# data[].attributes.body
I included these learning targets because for studying for this test and the final, I have specific examples and resources I can use. On the quiz for this chapter, I did not do so well, but that was because I did not separate each of the properties and definitions from each other and so I have improved through these writing these learning targets and writing down all of them on a sheet to keep track of them. From writing down the definitions, it became easier to apply them in proofs and with the help of these sources, I have a greater understanding. I am now more confident in my abilities to apply them and for some of them such as complementary and supplementary angles (I was able to understand this right away), I still have a great understanding of this.

# data[].attributes.headline
LT #5 - I can prove angle congruent by using complementary and supplementary angle theorems

# data[].attributes.subject
LT #5 - I can prove angle congruent by using complementary and supplementary angle theorems

# data[].attributes.body
For problems involving complementary and supplementary angles, there equations that must be used. For finding the complement, 90-x=m< must be used and for the supplement, 180-x=m must be used (x=the angle). In #21 and #22, these are used to find either the complement or supplement along with other things such as 4 times the complement. These problems are a good examples because it makes me apply what I already know and put it into an algebraic equation to figure out what the comp/supp angle is. In proofs it can be useful to use these theorems to prove something is a right or straight angle.

# data[].attributes.body
Opposite rays are two collinear rays that share a common endpoint and extend in different directions. They must follow all of those characteristics or else they cannot be opposite rays. This section from our notes does a nice job explaining opposite rays and lists different characteristics as to why the examples are not opposite rays. I can use this information and apply it in proofs when looking at diagrams (because it can't be used as a step in proofs).

# data[].attributes.headline
LT #8 - I can apply the multiplication and division properties of segments and angles

# data[].attributes.subject
LT #8 - I can apply the multiplication and division properties of segments and angles

# data[].attributes.body
The multiplication property is: if angles are congruent, then their like doubles/triples are congruent. The division property is: if angles are congruent, then their like halves/thirds are congruent. The multiplication property starts small and ends big; division property is the opposite. This is used with bisectors and trisectors. This source helped me because the examples cleared confusion and it went over the fundamentals which are essential to understanding how to do more complex problems.

# data[].attributes.attachment_link.readable_text
Using Theorems of Like Multiples and Like Divisions in Proofs

By Mark Ryan

The Multiplication and division theorems are based on very simple ideas, but they do trip people up from time to time, so pay careful attention to how these theorems are used in the example proofs.

Like Multiples: If two segments (or angles) are congruent, then their like multiples are congruent. For example, if you have two congruent angles, then three times one will equal three times the other.

Like Divisions: If two segments (or angles) are congruent, then their like divi sions are congruent. If you have, say, two congruent segments, then 1/4 of one equals 1/4 of the other, or 1/10 of one equals 1/10 of the other, and so on.

Look at the above figure.

Those congruencies follow from the definition of bisect.

People sometimes get the Like Multiples and Like Divisions Theorems mixed up. Here's a tip that'll help you keep them straight: In a proof, you use the Like Multiples Theorem when you use congruent small segments (or angles) to conclude that two big segments (or angles) are congruent. You use the Like Divisions Theorem when you use congruent big things to conclude that two small things are congruent. In short, Like Multiples takes you from small to big; Like Divisions takes you from big to small.

When you look at the givens in a proof and you see one of the terms midpoint, bisect, or trisect mentioned twice, then you'll probably use either the Like Multiples Theorem or the Like Divisions Theorem. But if the term is used only once, you'll likely use the definition of that term instead.

You see how to use the Like Multiples Theorem in the next proof.

Game plan: Here's how your thought process for this proof might go: Ask yourself how you can use the givens. In this proof, can you see what you can deduce from the two pairs of congruent angles in the given? If not, make up arbitrary measures for the angles.

Then, when you see trisect mentioned twice in the other givens, that should ring a bell and make you think Like Multiples or Like Divisions.
Statement 1:
Reason for statement 1: Given.
Statement 2:
Reason for statement 2: Given.
Statement 3:
Reason for statement 3: If two congruent angles are subtracted from two other congruent angles, then the differences are congruent.

Statement 4:

Reason for statement 4: Given.

Reason for statement 5: Given.

