1. Module 5 Lesson 2 – Part 2 Writing Proofs
Proving Triangles Congruent
(Remember to print the Learning Guide notes that go with this lesson so you can use them as you follow along.)
This powerpoint is about module 5 lesson 2 – part 2 – writing proofs. Remember to print the Learning Guide notes for this lesson so that you can use them to follow along with parts of this powerpoint.

2. You have studied proofs in earlier modules.
Remember that you always start with the given information.
MARK that information on the diagrams. This is KEY! That is the only way you will know if it is SSS or SAS, etc.
For each step in the proof, you must give a reason.
You studied proofs in earlier modules. Remember that the first step is always the given information. Before you even begin working on the proof, mark that given information in the diagrams. This is KEY in this modules because that is the only way you can determine how the triangles are congruent (if it is SSS or SAS or ASA, etc.). For each step in the proof, you must give a reason.

3. Possible reasons Given Properties Definition of _________ Theorems
Reflexive (AB = AB) (very useful when triangles share a side)
Symmetric
Transitive
Definition of _________
Definition of perpendicular lines
Definition of midpoint
Definition of segment bisector
Definition of angle bisector
Definition of congruent segments
Theorems
Vertical angles are congruent.
All right angles are congruent.
If 2 parallel lines are cut by a transversal, alternate interior angles are congruent.
If 2 parallel lines are cut by a transversal, corresponding angles are congruent.
SSS, SAS, ASA, AAS and HL
There are many possible reasons for each step in a proof. The reason for the first step is always given. Then you might use a property like those listed here. Or you might use a definition of a word and some examples are listed here. You might use a theorem and several of those that you have already learned are listed here. The tests we learned for congruent triangles (like SSS) are theorems too.

4. 1. Look at the given information and MARK it in the diagram.
How do I start a proof??
1. Look at the given information and MARK it in the diagram.
What you are given can lead to more steps in the proof (everything is given for a reason!)
For example, if you are given an angle bisector, you will have a pair of congruent angles.
You need to MARK them in the diagram and also state that the angles are equal in the proof.
Many students struggle with proofs and do not know how to start. Always look at the given information and make sure you MARK it in the diagram. Then look at the given again and see what other information you know from that. What you are given will lead to more steps in the proof. For example, if you are given an angle bisector, you know that the angles on each side of the bisector are congruent. You will have to MARK them in the diagram and then STATE that they are equal in the proof.

5. 2. Look for shared sides.
If the two triangles share a side, MARK THEM IN THE DIAGRAM and state it in the proof. Statement: OM = OM Reason: Reflexive Property
Once you have finished the given, then there are a couple more things you might see that can be useful. Be sure to look and see if the triangles share a side. If they do, MARK IT in the diagram and then that will be a statement in the proof. For example, look at this picture. The side OM is in both triangles so they share it. Put a mark on it (Or two marks) to show that it is congruent to itself. The statement in the proof will be OM = OM and the reason will be the reflexive property (since the segment is equal to itself).

6. 3. Look for vertical angles.
If you see an X in the diagram anywhere, those are vertical angles. MARK THEM IN THE DIAGRAM and state it in the proof.
Statement: <3 = <4
Reason: Vertical angles are congruent
One final thing to look for is vertical angles. Remember that vertical angles come from an X in the diagram. If you see them in the diagram you are given, mark them and then state it in the proof. So in this diagram, <3 and <4 are vertical angles. They are congruent so MARK THEM FIRST – don’t forget that or it will cause you problems later on. The statement here would be <3 =<4 and the reason would be vertical angles are congruent.

7. 4. You should be ready to finish the proof.
Look at the diagram and all the things you have marked. Now decide if it is SSS, SAS, ASA, AAS or HL.
Statement: Name the triangles (remember to make sure the letters are in the right order).
Example: ∆ABC= ∆ DEF
Reason: Look at the diagram to determine if it is SSS, SAS, ASA, AAS or HL
Whew! By now we should be able to finish the proof. Look at all the marks you have made on the diagram and decide if it is SSS or ASA, etc. Then name the triangles (remember to match up the angles in the diagram to be sure you have the correct name) and then give a reason.

