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Proving Segment Relationship
 A good proof is made of five essential parts:
 State the theorem to be proved
 List the given information
 If possible, draw a diagram to illustrate the given information
 State what is to be proven
 Use a system of deductive reasoning to complete the proof
 Reasons in a proof include:
 Undefined terms
 Definitions
 Postulates
 Previously proven theorems
Proving Segment Relationship
If AB = CD, CD = EF, then AB = EF.
―XY @―YX
If ―AB = ―CD , ―CD = ―EF , then ―AB = ―EF .
An acute angle's supplementary angle is an obtuse angle.
Given: ∠ABC and ∠CBD are supplementary angles, and ∠ABC is an acute angle.
Prove: ∠CBD is an obtuse angle.
The measures of obtuse angles are larger than acute angles.
Given: ∠ABC is an obtuse angle, ∠CBD is an acute angle.
Prove: m∠ABC > m∠CDF.
Two adjacent complementary angles form a right angle.
Given: ∠MON and ∠NOP are adjacent, and they are also complementary angles.
Prove: ∠MOP is a right angle.
If two lines intersect but not perpendicular to each other, then they can form 2 pairs of congruent angles.
Given: Line AB and line CD intersect at point O, and they are not perpendicular to each other.
Prove: There are two pairs of congruent angles.
If MN + CB = 5, and CB = 2x + 1, then MN + 2x + 1 = 5.
Adjacent angles share a votex and a side.
Given: ∠AOB and ∠BOC are adjacent.
Prove: ∠AOB and ∠BOC share a votex and a side.
Given: D is the midpoint of ―AB
Prove: ―AD ≅―BD .


*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Proving Segment Relationship
Lecture Slides are screencaptured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
 Intro 0:00
 Good Proofs 0:12
 Five Essential Parts
 Proof Reasons 1:38
 Undefined
 Definitions
 Postulates
 Previously Proven Theorems
 Congruence of Segments 4:10
 Theorem: Congruence of Segments
 Proof Example 10:16
 Proof: Congruence of Segments
 Setting Up Proofs 19:13
 Example: Two Segments with Equal Measures
 Setting Up Proofs 21:48
 Example: Vertical Angles are Congruent
 Setting Up Proofs 23:59
 Example: Segment of a Triangle
 Extra Example 1: Congruence of Segments 27:03
 Extra Example 2: Setting Up Proofs 28:50
 Extra Example 3: Setting Up Proofs 30:55
 Extra Example 4: TwoColumn Proof 33:11
Geometry Online Course
Transcription: Proving Segment Relationship
Welcome back to Educator.com.0000
This next lesson is on proving segment relationships.0002
From the previous lesson, the concept of proofs was introduced.0007
And for this lesson, since we are going to be proving segment relationships, we are going to try and start setting up proofs and get you more familiar with them.0014
First, let's talk about what makes a good proof: a good proof is made of five essential parts.0031
First, state the theorem to be provedyou are going to state what you are going to prove.0041
List the given informationyou are going to list everything that is given; make sure that it is good information and that it is all listed.0055
#3: If possible, draw a diagram to illustrate the given informationagain, a diagram or some kind of picture or drawing0064
to show what you are dealing with, the statements and stuff.0075
State what is to be proven: you have a given statement, and you have a "prove" statement.0082
Use a system of deductive reasoning to complete the proofyou are going to go from point A to point B to complete the proof.0087
We have, on the twocolumn proof, the first column, which are all statements, and the second column, which are all reasons.0101
So, some of the reasons that you can list under that column can include undefined termsjust terms that you have previously learned;0107
definitionsthis includes any type of definition of the terms, like the definition of...and you would just write definition as def. for short...0127
of maybe congruent segments, or maybe the definition of supplementary angles, and so on.0141
When you use definitions, you are just going to write "definition of," and then whatever it is.0157
Postulates: an example of a postulate would be the Angle Addition Postulate; so you can write Angle Addition Postulate.0163
And for most of these, you can make them shorter; if there is no name for it, then you can just write out the statement in a short way0177
like, instead of writing out "congruent," you can just write this sign; instead of writing the whole word "definition," you can just write "def."0189
For the Angle Addition Postulate, it would just be Angle Addition Postulate; that is one postulate that you can use in your reasons.0196
Previously proven theorems: Theorems, unlike postulates...0205
Remember: with postulates, we talked about how they don't have to be proven; we can just assume them to be true.0212
So, once a postulate is introduced, we can just go ahead and start using them for our reasons, if we need to.0218
Theorems, however, have to be proven; once the theorems are proven (and they are usually proven in your textbooks), you can use them.0224
Once they are proved to be true, then theorems can also be used in your reasons.0238
Here is a proof on the congruence of segments.0252
Remember: we talked about, last lesson, our properties of equality.0259
We talked about the reflexive property, the symmetric property, and the transitive property.0265
