## Oh geometry proofs–how can you conquer these with a winning game plan? You’re given facts and a goal (something to prove), so you must figure out the step-by-step path supported by evidence. Follow these steps to solve any geometry proof more easily.

[2 minute read]

## 1. Know the rules of the game

My geometry teacher, Coach Pete, was also the head of our football team. He always told us, “You need a game plan!” You also need to know the rules that you’ll use.

• Do what we did and make your own playbook of each property, postulate, theorem, and definition.

• Write the relevant topic (such as Right Triangles) at the top of each page so you can quickly flip to the topic you need.

• Add simple drawings illustrating the rules–and use colored pencils, highlighters, or pens to indicate important pieces such as line segments and angles.

• Before you start practicing new problems, see if you can write the important rules from memory. Do your best! Then review your playbook to see how well you did. Always start with this kind of review to remind yourself of what you already know.

## 2. Label your drawing well

If the diagram is provided, great, label it carefully. This is where color helps make the shapes and rules easier to see and understand. By writing on paper, you’ll trigger other ideas and associations much faster than just staring and thinking about the proof.

• If you aren’t given a diagram, create your own.

• Draw large enough so you can clearly label all the detailed info.

• Label points with letters and make sure they match what you’re given.

## 3. Write the Theorem statement that needs to be proved

This statement is what you’ll prove–but it might not be written for you in the problem. That’s quite okay because you can create it by writing what must be proven in plain English.

• Write in plain English off to the side, not within your proof table. This will help you realize what your goal really is.

• Think in plain English about how you might arrive at the conclusion before you think in terms of math and symbols.

• Make up simple numbers for segments and angles to test them out. Do simple math operations (add, subtract, multiply, etc.) to better understand how the pieces work together.

## 4. Write your final goal

Now you’re ready to write in your two-column table. The last line in your Statement column will match the prove statement.

• Include the info from the prove statement in your drawing just like you did with the givens.

• If you ever feel stuck, jump to the end of the proof and work backwards, or upwards in this case. Guess the possible reason for the conclusion. Then see if you can guess the penultimate (next-to-last) statement that would lead to the conclusion.

## 5. State the given(s)

The given(s) have the facts you’ll use as statements. No irrelevant givens will appear, so ask yourself, why is each one provided?

• They’re given to you for a reason so expect to use each one.

• You may not end up using *every given in the *beginning of your proof. The givens will fit wherever they make the most sense.

• Try writing each given in the Statement column with plenty of space in between each one. Then write another statement that reasonably follows from the given. Even if you’re unsure how they’ll all come together, you’ll start seeing the path from one statement to another. Think of this as your rough draft–and it might work!

## 6. Fill in the logically deduced statements

Now you’re ready to go step-by-step from the first statement through the geometry rules from your playbook: properties, definitions, postulations, and other theorems.

• Every single step in this logical chain must be written, even if it seems super obvious to you. Imagine you’re talking to a computer that will only understand each little step if you spell it out simply and precisely.

• Check one reason at a time. The idea in the **IF** phrase must appear **above** in your Statement column.

• The idea in the **THEN** phrase should appear in the Statement column in the **same** line.

## Bonus Tips!

• Look for parallel lines. If you find some, you may use one or more of the parallel-line theorems.

• Look for congruent triangles. If you’ve learned CPCTC, see if you can use it. It’s usually used on the line right after you prove that two triangles are congruent.

• Look for isosceles triangles. If you find one, you may use the If Sides Then Angles or If Angles Then Sides theorems.

• With circles, look for radii. Draw new radii to important points on the circle where other curves or lines touch. Mark all radii congruent: they’re all the same length.

Make a good game plan, write neatly, and you’re bound to reach your goal more easily. Coach Pete would definitely be proud of your winning strategy!

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If this helped you, can you please share with a friend? Let me know what’s working for you with your proofs. As an academic coach, I greatly appreciate your feedback and comments.

Hi Dan – thanks for commenting. I looked up Ptolemy’s theorem, which is an historic relation between the four sides and two diagonals of a special type of quadrilateral — one whose vertices lie on a common circle.

Have you found the English statement of the converse of the theorem? From Wikipedia (which is often reliable for math and the natural sciences),

You mentioned you have two pairs of sides of a pair of triangles (within the quadrilateral) proportional. It sounds like you’re close to the end, since those sides of the triangles may be the diagonals or the opposite sides of the quadrilateral. Without seeing your diagram, I’d suggest visiting this thread on the Math Stackexchange, which includes diagrams and thorough step-by-step responses:

math.stackexchange.com/questions/

Hi Jess, what types of proofs or geometry problems have you been working on? Have you found any similar diagrams by doing an Internet search for Images, instead of websites? Oftentimes, my students and I are able to find the same exact diagram, or similar problem, by running an Image search for something such as “triangle SAS proof.”

I’m working on a proof of the converse of ptolemys Theorem. I’m stuck at a point of having two pair of sides of a pair of triangles proportional.

i am so lost

Thanks for your comment – if you apply this general method to your proofs, see how they become easier to complete. If you aren’t sure about your solution, have you ever searched for the problem or similar problem on sites such as Yahoo Answers or Quora? Oftentimes, the same problem has been asked by another student and someone else has posted an answer. This way, you don’t need to actually ask your question and wait for a reply. Let me know if this works for you!

Really nice tips.They might help me