1. Proving Triangles Congruent
Module 5 Lesson 2 – Part 1
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 is part 1 of the notes on Module 5 lesson 2. These notes are about what it means for triangles to be congruent and 5 shortcuts for proving that they are congruent. It would be a good idea to print the Learning guide notes for this lesson so you can use them to follow along with the notes.

2. Triangles are congruent if they have the same shape and the same size.
Since triangles have 6 parts (3 angles and 3 sides), all 6 corresponding parts must be the same. For example,
Triangles are congruent if they have the same shape and size. All three angles must be the same and all three sides must be the same. For example, in this picture, the triangles are congruent because all the sides and all the angles will match up. So you say triangle ABC = triangle EFG.

3. The order of the letters makes a HUGE difference so be careful!
For these two triangles, we said because the letters matched up (<A <E, <B <F, <C <G) We could have also said since the letters still match up (just in a different order). But this statement would be incorrect because the letters do not match. (<C does NOT equal <F so they can’t be in the same spot.)
The order of the letters makes a huge difference so be careful. For these triangles, we can say that triangle ABC = EFG because the angles match up. <A and <E both have 1 mark. <B and <F both have 2 marks. <C and <G both have three marks. We could have also said that triangle BAC = FEG since the letters still match. But if we had said CBA = FGE, that would be wrong. <C and <F are both first in that name, but <C has 3 marks and <F only has 2 marks so they don’t match up.

4. Example
Identify the congruent triangles in the picture (in other words, write a congruency statement).
So one answer is
In this example, you want to identify the congruent triangles in the picture, which means to name them. Notice that <J is congruent to <M (they each have 1 curved mark).In the top triangle, <K is the same as <L in
the bottom triangle (since they are both right angles).Finally, <L in the top triangle is the same
as < K in the bottom triangle. SO one answer is triangle JKL = triangle MLK. Another correct answer would be triangle JLK = triangle MKL since the letters still match up but are just in a different order.
Another CORRECT answer would be since the letters still match up correctly (just in a different order).

5. Another example Find x if
Step 1: Draw the two triangles and label everything.
Step 2: Remember that <T = <F so you can fill in 5x + 45 for <T also.
Step 3: Remember the 3 angles of a triangle always add up to 180.
<R + <S + <T = 180 so
x + 45 = 180
x = 180
5x = 35
x = 7 is the answer! R S T D E F 75 25
5x+45 R S T D E F 75 25
5x+45
5x+45
In this example, we know triangle RST = triangle DEF and we know three of the angle measures. First, draw the two triangles and label everything. Be sure to put R and D in the same spot, S and E in the same spot, and T and F in the same spot. Step 2, since the triangles are congruent, <T = <F so we can put 5x + 45 in for <T also.
Step 3: Now we have all 3 angles of triangle RST so we can add all three angles together and set = 180 and then solve it.

6. 5 shortcuts for proving triangles congruent
You don’t always have to prove all 3 angles and all 3 sides are equal.
You can use one of these shortcuts:
Side-Side-Side (SSS)
Side-Angle-Side (SAS)
Angle-Side-Angle (ASA)
Angle – Angle-Side (AAS)
Hypotenuse – Leg (HL)
Fortunately, we do not always have to match all 3 angles AND all 3 sides. There are 5 shortcuts for proving that triangles are congruent. We can use one of these shortcuts: SSS, SAS, ASA, AAS, and HL. To know which one to use, you MUST look at the picture.
You MUST look at a picture to determine which one you can use!
Let’s look at an example of each one.

7. SSS – Side-Side-Side
This is the easiest one to spot! The three sides of one triangle are congruent to the three sides of the other.
The sides are either already marked like this:
OR you will be given information to use to mark them yourself.
For example, for SSS, all three sides of one triangle are congruent to all three sides of the other triangle. The sides will be marked with tic marks like this or you will be given information to use to mark them yourself.