Reason for statement 6: If two angles are congruent (angles EHN and JMI), then their like multiples are congruent (three times one equals three times the other).

Now for a proof that uses Like Divisions:

Here's a possible game plan: What can you do with the first given? If you can't figure that out right away, make up lengths for line segments ND, EL, and DE. Say that line segments ND and EL are both 12 and that line segment DE is 6. That would make both line segments NE and DL 18 units long. Then, because both of these segments are bisected by their midpoints, line segments NO and AL must both be 9. That's a wrap.
Statement 1:
Reason for statement 1: Given.
Statement 2:
Reason for statement 2: If a segment is added to two congruent segments, then the sums are congruent.

Reason for statement 3: Given.

Reason for statement 4: If two segments are congruent (line segments NE and DL), then their like divisions are congruent (half of one equals half of the other).

The Like Divisions Theorem is particularly easy to get confused with the definitions of midpoint, bisect, and trisect, so remember this: Use the definition of midpoint, bisect, or trisect when you want to show that parts of one bisected or trisected segment or angle are equal to each other. Use the Like Divisions Theorem when two objects are bisected or trisected (like line segments NE and DL in the preceding proof) and you want to show that a part of one (line segment NO) is equal to a part of the other (line segment AL).

# data[].attributes.attachment_link.title
Using Theorems of Like Multiples and Like Divisions in Proofs - dummies

# data[].attributes.attachment_link.description
The Multiplication and division theorems are based on very simple ideas, but they do trip people up from time to time, so pay careful attention to how these theorems are used in the example proofs. Like Multiples: If two segments (or angles) are congruent, then their like multiples are congruent.

# data[].attributes.headline
LT #10 - I can apply the Substitution Property

# data[].attributes.subject
LT #10 - I can apply the Substitution Property

# data[].attributes.body
The substitution property includes the transitive property, which is a special case. For a proof, this is the statement you can use in the reasoning column (it is okay to put "substitution"): If two angles are congruent, then they are complementary to the same angle. This property can be used in algebra and in geometry, but in geometry it is used when variables are not present and when sums/equations are referenced (not explicitly too). It can be used when you are trying to prove an angle is supplementary/complementary to another angle and both of those angles are congruent to the same angle. This source is useful because it shows examples and explains them. It also distinguishes between the transitive and substitution properties and when to use them.

# data[].attributes.attachment_link.readable_text
By Mark Ryan

You're probably already familiar with the Transitive Property and the Substitution Property from algebra. If a = b and b = c, then a = c, right? That's transitivity. And if a = b and b < c, then a < c. That's substitution. Easy enough. Below, you see these theorems in greater detail:

Transitive Property (for three segments or angles): If two segments (or angles) are each congruent to a third segment (or angle), then they're congruent to each other.

The Transitive Property for three things is illustrated in the above figure.

Transitive Property (for four segments or angles): If two segments (or angles) are congruent to congruent segments (or angles), then they're congruent to each other.

The Transitive Property for four things is illustrated in the below figure.

Substitution Property: If two geometric objects (segments, angles, triangles, or whatever) are congruent and you have a statement involving one of them, you can pull the switcheroo and replace the one with the other. (Note that you will not be able to find the term "switcheroo" in your geometry glossary.)

A figure isn't especially helpful for this property, so one isn't included here.

To avoid getting the Transitive and Substitution Properties mixed up, just follow these guidelines:

Check out this TGIF rectangle proof, which deals with angles:

-1 @ -2

No need for a game plan here because the proof is so short - take a look:
Statement 1:
Reason for statement 1: Given.

Reason for statement 2: If two angles form a right angle, then they're complementary (definition of complementary).

Reason for statement 3: Given.

Reason for statement 4: Substitution Property (statements 2 and 3; angle 2 replaces angle 1).

And for the final segment of the program, here's a related proof, OSIM (Oh Shoot, It's Monday):

This is another incredibly short proof that doesn't call for a game plan.
Statement 1:
Reason for statement 1: Given.

Reason for statement 2: A midpoint divides a segment into two congruent segments.

Reason for statement 3: Given.

Reason for statement 4: Transitive Property (for four segments; statements 2 and 3).