We talked about these properties, but they are of equality; so they are good only when you are using the equals sign for them.0270
The reflexive property, remember, was that if you have a number a, then a = a; it equals itselfthat is the reflexive property of equality.0278
The symmetric property is when you have that if a = b, then b = a; you flip them, and then that is the symmetric property of equality.0289
The transitive property was when you do that if a = b and b = c, then a = c; and that is the transitive property of equality.0299
So, these three properties can also be used for congruent segments.0309
But congruence is a little bit different; when you talk about segments that are congruent, it is a little bit different than equal.0317
If you are going to use them for segments, to show their congruence, then you can't write "reflexive property of equality."0328
So, instead of writing "reflexive of equality," like you did for the last lesson,0338
if it is for equality, you are going to write it as "reflexive property of congruent segments."0345
So, depending on how you use the reflexive property, you are going to write it in different ways.0359
If you are going to write it using an equals sign, then you can write "reflexive property of equality," like this.0366
But if you are going to use it for congruence of segments, then you are going to have to write "reflexive property of congruent segments"this or this.0372
And the same thing for the symmetric property; instead of "symmetric property of equality,"0385
like this, using the equals sign, you are going to write "symmetric property of congruent segments."0390
Or you can write out "segments," too.0399
For the transitive property, it is "transitive property of congruent segments," also.0403
This is the theorem; the theorem says that congruence of segments is reflexive, symmetric, and transitive.0408
The theorem is saying that you can use these three properties for the congruence of segments.0414
So, before we can actually use this and this and the transitive property of congruent segments, we have to prove this theorem.0420
Remember, theorems have to be proved; so before we can actually use this, we have to prove this theorem.0429
This is the proof to prove this theorem; and there are three of themthere is reflexive, symmetric, and transitive.0437
But we are only going to look at the symmetric one.0448
For the other ones, your book should show you some kind of proof to the theorem.0454
Or if you want to look at any online things, you can just look that up for the theorem to prove the other two.0460
For this one, the congruence of segments is symmetric; that is what we are trying to prove.0472
We are proving one part of this theorem: AB is congruent to CD, and we want to prove that CD is congruent to AB.0477
So again, this is to prove the congruence of segments; see how this little symbol right here means "congruent."0488
And these are all segments: AB and CD, both of these.0497
We are trying to show that it can be symmetric; that meanssee how this shows the symmetric property?0502
Congruent segments can also use the symmetric property; that is what you are proving.0510
The first statement: AB is congruent to CD: the reason for that is "Given"; the first step is always "Given."0516
The next step, AB = CD: from here to here, all that has changed from this to this is that congruence went to equality.0526
This is actually that, whenever you go from equal to congruent, or from congruent to equal, this is "definition of congruent"...0540
and since we are dealing with segments..."segments."0549
If you use angles, then you would say "definition of congruent angles."0553
And then, here, from here to here, what happened? It just flipped.0559
And we know that we can use (since we already used it for the last lesson)this is the symmetric property of equality.0567
We know that we can use the equality one; so we are going to use that for the reason.0577
And then, the next step: see how it just went from equal to congruent now?0581
Then again, this is "definition of congruent segments."0588
And this is what we are trying to prove: that we can use the symmetric property for congruent segments.0595
Now that we have proven this theorem, we can now use these statements as our reasons.0606
Here is an example of a proof: Given is AB congruent to CD.0618
Now, you use your diagram; if you are given this, and you don't have a diagram, then draw in a diagram.0629
And then, this little mark right here, this hash mark, is just to show that this is congruent.0637
So, if you mark this once, and you mark this once, then they are congruent.0642
The next two that you want to show congruent: BD is congruent to DE.0649
I can't mark it once; if I mark it once, then that is saying that this and this are the same.0656
So then, you have to mark this twice; so this is a different one...and then DE...you mark this twice.0662
So, the segments with the same number of hash marks are congruent.0668
And you are proving that AD, this whole thing, is congruent to CE, that whole thing.0675
We know that their parts are congruent: AB is congruent to CD, that part; and then, this small part is congruent to the other small part.0682
And we want to prove that the whole thing, AD, is congruent to CE.0690
Now, remember: if you are given parts, and you have to try to prove something with the whole thing,0697