8. Ready for some examples?
Let’s try some examples!

9. Given Reflexive Property SAS Right angles are congruent.
For this example, you are given that <1 and <2 are right angles and that ST = TP. Remember to mark this in the diagram. So I will put a box on <1 and <2 to show they are right angles. Then I will put one mark on ST and one on TP to show they are congruent. Now let’s try to fill in the reasons. The first step is always to write the given information so the first reason is always given. Step 2 says <1 = <2. How did we know that? Go back to the given and you will see that they are right angles. We know right angles are congruent so that is our reason. Look at step 3. It says TR = TR. Look at the diagram and you will see that the triangles share that side TR. So, make sure to mark that in the picture. The reason that TR = TR is the refexive property. Now for the last statement. Look at the picture to see which of the 5 tests work here. We have 2 sides and one angle. The angle is right in between the two sides so the answer is SAS.
SAS

10. Given Defn. of angle bisector Reflexive property SAS
The Given states that segment OM bisects angle LMN. That means that <1 = <2 so mark that in the picture. Also in the given is that LM = NM so mark those two segments. The first reason is Given. The 2nd statement is that <1 = <2 which we knew because of the word “bisects” in the given. So the reason is definition of angle bisector. Now in step 3, we see that OM = OM. We know that because they share a side. So that is the reflexive property. Be sure to mark that side in the picture. Now look at the picture again. There are two sides marked in each triangle and the angle in between them is marked. So the reason is SAS.
Reflexive property
SAS

11. Given ASA Alternate interior angles are congruent.
Vertical angles are congruent.
For this proof, we are given that VZ is parallel to WX. Be sure to mark those with arrows so that you remember they are parallel. It does not mean they are congruent so we can’t put tic marks there. Now the rest of the given says VY = XY so we will mark that in the diagram. The first reason is Given. For the 2nd statement we see that <1 = <2. How would we know that? If you connect the two parallel lines, you make a Z shape and <1 and <2 are in the corners of the Z. That means they are alternate interior angles and they are always congruent. So that is our reason for step 2. Step 3 says that <3 = <4 and those are vertical angles so that is our reason. Be sure to mark that in the diagram. Now we can look and see that the reason the triangles are congruent is ASA.
ASA

12. You try some!
There are some “You Try” problems in the learning guide notes.
Try those and then check your answers.
If you have any questions, contact your teacher so that he/she can help you.
Time for you to try some. There are some “You try” problems in the learning guide notes. Try those and then check your answers at the end of the learning guide sheet. If you have any questions, contact your teacher so that you can get some help.

13. Last idea: We know that parts of congruent triangles match up.
For example, if ∆RST = ∆FED, then we know that
RS = FE (since they are the first two letters of each)
<S = <E (since they are the middle letters of each)
TR = DF, etc.
This is called CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Last idea: We know that parts of congruent triangles match up. For example, if triangle RST = triangle FED, then we know that RS = FE. We also know that <S = <E , TR = DF, etc. This process of matching up parts once you have the congruent triangles is called CPCTC. That stands for corresponding parts of congruent triangles are congruent.

14. Last proof. This one will use CPCTC
Last proof! This one will use CPCTC. Notice how it is set up the same, but the “prove” is different. 1 2
Given
Vertical angles are congruent
SAS
Last proof. This one will use CPCTC. It looks similar to the others, but if you look at the “prove” you are proving that two angles are equal instead of the triangles as we had been doing so far. Let’s get started. First, we will mark the given information on the diagram and then put given as our reason. Next, we will see that <1 = <2 so those are vertical angles. Don’t forget to mark them in the diagram. Now we can look at the diagram and see that the reason for step 3 is SAS. The last step says that <B = <C. Notice that those are the middle letters in both of the triangles from step 3. So the letters match up and that means the reason is CPCTC.
CPCTC

15. You try one more!
There is one more “You Try” problem in the learning guide notes.
Try it (remember it will use CPCTC) and then check your answers.
If you have any questions, contact your teacher so that he/she can help you.
You try one more! There is one more “You Try” problem in the learning guide notes.
Try it (remember it will use CPCTC) and then check your answers.
If you have any questions, contact your teacher so that he/she can help you.