# data[].attributes.attachment_link.title
The Transitive and Substitution Properties - dummies

# data[].attributes.attachment_link.description
You're probably already familiar with the Transitive Property and the Substitution Property from algebra. If a = b and b = c, then a = c, right? That's transitivity. And if a = b and b < c, then a < c. That's substitution. Easy enough.

# data[].attributes.headline
LT #7 - I can apply the subtraction properties of segments and angles

# data[].attributes.subject
LT #7 - I can apply the subtraction properties of segments and angles

# data[].attributes.body
The subtraction property is: if coungruent angles/segments are subtracted from congruent angles/segments, then their differences are congruent. The reflexive property (where an angle is congruent to itself) can come into play here to help prove how two angles are congruent. This source is useful because it has problems that were beneficial to my learning and helped clear my confusion.

# data[].attributes.attachment_link.readable_text
By Mark Ryan

There are four subtraction theorems you can use in geometry proofs: two are for segments and two are for angles. Each of these corresponds to one of the addition theorems.

Here are the subtraction theorems for three segments and three angles (abbreviated as segment subtraction, angle subtraction, or just subtraction):

Segment subtraction (three total segments): If a segment is subtracted from two congruent segments, then the differences are congruent.

Angle subtraction (three total angles): If an angle is subtracted from two congruent angles, then the differences are congruent.

Check out the above figure, which provides the visual aids for these two theorems.

Last but not least, here are the subtraction theorems for four segments and for four angles (abbreviated just like the subtraction theorems for three things):

Segment subtraction (four total segments): If two congruent segments are subtracted from two other congruent segments, then the differences are congruent.

Angle subtraction (four total angles): If two congruent angles are subtracted from two other congruent angles, then the differences are congruent.

The above figure illustrates these two theorems.

Before reading the formal solution of the next proof, try to think through your own game plan or commonsense argument about why the prove statement has to be true.

Statement 1:
Reason for statement 1: Given.

Reason for statement 2: If an angle (angle QUR) is subtracted from two congruent angles (angles PUR and SUQ), then the differences are congruent.
Statement 3:
Reason for statement 3: Given.
Statement 4:
Reason for statement 4: If a ray bisects an angle, then it divides it into two congruent angles (definition of bisect).

Reason for statement 5: If two congruent angles (the angles from statement 2) are added to two other congruent angles (the ones from statement 4), then the sums are congruent.

# data[].attributes.attachment_link.title
Using Subtraction Theorems in Proofs - dummies

# data[].attributes.attachment_link.description
There are four subtraction theorems you can use in geometry proofs: two are for segments and two are for angles. Each of these corresponds to one of the addition theorems. Here are the subtraction theorems for three segments and three angles (abbreviated as segment subtraction, angle subtraction, or just subtraction): Segment subtraction (three total segments): ...

# data[].attributes.headline
LT #6 - I can apply the addition properties of segments and angles

# data[].attributes.subject
LT #6 - I can apply the addition properties of segments and angles

# data[].attributes.body
The addition property theorem is: if congruent angles/segments are added to congruent angles/segments, then their sums are congruent. If you wanted to figure out if an angle was congruent to another one and you had parts connected through the given, you could possibly use this property (as long as it is not multiplication/division/subtraction). This source is useful because it helps you plan out your proof and gives different examples that were beneficial to my learning.

# data[].attributes.attachment_link.readable_text
By Mark Ryan

There are four addition theorems: two for segments and two for angles. They are used frequently in proofs.

Use the following two addition theorems for proofs involving three segments or three angles:

Segment addition (three total segments): If a segment is added to two congruent segments, then the sums are congruent.

Angle addition (three total angles): If an angle is added to two congruent angles, then the sums are congruent.

After you're comfortable with proofs and know your theorems well, you can abbreviate these theorems as segment addition or angle addition or simply addition; however, when you're starting out, writing the theorems out in full is a good idea.

The above figure shows you how these two theorems work.

In other words, 8 + 2 = 8 + 2. Extraordinary!

Brilliant!

Note: In proofs, you won't be given segment lengths and angle measures like the ones in the above figure. They are added to the figure so you can more easily see what's going on.

As you come across different theorems, look carefully at any figures that accompany them. The figures show the logic of the theorems in a visual way that can help you remember the wording of the theorems. Try quizzing yourself by reading a theorem and seeing whether you can draw the figure or by looking at a figure and trying to state the theorem.