any time that you have to go from parts to whole or whole to parts, remember,0706
you are going to use the Segment Addition Postulate.0711
So, if you are dealing with segments (which we are now), then it is going to be the Segment Addition Postulate.0716
If these are angles, and you go from partial angles to the whole angle, or the whole angle to its parts, then you are going to use the Angle Addition Postulate.0722
Just so you are aware of your steps...the first step I have to fill in, and the reason is "Given."0734
Remember: I am only writing this down for my given statements.0748
AB is congruent to CD, and BD is congruent to DE.0753
Now, step 2...and I like to, for my statements and reasons, keep the same statement and the same reason on a level, so that it is easy to see.0769
From step 2's statement to step 2's reason...number 2: AB = CD, and BD = DE.0784
How did I come up with this? Well, all that has changed is going from "congruent" to "equal."0796
AB = CD right here, and BD = DE.0807
So, we just went from congruence to equality; and that means that it is "definition of congruent segments."0811
You can also say "definition of congruency"; sometimes that is used, but for this, we are just going to use "definition of congruent segments."0822
Then, this next step: here, since BD is equal to DE, this was added to AB, and this is added to this.0833
AB + DE = CD + DE; so it is just added onto each of those steps.0856
If you want, if this step is a little bit confusing, you can go on to the next step first,0873
because the next step, right here, uses the Segment Addition Postulate, AB + BD = AD, and CD + DE = CE.0880
We know that step 4 is "Segment Addition Postulate"; and then, from there, you can move on to step 3.0898
Now, proofs can be a little bit different; remember how, last lesson, I gave you the analogy of driving the car from point A to point B0910
or maybe not drivingmaybe you can't drive; but if you are going from point A to point B, you have a series of steps to get there.0920
You can make a left here, and make a right here, and so on.0932
Now, usually, from point A to point B, there might be a couple of different ways that you can get there.0936
So, in the same way, with proofs, it is not always going to be the same.0943
For the most part, there is a best way to get from point A to point B.0950
But you might have a few more steps, and that just means that you took a longer route to get to point B, your destination.0955
So, it doesn't mean that it is incorrect; as long as you correctly get from point A to point B, you have a correct proof.0965
For this one, again, you can just write, or you can just have this before,0980
because that is going to help you move on to the next step, which is AB + DE, and then this.0987
So then, 3 and 4 can be switched around; but you can also think of it as AB, and then you are adding on this BD.0996
And then, you are just taking the same equation right here, and you are just including BD and DE into it.1004
You are just including it in; now, for this one, you can do this in two steps, too.1013
You can say, since BD = DE, to add BD here and add BD here, because then it would be the Addition Postulate.1023
Here, we can also say that this is the Addition Postulate, because, since they are the same, we are just adding it onto this whole equation.1040
Let's just call it the...I'm sorry, not the addition postulate; the addition property of equality.1048
And then, this right here is the Segment Addition Postulate.1060
Then, since we know that all of this equals all of thislook here, I have all of that and all of that.1067
If this equals this, then that is the same thing as this equaling this, which means that AD is going to equal CE.1079
So, my next step is going to be AD = CE, and that is the substitution property of equality, because it is just substituting all of this from step 3.1092
If step 3 is my equation, all I did was replace this part, the whole thing, with AD, and this whole thing with CE; so it is just the substitution property.1113
And then, step 6: my step 6 has to be my "prove" statement.1123
From here to here, it went from equal to congruent; so "definition of congruent segments."1131
And we are done with this proof; so then, from that, we just were able to prove that AD, the whole thing, is congruent to CE.1144
OK, setting up proofs: I want you to practice setting up a proof.1155
You don't have to actually solve the proofs yet; don't worry about the proof itself; we will start with just setting them up first.1160
If you are given a statement or a conditional (an if/then statement), then it is just setting it up, and then having a diagramhaving a drawing.1171
This first one: If two segments have equal measures, then the segments are congruent.1187
If you have two segments (let's say AB is one segment, and CD is another segment), and they are the same1196
(let's say this is 10 and this is 10), then the segments are congruent.1209
So, the "if" part, my hypothesis, is going to be my given; that is what is given to me, "if two segments have equal measures."1214
My given is going to be that...now, I am not writing it in words; I am going to write it using my actual segments that I drew out;1225
the given is that two segments have equal measures, and my two segments are AB = (because it is equal to) CD.1237
