How to Write a Congruent Triangles Geometry Proof

Reviewed by Joseph Meyer

Last Updated: December 15, 2023 Fact Checked

Congruent triangles are triangles that are identical to each other, having three equal sides and three equal angles.^{[1]
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Research source} Writing a proof to prove that two triangles are congruent is an essential skill in geometry. Since the process depends upon the specific problem and givens, you rarely follow exactly the same process. This can be frustrating; however, there is an overall pattern to solving geometric proofs and there are specific guidelines for proving that triangles are congruent. Once you know them, you’ll be able to prove them on your own with ease.

Part 1

Part 1 of 2:

Proving Congruent Triangles

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1
Draw a diagram. A diagram may already be provided, but if one is not, it’s essential to draw one. Try to draw it as accurately as you can. Include all of the given information in your diagram. If two sides or angles are congruent (equal), mark them as such.^{[2]
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- 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.
2
Identify the known information. Using the givens and your knowledge of geometry, you can start to prove some things and determine if any sides and/or angles of two triangles are congruent. Think about the parts of the proof logically and determine step-by-step how to get from the givens to the final conclusion.^{[3]
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- 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.
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3
Choose the correct theorem to prove congruency. There are five theorems that can be used to prove that triangles are congruent. Once you have identified all of the information you can from the given information, you can figure out which theorem will allow you to prove the triangles are congruent.^{[4]
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- 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.

Part 2

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Writing a Proof

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The most common way to set up a geometry proof is with a two-column proof. Write the statement on one side and the reason on the other side. Every statement given must have a reason proving its truth. The reasons include it was given from the problem or geometry definitions, postulates, and theorems.
2
Write down the givens. The easiest step in the proof is to write down the givens. Write the statement and then under the reason column, simply write given. You can start the proof with all of the givens or add them in as they make sense within the proof.
- 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.
3
Use the appropriate theorems, definitions, and postulates as reasons. When developing a proof, you need a solid foundation in geometry before you can begin. Knowing the relevant theorems, definitions, and postulates is essential. A working knowledge of these will help you to find reasons for your proof.^{[5]
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- 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.
4
Order the proof logically. When constructing a proof, you want to think through it logically. Try to order all of your steps so that they naturally follow each other. Sometimes it helps to work the problem backwards: start with the conclusion and work your way back to the first step.^{[6]
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- 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.

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In s-s-s, are the 3 sides congruent?

Donagan

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Yes, you can prove congruency if you can show that each of the three sides of a triangle is congruent (equal in length) respectively to a side of the other triangle.

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My teacher will never give marks if I follow these steps. He just wants exactly the same written in classwork. If I solve at least half, and it's correct, teachers are supposed to give marks but our teacher will give a 0. What do I do?

Donagan

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Give your teacher what s/he wants. You won't have to put up with that forever.

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What do I write if all three sides are not congruent when doing a geometry proof?

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It will always be a congruent if you are to prove any (angle/Side) provided you take the right triangle.

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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."

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