Geometry Proofs Explanation

Remember geometry proofs? For many students, they are the basis of recurring nightmares.

Here is good news; they don’t have to be. Just do them as if you were telling a story.


Bear with me. If you were telling a story about your trip to a beach, is there any way you would mess things up like: “1. First I dried off with a towel, 2. then I dove into the water, 3. then I was wet and cold so I stepped out of the water.”?

Of course not; it isn’t logical. Well, the same applies to geometry proofs; they have to be logical. (By the way, logical thinking is precisely what geometry proofs are attempting to develop in children).

STORY METHOD: Do a Casual Proof First

STORY METHOD. Before writing all the statements and reasons, just tell yourself a story. Make little notes and marks as needed on a simplified diagram (that you can freehand or trace from the original).

The story you tell yourself might go something like this: “This angle equals that one (mark them); that angle equals the other one (mark them); this line segment equals that one (mark them); that little line segment equals itself, so when I add it to the other pieces, the results are equal (mark the bottom sides as equal); and so this triangle is congruent to that one.”

Now, add some detail to your story. Just go step by step doing what I like to call a casual proof. That means using almost no writing, just make light pencil marks and some abbreviated comments as you tell your story. Once you’ve finished the casual proof (story), now write each step, statement and reason. And, to make it easier to follow, as you go through the steps, darken the pencil marks you made on the casual diagram.

Why do it this way? Because once you start writing long, formal statements, it’s hard to concentrate on your logical overall plan. By doing a casual proof first, it is less likely that you will get lost.

And, for Pete’s sake (who the heck is Pete, anyway?), don’t write all the givens first. That’s not logical. Instead, write each one as you need it in your “story.” They don’t make logical sense all bunched up at the beginning of the proof. (The reason teachers tell students to put all the givens first, is to make sure the students don’t leave any out. That is easy to do by checking them off one-by-one as you use them).

And finally, relax in the formality when it comes to reasons. Instead of “When two straight lines intersect, they create pairs of vertical angles whose measures are congruent,” simplify the reasons. Make them short and punchy, like “All vertical angles are equal.” When you make the reasons simple and less formal, they are shorter and easier to remember.

Have fun doing proofs, after all, they are a lot like puzzles.

P.S. Do you know what geometry means? “Geo” means Earth, and “metry” means measuring. Geometry was created long ago to measure the earth, or pieces of land on the earth. Pretty cool.


  1. in geometry proofs, if you prove certain angles are equal or sides equal, does the tests for congruency in the geo. parts given have to be stated so that the theorems, axioms or postulates tie into proving the original statement is true. I guess what I’m asking is do you have a step by step format you follow in proving the given theorem?

    • Dr. Al Silbert says:

      I’ll answer this using triangles so it’s easy to follow. However, the process is similar for any proof. Your teacher has shown you (or proved to you) the various ways to prove two triangles congruent. They include ASA, SAS, SSS, SAA, and Hypotenuse Leg. Be assured that you don’t know which one to use until you’ve gotten pretty far into the proof. So, just start.

      You start with any one of the “givens” and, if appropriate, apply what you know to move you forward. Next, introduce another “given” and do the same. Then another. As you mark up your diagram doing a “casual proof” you’ll soon see which of the statements you need to use to prove the two triangles congruent.

      For example, if you see that you have two sides and the included angle in one triangle equal respectively that in the other triangle, you use SAS=SAS to prove the triangles congruent. If all you’re asked to do is prove the triangles congruent, ta-da!, you’re done. All you have to do is to write it up following your causal proof.

      However, if the item asks you to prove something like the third side in one triangle equals that in the other triangle, after proving the triangles congruent (meaning every part in one triangle is CONGRUENT to the corresponding part in the other,) you can now say those sides are congruent because of CPCTC. I suggest you NOT abbreviate this. Instead, write (or at least say to yourself) “Corresponding Parts of Congruent Triangles are Congruent.”

      It’ll help you if you really understand what that means: ” corresponding parts (parts that play the same role in both triangles) of congruent triangles (triangles you just proved congruent) are congruent (meaning, exactly the same).

      I think this is the “format” you referred to in your comment. I hope this helps.

  2. hello im recently on the triangle proofs in geometry. my teacher julie hatmaker is an amazing teacher and teaches very well im having trouble with the statements and reason for example: Given:< PSU is congruent to <PTR SU congruent to TR Prove SP congruent to TP. this is the problem to an overlapping triangle i understand how to break it apart but my problem to solving it is hard the tick marks we use and how to simplify it is extremely difficult to me. Im in need of help

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