Use these addition theorems for proofs involving four segments or four angles (also abbreviated as segment addition, angle addition, or just addition):

Segment addition (four total segments): If two congruent segments are added to two other congruent segments, then the sums are congruent.

Angle addition (four total angles): If two congruent angles are added to two other congruent angles, then the sums are congruent.

Check out the above figure, which illustrates these theorems.

Now for a proof that uses segment addition:

You will find what amounts to a game plan for this proof inside the following solution, between the numbered lines.
Statement 1:
Reason for statement 1: Given.
Statement 2:
Reason for statement 2: Given.

You probably know what comes next: Statement 3 has to use one or both of the givens. To see how you can use the four segments from the givens, make up arbitrary lengths for the segments:

So now you've got line 3.
Statement 3:
Reason for statement 3: If two congruent segments are added to two other congruent segments, then the sums are congruent.

This is the three-segment version of segment addition, and that's a wrap.
Statement 4:
Reason for statement 4: If a segment is added to two congruent segments, then the sums are congruent.

By the way, did you see the other way of doing this proof? It uses the three-segment addition theorem in line 3 and the four-segment addition theorem in line 4.

Before looking at the next example, check out these two tips - they're huge! They can often make a tricky problem much easier and get you unstuck when you're stuck:

Use every given. You have to do something with every given in a proof. So if you're not sure how to do a proof, don't give up until you've asked yourself, "Why did they give me this given?" for every single one of the givens. If you then write down what follows from each given (even if you don't know how that information will help you), you might see how to proceed. You may have a geometry teacher who likes to throw you the occasional curveball, but in geometry books, the authors generally don't give you irrelevant givens. And that means that every given is a built-in hint.

Work backward. Thinking about how a proof will end - what the last and second-to-last lines will look like - is often very helpful. In some proofs, you may be able to work backward from the final statement to the second-to-last statement and then to the third-to-last statement and maybe even to the fourth-to-last. This makes the proof easier to finish because you no longer have to "see" all the way from the given to the prove statement. The proof has, in a sense, been shortened. You can use this process when you get stuck somewhere in the middle of a proof, or sometimes it's a good thing to try as you begin to tackle a proof.

The following proof shows how you use angle addition:

This proof includes a partial game plan that deals with the part of the proof where people might get stuck. The only ideas missing from this game plan are the things (which you see in lines 2 and 4) that follow immediately from the two givens.
Statement 1:
Reason for statement 1: Given. (Why would they tell you this? See statement 2.)
Statement 2:
Reason for statement 2: If an angle is bisected, then it's divided into two congruent angles (definition of bisect).
Statement 3:
Reason for statement 3: Given. (And why would they tell you that?)
Statement 4:
Reason for statement 4: If an angle is trisected, then it's divided into three congruent angles (definition of trisect).

Say you're stuck here. Try jumping to the end of the proof and working backward. You know that the final statement must be the prove conclusion,

Now ask yourself what you'd need to know in order to draw that final conclusion. To conclude that a ray bisects an angle, you need to know that the ray cuts the angle into two equal angles. So the second-to-last statement must be

And how do you deduce that? Well, with angle addition. The congruent angles from statements 2 and 4 add up to

That does it.
Statement 5:
Reason for statement 5: If two congruent angles are added to two other congruent angles, then the sums are congruent.

Reason for statement 6: If a ray divides an angle into two congruent angles, then it bisects the angle (definition of bisect).

# data[].attributes.attachment_link.title
Using Addition Theorems in Proofs - dummies

# data[].attributes.attachment_link.description
There are four addition theorems: two for segments and two for angles. They are used frequently in proofs. Use the following two addition theorems for proofs involving three segments or three angles: Segment addition (three total segments): If a segment is added to two congruent segments, then the sums are congruent.

# data[].attributes.headline
LT #9 - I can apply the transitive properties of angles and segments

# data[].attributes.subject
LT #9 - I can apply the transitive properties of angles and segments

# data[].attributes.body
The transitive property is a special case of the substitution property. It can be used to show congruence of angles or segments. For a proof in the reasoning column you can use this: If two angles are congruent to the same (or congruent) angles (or segments), then they're congruent to each other. In problem five, I used that in my final reasoning to explain how angles two and three are congruent to each other.

