Writing math proofs can be the hardest part of geometry, but they are essential. Every geometry book has a proofs section, and congruent triangle proofs take up a large portion. Since the process may depend on your triangles and your givens, you rarely follow exactly the same process, which can get very frustrating. But there is an overall pattern, and following these guidelines will help.
1Set up a two-column proof format. The the left-hand column should be labeled "Statements" and the right-hand column labeled "Reasons."Ad
2Copy the diagram onto your paper, especially if the teacher requires you to do so.
3List your givens as the first step in your statements column. Under the reasons column, you should write "given."
4Translate the givens into useful information. For each given, you'll want to think what you could do with that knowledge. At this point in the proof, most of your reasons will be definitions.
- If the one of the givens is "Point C is the midpoint of AB," your statement would read "AC = CB" and your reason would be "Definition of midpoint," or the full theorem: If a segment has a midpoint, then it divides the segment into two congruent parts.
- If segment XY bisects segment NM, and P is the point where XY intersects NM, your statement would read "NP = PM" and your reason would be "Definition of bisector," or the full theorem: If a segment is bisected, then it is divided into two congruent parts.
- If the givens say "KL is perpendicular to HJ," your statement would be "Angle KLH is a right angle." Your reason would then be: If two perpendicular lines intersect, then they form right angles. After this, you would reference the step that says "KL is perpendicular to HJ," and write the number in parentheses next to the reason. The letters will depend on the diagram, which is one reason why the diagram is important.
5Look for any isosceles triangles in the diagram based on your givens. Isosceles triangles are triangles that have two legs of equal length. Write the two legs are congruent as a statement citing "Definition of isosceles" as the reason.
6Look for parallel lines. If a third line runs through both of them, it will form alternate interior angles and corresponding angles. These are quite common in triangle proofs. The overall shape formed by the angles looks like a the letter "z."
- Write in your statements column that those angles are congruent to each other with the reason being "Alternate interior angles are congruent" or "Corresponding angles are congruent."
- Do not use reasons such as "Definition of parallel" or "Definition of alternate interior angles." These are not valid proof reasons.
7Look in the diagram for vertical angles. Vertical angles will form an X and are the angles across from each other, touching at the center but not along their sides. These vertical angles are congruent. An example would be to use the statement "Angle P is congruent to angle R" with the reason "Vertical angles are congruent."
8Look in the diagram for any lines or angles shared by triangles. This is where the reflexive property comes into play.
- Your statement should read "XY = XY" or "Angle A = angle A" and the reason would be "Reflexive property of congruence."
9Transfer every congruency statement you've found so far, including the givens, into the diagram. Congruent sides get marked with hash marks; congruent angles are marked with arcs.
10Look at one of the triangles in the picture. Note all the marking you just made and the ones that were given to you. How many angles are congruent? How many sides? Match this information with one of the triangle congruency theorems. Your statement should read "Triangle ABC is congruent to triangle XYZ" and the reason would be the appropriate choice between AAS, ASA, SAS, SSS, HL, etc.
- SSS stands for side, side, side. All three sides must be congruent.
- Two sides and an angle must be congruent for it to be SAS (side, angle, side).
- See if your triangles have congruent right angles, and two congruent sides, one being the hypotenuse (which is the side directly across from the right angle). If they do, then state them as being congruent with the reason HL (Hypotenuse Leg Theorem).
- It can be ASA or AAS if two angles are congruent, but only one side.
11Take another look at the "Prove" line of your problem. If it's the same as your last statement, then you are finished.Ad
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- If your diagram has two overlapping triangles, try redrawing them as separate triangles. It will be much easier to find and mark the congruent pieces.
- If your diagram does not have two triangles, you might have a different kind of proof. Double check to make sure the problem asks you to prove congruency of two triangles.
- If your givens include the word "perpendicular," do not say that an angle is 90 degrees due to definition of perpendicular lines. Instead, write a statement saying such angle is a right angle because of "definition of perpendicular lines and then write another statement saying said angle is 90 degrees because of "definition of right angle."
- You cannot prove a theorem with itself. If you're trying to prove that base angles are congruent, you won't be able to use "Base angles are congruent" as a reason anywhere in your proof.