It may be beneficial to sketch a first diagram that is not accurate and re-draw it a second time to look better. 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.

For example: Using the following givens, prove that triangle ABC and CDE are congruent: C is the midpoint of AE, BE is congruent to DA. If C is the midpoint of AE, then AC must be congruent to CE because of the definition of a midpoint. This allows you prove that at least one of the sides of both of the triangles are congruent. If BE is congruent to DA then BC is congruent to CD because C is also the midpoint of AD. You now have two congruent sides. Also, because BE is congruent to DA, angle BCA is congruent to DCE because vertical angles are congruent.

Side-side-side (SSS): both triangles have three sides that equal to each other. Side-angle-side (SAS): two sides of the triangle and their included angle (the angle between the two sides) are equal in both triangles. Angle-side-angle (ASA): two angles of each triangle and their included side are equal. Angle-angle-side (AAS): two angles and a non-included side of each triangle are equal. Hypotenuse leg (HL): the hypotenuse and one leg of each triangle are equal. This only applies to right triangles. For example: Because you were able to prove that two sides with their included angle were congruent, you would use side-angle-side to prove that the triangles are congruent.

Write down what you are trying to prove as well. If you want to prove that triangle ABC is congruent to XYZ, write that at the top of your proof. This will also be the conclusion of your proof.

Some good definitions and postulates to know involve lines, angles, midpoints of a line, bisectors, alternating and interior angles, etc. 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.

Every step must be included even if it seems trivial. Read through the proof when you are done to check to see if it makes sense.