking me to prove that AB, this side, is congruent to this side.
00:32:30.900 --> 00:32:35.800
But is there any way for me to be able to prove that those two sides are congruent?
00:32:35.800 --> 00:32:52.800
I don't see a way to say that these two sides are congruent; there is nothing here that allows me to prove that AB is congruent to ED, except for CPCTC.
00:32:52.800 --> 00:33:03.100
Remember: it is a random side that I have to prove congruent; whenever it is just a random side,
00:33:03.100 --> 00:33:11.400
a random part, that you have to prove congruent, then you have to first prove that the triangles are congruent,
00:33:11.400 --> 00:33:21.700
which we can do by Angle-Angle-Side; and then, once the triangles are congruent, you can say that corresponding parts are congruent.
00:33:21.700 --> 00:33:27.800
So then, once the triangles are congruent, I can say that AB is congruent to ED,
00:33:27.800 --> 00:33:33.100
because corresponding parts of congruent triangles are congruent.
00:33:33.100 --> 00:34:08.500
I know (wrong one!)...statements/reasons...1: BD bisects AE; angle B is congruent to angle D; that is given.
00:34:08.500 --> 00:34:18.000
OK, so one of the parts is already stated out: angle B is congruent to angle D--that is an angle.
00:34:18.000 --> 00:34:25.900
Now, if you are going to do this like how I am doing it, how you are writing out what you are showing on the side,
00:34:25.900 --> 00:34:32.000
that is good; but just be careful when it comes to your included angle or your included side,
00:34:32.000 --> 00:34:39.600
because if you are not doing the right order--let's say I mention the angle--see how this angle is mentioned first,
00:34:39.600 --> 00:34:44.600
and then maybe the next step--what if I mention the sides?
00:34:44.600 --> 00:34:49.200
Then, be careful so that it is not going to be in that order.
00:34:49.200 --> 00:34:54.600
I will show you when we get to that.
00:34:54.600 --> 00:35:03.500
Here, the next step: I am going to say that AC is congruent to...now, if I am going to say AC,
00:35:03.500 --> 00:35:12.200
I can't say CE; I have to say EC; remember corresponding parts--with AC, what is congruent to A?
00:35:12.200 --> 00:35:18.400
E is, so then, if I say AC, then I have to say EC.
00:35:18.400 --> 00:35:33.300
What is the reason--why are those two sides congruent?--because this is "definition of segment bisector."
00:35:33.300 --> 00:35:38.700
If it was an angle that was bisected, then it would be "angle bisector."
00:35:38.700 --> 00:35:44.100
But since this is a segment, it is "segment bisector."
00:35:44.100 --> 00:35:57.600
OK, so then, this one is the side that is mentioned; and then, angle 1 is congruent to angle 2.
00:35:57.600 --> 00:36:08.700
What is the reason for that?--"vertical angles are congruent," so that is my angle.
00:36:08.700 --> 00:36:23.100
I have three parts mentioned, so now I can say that my triangles are congruent; triangle ABC is congruent to...
00:36:23.100 --> 00:36:34.300
A is corresponding to E, so it has to be triangle E...what is corresponding with B?--D; C.
00:36:34.300 --> 00:36:50.000
What is the reason? Now, ASA is one method, but we didn't use ASA--we used Angle-Angle-Side.
00:36:50.000 --> 00:36:53.100
Now, that is what I was talking about earlier.
00:36:53.100 --> 00:37:05.800
Be careful, because I didn't mention it in the order of AAS; I mentioned an angle, and then I mentioned the side, and then I mentioned the angle.
00:37:05.800 --> 00:37:13.000
It is OK if this is out of order, but just be careful if you are going to write it out on the side like this, like how I am doing.
00:37:13.000 --> 00:37:19.500
Then, you don't put it in that order, ASA; you have to look at the diagram and see what the order is.
00:37:19.500 --> 00:37:22.400
It is Angle-Angle-Side--it is not Angle-Side-Angle.
00:37:22.400 --> 00:37:28.700
I just mentioned it in this order, but it is not the actual order of the diagram.