That is my given: two segments having equal measures.1249
Then, the segments are congruentI am trying to prove that the segments are congruent.1255
This is my given statement; this is my "prove" statement, because this is the conclusion.1264
So then, the conclusion is the "prove" statement; both segments are congruent, so AB is congruent to CD.1273
When you write segments, only when you are dealing with congruent segments do you write the segment up here.1285
If AB = CD, then you are proving that AB is congruent to CD.1295
And that is all you need to do for nowjust setting up the proof in this way: the given statement, the "prove" statement, and then the diagram.1299
Write the given and "prove" statements again, and draw the diagram.1311
Vertical angles are congruent: now, this one is not a conditional statementit is not written in if/then form.1314
You can rewrite it if you want; or if you are able to figure out1323
what the hypothesis and the conclusion are by looking at this, then you can just go ahead and set up.1327
Vertical angles are congruent: if I draw my diagram, I have vertical angleslet's say 1 and 2.1333
If I rewrite this as a conditional, it is "If angles are vertical," or you can say "if two angles are vertical angles, then they are congruent."1349
My given is that angles are vertical; now, I am not writing this out; I have to write it out using my diagram.1377
That means...well, in this case, I can say that angle 1 and angle 2 are vertical angles; I am using my actual angles.1388
And then, the "prove" statement is that they are congruent; that means that I have to say that angle 1 is congruent to angle 2.1404
Now, you might not use angles 1 and 2; maybe you can label these points, and then use angles that way.1420
You can use whatever angles you want; you are setting it up the way you want to set it up.1427
But just make sure: it has to be written like this; you can just label the angles differently.1431
The next one: If a segment joins the midpoints of two sides of a triangle, then its measure is half the measure of the third side.1440
We have a triangle; if a segment joins the midpoints of two sides...so then, I need midpoints.1450
A segment joins the midpoints: here is one midpoint; that is a midpoint; here is another midpoint.1470
If a segment joins those two midpoints together, that is the segment that this is referring to.1480
Then, its measure, the measure of this (let me just label this: A, B, C, D, E), DE, is half the measure of the third side.1490
So then, these were the first two sides that it was talking about; the third side would be BC.1506
They are saying that, if this segment joins the midpoint of this and the midpoint of this,1511
from the triangle, then the measure of that segment is half the measure of the third side, this unmentioned one.1518
My given is a few different things; my given can be that...1529
Well, first of all, I have a triangle; I need to mention that, so a given is my triangle ABC; that is the triangle.1536
Then, I say that D (because I need to mention its midpoints) is the midpoint of AB, and E is the midpoint of AC.1544
And then, my "prove" statement will be that its measure is half the measure of the third side.1576
So then, DE is half the measure of BC.1583
So, given my triangle ABC, where D is the midpoint of AB and E is the midpoint of AC, then I am proving that DE is half the measure of BC.1598
DE is half the measure of ("of" means "times") BC.1610
If you divide BC by 2, then that is going to be DE.1617
OK, let's do a few more examples: for these examples, you are going to justify each statement with a property from algebra or congruent segments.1623
The first one: if 2AC = 2BD, then AC = BD.1634
This onewhat did you do here? You just divided the 2 to get AC and BD, so this one is the division property.1643
Now, it is the division property of equalitywe can just write the equals sign.1652
The next one: XY is congruent to XY; remember that this is a theorem that we went over at the beginning of the lesson, using congruency.1661
For this, this one, if you remember, is the reflexive property, but it is not of equality,1674
because it is not an equals sign; it is the reflexive property of congruent segments.1688
The next one: If PQ is congruent to RS, and RS is congruent to UV, then PQ is congruent to UV.1700
See how these are the same, so PQ is congruent to UV; this is the transitive property.1710
Is it of equality? No, it is the transitive property of congruent segments.1720
The next example: Write the given statement, the "prove" statement, and the drawing to set up a proof.1731
Acute angles have measures less than 90 degrees.1745
This is an acute angle; my given statement is going to be something with being acute, so I am going to say that angle A is an acute angle.1759
Remember from this: you are drawing a diagram and labeling it, and you are going to use what you labeled in these statements.1783
The given is that angle A is an acute angle, and then, what you have to prove is that this angle measures less than 90 degrees.1792
To write this as a conditional, you can say, "If the angles are acute, then they measure less than 90 degrees," and that can be our conditional.1803
Angle A is an acute angle, and then you are proving that it measures less than 90 degrees.1837