8. SAS – Side – Angle- Side
You will need 2 sides that are congruent to side in the other triangle AND you need 1 angle.
The angle MUST be right in between the two sides like this:
For SAS, you will need two sides and the angle in between. The angle must be right in between the two sides as shown in the diagram.
Side 1
Side 2
The angle is right in between the two sides,where the two sides TOUCH!

9. ASA – Angle – Side - Angle
You will need 2 ANGLES this time and only 1 side.
BUT, the side must be right in between the two angles.
For ASA, you have two angles and one side marked. However, the side must be right in between the angles as shown in this diagram.
If you look at the side, there is one angle marked at EACH END of the side.
This is ALWAYS what ASA looks like.

10. AAS – Angle – Angle - Side
This is easily confused with ASA because there are 2 angles and 1 side marked.
For AAS, the side comes AFTER the two angles (it is NOT in between them).
AAS looks like this:
For AAS, you have two angles and one side, just like ASA. BUT the side must come after the angles (it can NOT be in between them). AAS will look like the diagrams shown here.

11. HL – Hypotenuse Leg HL only works with RIGHT triangles.
So you need 3 things:
A right angle box in each triangle
The hypotenuse of each triangle is marked the same
One of the legs is also marked the same
The last shortcut is HL which stands for hypotenuse leg. This only works for right triangles. So you need 3 things: a right angle, the hypotenuse of each triangle to be the same and one of the legs must be the same. It will look like this diagram shown here. Notice that some students think this is SSA, but there is no such shortcut.
NOTE: Some students think this looks like SSA but remember there is no such shortcut!

12. All the sides and angles must be marked before you answer the question.
You are NOT allowed to make “extra” marks on the diagrams except for 3 situations.
The following slides show an example of each situation in which you are allowed to make extra marks.
Usually, all the sides and angles will be marked for you already. You are not allowed to make “extra” marks on the diagrams except in 3 situations. The following slides show an example of each of those situations.

13. 1. You can mark Vertical angles. Example:
This looks like only 2 sides are marked BUT you can mark the vertical angles in the middle. So the answer is SAS.
You are allowed to mark vertical angles. For example, if you have this problem, it looks like the triangles are not congruent because only 2 sides are marked. But you know the vertical angles in the middle are congruent so mark them. Now you can see that the answer is SAS.

14. 2. You can mark the side if the two triangles share it
2. You can mark the side if the two triangles share it. (That is because the side is congruent to itself –the reflexive property.)
Example:
This looks like 1 side and 1 angle. BUT, they share the side in the middle so you can mark that. (We will use 2 tic marks since the other sides are already using 1 mark.)
You can also mark a side if two triangles share it. For example, if you were given this problem, you will notice that there is only one angle marked and one side. But the triangles share that side in the middle so you can mark it. Now you can see that the answer is SAS.
So the answer is really SAS.

15. 3. You can mark angles if you have parallel lines (angles like alternate interior).
Remember with parallel lines, the angles that make a Z shape are alternate interior angles and they are equal. So for this problem, The arrows mean you have parallel lines. Connect them to make a Z or backwards Z shape. The angles inside the Z are congruent so mark them. You can also mark the side in the middle because they share it. ANSWER: SAS
Finally, you can mark angles if you have parallel lines. Remember that with parallel lines, the angles that make a Z or backwards Z shape are alternate interior angles and they are equal. So if you are given this problem, the arrows mean that you have parallel lines. Connect them to make a Z or backwards Z. The angles inside the Z shape are alternate interior and they are congruent so mark them. You can also mark the side in the middle since they share it. Now that everything is marked, the answer is SAS.

16. Review SSA or AAA. There are only 5 ways to prove triangles congruent:
SSS
SAS
ASA
AAS HL
If you can’t use one of these ways, then the triangles are NOT CONGRUENT!
You are NOT allowed to use
SSA or AAA.
Those are not valid shortcuts!
Remember, there are only 5 ways to prove triangles are congruent: SSS, SAS, ASA, AAS or HL. If you can’t use one of these ways, then they are not congruent. You are NOT allowed to use SSA or AAA. These do not work for congruent triangles.