00:37:28.700 --> 00:37:34.000
Just be careful with that; it is OK to list them out like this, but then, when it comes to the order,
00:37:34.000 --> 00:37:42.500
look back and say, "Is it ASA? No, it is AAS."
00:37:42.500 --> 00:37:47.100
And then, I am done with the proof, right?
00:37:47.100 --> 00:37:59.200
The whole point of proving these two triangles congruent is so that I can prove that parts of the triangles are congruent.
00:37:59.200 --> 00:38:11.000
So then, now that it is stated that the triangles are congruent, I can now state any of the corresponding parts congruent.
00:38:11.000 --> 00:38:21.100
Now, I can say that AB is congruent to ED, because these are parts of these congruent triangles.
00:38:21.100 --> 00:38:30.500
What is my reason?--"corresponding parts of congruent triangles are congruent."
00:38:30.500 --> 00:38:36.300
The corresponding parts are congruent, as long as they are from congruent triangles.
00:38:36.300 --> 00:38:48.500
OK, we are going to do one more proof on this.
00:38:48.500 --> 00:39:00.600
Let's see what we have: AB is parallel to DC, and then, AD is parallel to BC.
00:39:00.600 --> 00:39:08.800
Now, just like those slash marks, I have to write this out twice.
00:39:08.800 --> 00:39:14.400
And then, I want to prove that angle A is congruent to angle C.
00:39:14.400 --> 00:39:23.100
Is there any way that I can prove that those two angles are congruent?
00:39:23.100 --> 00:39:31.200
No, I don't think that there is anything; how would you prove that those two angles are congruent?
00:39:31.200 --> 00:39:39.000
Well, then, can I do it in two steps, where I can prove that these two triangles are congruent,
00:39:39.000 --> 00:39:44.800
and then use CPCTC to say that these parts are congruent?
00:39:44.800 --> 00:39:53.600
Let's say if I can prove the triangles congruent: well, if these two are parallel (remember parallel lines?),
00:39:53.600 --> 00:39:59.900
here is my transversal; see, extending it out makes it easier to see.
00:39:59.900 --> 00:40:07.600
Then, alternate interior angles, that angle with this angle of this triangle, are going to be congruent.
00:40:07.600 --> 00:40:17.200
And then, since these two lines are parallel, the same thing here: angle 1 is going to be congruent to angle 4.
00:40:17.200 --> 00:40:25.600
If you want to see that again, these are the two parallel lines; this is AB, and this is DC.
00:40:25.600 --> 00:40:37.800
Here is the transversal; this is 3, and this is 2; see if they are parallel--then the alternate interior angles are going to be congruent.
00:40:37.800 --> 00:40:47.100
The same thing is going this way: my transversal...here is angle 4; here is angle 1;
00:40:47.100 --> 00:40:53.400
as long as they are parallel, then these two alternate interior angles are congruent.
00:40:53.400 --> 00:41:00.700
So then, I have two angles; I have Angle-Angle, but then I need one more; I need three.
00:41:00.700 --> 00:41:11.400
Remember: earlier, we looked at a diagram similar to this, where we have two triangles, and they share a side.
00:41:11.400 --> 00:41:19.600
If they share a side, then automatically, I can say that this side to this triangle is congruent to this side to this triangle.
00:41:19.600 --> 00:41:31.100
That is another one of those automatic things: you have vertical angles that are automatically congruent, and you have a shared side that is automatically congruent.
00:41:31.100 --> 00:41:40.700
Now I have three parts: I have an angle; I have a side; and I have angles.
00:41:40.700 --> 00:41:51.800
Now, in order to prove that this angle is congruent to this angle, I can first say that this whole triangle is congruent to this whole triangle.
00:41:51.800 --> 00:42:01.600
And then, these parts of those congruent triangles are going to be congruent.
00:42:01.600 --> 00:42:23.700
Statements/reasons: 1: AB is parallel to DC, and AD is parallel to BC; "Given."
00:42:23.700 --> 00:42:41.900
Step 2: Angle 1 is congruent to angle 4; and then, my reason for angle 1 being congruent to angle 4,
00:42:41.900 --> 00:42:46.000
and angle 2 being congruent to angle 3, is going to be the same.