So, you are going to say, "The measure of angle A is less than 90 degrees."1842
The next example: If two segments are perpendicular, then they form 4 right angles.1856
If I have segments AB and CD, then you are saying that if they are perpendicular, and that means that all four are right angles.1871
We know that I can write that this is a right angle, because they are perpendicular.1897
But remember how this is a linear pair; so this angle and this angle are supplementary, because linear pairs are always supplementary.1902
That is how you would prove this; but again, we don't have to prove anything right now.1918
We are just going to write what is given and what we have to prove.1922
The given statement is that these are perpendicular; AB is perpendicular to CD.1927
And then, prove that (I'll label this: 1, 2, 3, 4) angle 1, angle 2, angle 3, and angle 4 are right angles.1943
Given that AB is perpendicular to CD, because segments are perpendicular, you are proving that they form four right angles.1976
So, angles 1, 2, 3, and 4 are all right angles.1985
We are going to actually do the proof for the last example.1993
Write the twocolumn proof: Now, you can just write statements and make two columns for the statement and the reason;1999
or you can just draw out your actual twocolumn proof.2011
Then, let's look at this: GR is congruent to IL; GR, this whole thing, is congruent to this whole thing.2019
SR, this short one, is congruent to SL; I have to prove that GS is congruent to IS.2037
Now, I am not going to mark that yet, because I have to prove it.2050
It is not true yet, until I am done with my proof.2053
My statements...let's do it like that, and then here are my statements; here are my reasons.2061
Step 1 (I'll use a different color): GR is congruent to IL, and SR is congruent to SL; the reason is that it is given.2078
Step 2: Now, look at this again; whenever you deal with whole segments with its parts, we are going to use the Segment Addition Postulate.2102
You want to try to break it up into its parts.2116
GR = GS + SR, and IL = (I am just doing this to both segments) IS + SL; this is the Segment Addition Postulate.2120
Now, before I go on, notice how in step 1, I have congruent signs, and in step 2, I have equals signs.2167
For the Segment Addition Postulate, I have to make it equal, not congruent.2176
If I want to go on any further, using GR and IL or anything else, I need to change my step 1 to equals signs.2181
I could have done that for step 2, and then for step 3 done the Segment Addition Postulate;2194
but as long as you get it done before you use it, it is fine.2198
So then, in my step 3, I am going to write GR = IL, and SR = SL.2204
And then, since all I did was change it from congruent to equal, it is "definition of congruent segments."2215
Then, my next step: See how GR is right there, and then IL is right there;2228
so then, I can use this as my equation, and then substitute in the parts of GR, all of this, because that equals GR.2240
I can substitute all of that into GR, and then substitute all of this for IL.2253
I am going to make GS + SR equal to IS + SL; again, all we did was substitute in all of this stuff for GR, and substitute in all of this for IL.2261
And that is the substitution...of congruent segments?no, of equality, because it is equal.2283
Then my step 5: Since I know that...2296
Well, from here, let's go back to our previous statement: GS is congruent to IS.2304
Here is GS; here is IS; somehow, I have to get rid of SR and SL.2309
Luckily, I know that they equal each other; so I can substitute in one for the other.2317
GS +...it doesn't matter which one; I can change this to SL, or I can change this one to SR; so let me just change this to SL...equals IS + SL.2324
Step 5 (I am running out of room here): I use the substitution property.2341
Remember not to write "sub." for substitution, because that can be the subtraction property, too.2348
And then, step 6: For me to get rid of SL and SL, I just have to subtract it.2355
If I subtract, then I just get GS = IS; and that is the subtraction property of equality.2366
Now, I am almost done; right here, even though I have GS = IS, and this is GS converting to IS, I have to make it look exactly the same.2379
My last step that I am going to do is GS is congruent to IS; and that step is the definition of congruent segments.2390
And that is it; we have proved that GS is congruent to IS.2412
Remember: we used the Segment Addition Postulate, because I have this whole segment,2418
and then I need to use these parts; for parts to the whole, use the Segment Addition Postulate.2422
If you want, you can try to erase it or try not to look at this, and try working on this proof yourself.2434
Just go back and rewind it or whatever, and then just try to work on the proof on your own.2444
And then, you can come back to this and check your answers.2451
OK, well, that is it for this lesson; we will see you for the next lesson.2455
Thank you for watching Educator.com.2461
0 answers
Post by Kenneth Montfort on March 1, 2013
in previous lectures, you mentioned that if a segment doesn't have a bar over it, then we are talking about measures of segments, so wouldn't the reason be the definition of measure of AB instead of the definition of congruent segments?