Triangle Congruence Proofs

A couple years ago I found this ACT unit for triangle proofs...and LOVED it.

I start the unit with this sheet on labeling parts of triangle and recognizing the opposite side, opposite angle, included side, and included angle. This is the one thing I actually made myself. =)

Next comes this powerpoint. The first slide gets the whole class involved, immediately. Jealous. (By the way the left two cars are mine, the top right is my sister's, and they are all totaled. But save that for the end.)

Next students have to pick which questions need to be answered in order to know if a car is totaled. They usually pick all but 2 or 3. Then I show them the Merriam Webster definition of totaled "to damage so badly that the cost of repairs exceeds the market value of the vehicle". We revisit the questions and they usually narrow it down to 4. I tell them there are exactly 2 and they can figure it out.

From there we go to triangle diagrams with all angles and all sides marked. We match up the congruent sides and angles and then write a congruence statement.

Here's where the connection comes in...this is a lot of information to give. Just like the insurance questions, what is the least amount of information we need to know to decide if two triangles are congruent? And that is the set up for learning our postulates.

Then comes one of my favorite activities (excerpt from the ACT unit).

Print the first two pages on old school transparencies and the third and fourth on paper.

Students have to match a figure on transparency one with the same figure on transparency two. It can be flipped, upside down, or backwards. Using the third page, they write the letters in corresponding order. So clever. After we review correct answers, they work on the fourth page to reinforce the idea that ORDER AND MARKINGS MATTER.

This first pages gets handed out next for students to read and make their predictions of true and false.

We put these away for later and get to another one of my favorite activities. The second and third page of the document above gives students 8 scenarios to create triangles using a variety of angles and side lengths. I created sides from straws and angles from colored paper.

Students have to build the triangles as best they can and trace it on to the paper, labeling sides with lowercase letters and opposite angles with capital letters.

When all students are done, I give them eight one-fourth pieces of a transparency (cut ahead of time!), and they number them 1-8 and trace with a dry erase marker, the ENTIRE picture.

Now we bring back our true or false predictions. As a class (with my hints), we create a 'shortcut' name for each sentence. SS, SSS, AAA, etc.

Each number on this page matches up with the number on their transparencies. So each group compares their number one drawings by stacking them on top of each other. Are the triangles congruent? If so we mark True in the Actual column. Repeat for #2-8.

I have to help a lot on these because their drawings are somewhat ridiculous. So...there's that. And we end up seeing that the only true shortcuts are SSS, AAS, ASA, SAS, and HL.

Then I hand out this and students match the correct postulates to the correct markings.

And we work through this next worksheet together, reintroducing concepts like the reflexive property, alternate interior angles, and vertical angles. We practice marking, writing congruence statements, and determining postulates.

And finally, FINALLY, we're ready for real proofs.

The first page from this packet requires them to go back to their vocab notebook or their notes and fill in some definitions. I hope that this makes them notice more things when we start doing proofs. The next couple pages are the actual proofs.

I do algebraic proofs earlier in the year so students know what they look like and that they always start with the given.

From there I really emphasize that they should mark everything on the triangle first before writing anything down. The most common things I see go wrong are they they mark angles or segments that are bisected as congruent but then don't write it out in the proof. I've seen a lot less of that this year though and I've really hit on any time a word is used in the given that definition of that word had to be a reason in their proof.

It has also helped for students to realize that when an angle is bisected, we mark the angles on the letter in the middle. So many students still don't realize the different between bisecting a segment and bisecting an angle. I don't understand...they come with their own symbol, a picture of what it is! Craziness.

From there, I do lots of different activities to practice proofs (I only do congruent triangle proofs) but I just love this setup.

Hope you found something valuable from this looooooooooong post!


  1. Elissa, I can't see most of the pictures you have posted. I will try looking at this post from home, in case for some reason the domain where you host the pictures is blocked, but I thought I should let you know, just in case it is an issue for anyone else.

    I'm really interested in seeing the pictures so I have a better sense of what you are doing in terms of preparing students for proof.

    1. David,
      They are actually embedded documents from box.com rather than pictures. Let me know if you can't see them at home.

    2. That makes sense. I will be able to see them from home, but I won't be able to see them from work, since Box.com is blocked, both in the schools I work, and in my main office.

    3. That is unfortunate. I wonder what their reasoning is? It's just a place to share files. :S

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