00:42:46.000 --> 00:42:51.300
My reason is going to be the same, so I can just include it in the same step.
00:42:51.300 --> 00:42:57.200
I don't have to separate it: angle 2 is congruent to angle 3.
00:42:57.200 --> 00:43:09.500
And then, both of those are going to be...you can say "alternate interior angles theorem,"
00:43:09.500 --> 00:43:27.600
or you can just write it out: "If lines are parallel, then alternate interior angles are congruent"--you could just write it like that.
00:43:27.600 --> 00:43:40.300
Step 3: What do I have so far? I have my angle listed, an angle stated, and another angle stated; and now I have to state my side.
00:43:40.300 --> 00:43:58.000
DB is congruent to BD; now, notice how I didn't write BD and BD; I wrote DB, and then I wrote BD.
00:43:58.000 --> 00:44:15.800
If I draw this out again, if I separate out the two triangles, this is D, and this B; this is D, and this is B.
00:44:15.800 --> 00:44:21.000
This angle right here is actually corresponding with this angle right here.
00:44:21.000 --> 00:44:38.400
See how, if I flip it around, this angle and this angle are congruent; this angle and this angle are congruent,
00:44:38.400 --> 00:44:45.600
because this is angle 1, and then, this is angle 4; and then, we know that angle 1 and angle 4 are congruent.
00:44:45.600 --> 00:44:52.400
So then, this and this are corresponding; so then, I have to say B and D.
00:44:52.400 --> 00:44:58.600
So, if it is DB, then I have to say BD; does that make sense?
00:44:58.600 --> 00:45:05.600
Here is my side that I am sharing; that is this side right here.
00:45:05.600 --> 00:45:15.000
Since angle 1 is congruent to angle 4 here, this angle and this angle are corresponding parts.
00:45:15.000 --> 00:45:23.000
So, if I say DB, then I have to say BD, because this is corresponding to this, and this one is corresponding to this one.
00:45:23.000 --> 00:45:26.800
This one is congruent to this.
00:45:26.800 --> 00:45:36.400
Even though the letters are the same, DB and DB here, look at the angles: this one is corresponding to this angle,
00:45:36.400 --> 00:45:43.700
so then, if you mention D here first, you have to mention B first for the next one.
00:45:43.700 --> 00:46:01.000
Step 3: This is the reflexive property--any time something equals itself, this side equals the same side, then it is the reflexive property.
00:46:01.000 --> 00:46:14.800
Then, did I say all that I needed to say?--yes, so now I can say, since I have all three parts, that the triangles are congruent.
00:46:14.800 --> 00:46:33.000
Triangle ABD is congruent to triangle...what is corresponding with A? C; what is corresponding to B?--remember the angle, D; B.
00:46:33.000 --> 00:46:35.900
What is my reason? Is my reason Angle-Angle-Side?
00:46:35.900 --> 00:46:45.700
No, I have to look at the diagram: I used the Angle-Side-Angle Postulate.
00:46:45.700 --> 00:46:52.600
I used the angle, an angle, and a side, but not in that order; it is in this order.
00:46:52.600 --> 00:46:59.000
But that is not it; the whole point wasn't to just prove the triangles congruent.
00:46:59.000 --> 00:47:06.200
The whole point was to prove them congruent so that these parts would be congruent.
00:47:06.200 --> 00:47:15.700
So, angle A is congruent to angle C, and the reason is CPCTC.
00:47:15.700 --> 00:47:23.900
Now, remember again: if I want to use this CPCTC rule, first the triangles must be congruent.
00:47:23.900 --> 00:47:31.200
So, here this has to be stated somewhere before you use CPCTC.
00:47:31.200 --> 00:47:40.000
And once it is stated, then you can use it, saying that any corresponding parts will be congruent.
00:47:40.000 --> 00:47:45.500
That is it for this lesson; we will do a little more of this.
00:47:45.500 --> 00:47:50.500
We are going to go over more triangle stuff in the next lesson, so we will see you then.
00:47:50.500 --> 00:47:51.000
Thank you for watching Educator.com.