# data[].attributes.headline
LT #1 - I can recognize the need for clarity and concision in proofs

# data[].attributes.subject
LT #1 - I can recognize the need for clarity and concision in proofs

# data[].attributes.body
Proofs need to be clear because now as we have learned so many properties, it is important to know them to distinguish between them. You need to know when one is acceptable to use and when it is not. There are many clues in the problems when which property should be used (bisection/trisection indicates multiplication/division properties and not subtraction/addition properties). Also if the wording is off, it can be misinterpreted and points can be taken off. Here are five problems (#3-7) that include proofs. In #5, I should have used the division property and not the multiplication property to show that the segments are congruent. This homework is helpful because each of these problems requires being precise in order to prove what it is asking.

# data[].attributes.headline
LT #2 - I understand the concept of perpendicularity

# data[].attributes.subject
LT #2 - I understand the concept of perpendicularity

# data[].attributes.body
Perpendicularity is when two lines intersect at right angles. In a proof in the reasoning column you would say: if two lines (rays or segments) intersect at right angles, then the lines are perpendicular. This does not work for saying angles are perpendicular to each other because segments/rays/lines must intersect at right angles. Perpendicularity should not be assumed. In this homework, perpendicular segments are mentioned in #10. You can use the given that two perpendicular segments to go to the next statement that the angle is a right angle (because of the definition of perpendicular). This homework is helpful because I know when to apply perpendicularity and when I can't.

# data[].attributes.headline
LT #3 - I can recognize complementary and supplementary angles

# data[].attributes.subject
LT #3 - I can recognize complementary and supplementary angles

# data[].attributes.body
Complementary angles are two angles that sum to 90 degrees or a right angle. Supplementary angles are two angles that sum to 180 degrees or a straight angle. This video is helpful because it goes over the fundamentals and it is key to have strong fundamentals so you can build off of that and apply it to problems. He gives information on how to distinguish between complementary and supplementary angles and then he shows a diagram and explains why some angles are complementary/supplementary to each other and why others are not.

# data[].attributes.attachment_link.readable_text
Supplementary angles are two angles whose sum is 180 degrees while complementary angles are two angles whose sum is 90 degrees. Supplementary and complementary angles do not have to be adjacent (sharing a vertex and side, or next to), but they can be.

Two concepts that are related but not the same are supplementary angles and complementary angles. The difference is their sum.
Supplementary angles are two angles whose measures sum to a 180 degrees and complementary are the sum have to add up to 90 degrees. And I noted here that these do not have to be adjacent. So supplementary angles could be adjacent so if I had angles one and two those two would be supplementary. But I could also say if we had some angle here that we said three and let's say 3 was equal to 60 degrees and I had some other angle over here, let's say angle four was equal to 120 degrees, I could say that these two angles three and four are supplementary because they sum to 180 degrees.
The same is true for complementary angles. Let's look at a specific example where you might be asked to identify supplementary angles and complementary angles. Here we have five angles; 1, 2, 3, 4 and 5 and we're told that this angle 3 is 90 degrees, now one thing that you can assume is that 1, 2 and 3 are all linear, so if you add up 1, 2 and 3 it would be 180 degrees, which means that 1 and 2 must also sum to 90 degrees so I could label this as a right-angle. So complementary angles could be angles 1 and 2. So I could say angle 1 and angle 2.
Now, a supplementary pair could be angle 4 and angle 5 which are adjacent and they are linear. So notice that for a supplementary and for complementary you can't say that five angles are complementary but we're always talking about pairs or two's. So remember that when you're trying to evaluate your problems that supplementary sum to 180 degrees or they're linear and complementary angles sum to 90 degrees.

# data[].attributes.attachment_link.title
Supplementary and Complementary Angles - Concept - Geometry Video by Brightstorm

# data[].attributes.attachment_link.description
Supplementary angles are two angles whose sum is 180 degrees while complementary angles are two angles whose sum is 90 degrees. Supplementary and complementary angles do not have to be adjacent (sharing a vertex and side, or next to), but they can be.