What Are the 5 Ways to Prove Congruence?
Understanding how to prove congruence is a fundamental skill in geometry. When two geometric figures are congruent, they have exactly the same size and shape—every side and angle matches. For triangles specifically, there are five standard methods (often taught as four, with the fifth as a special case) that mathematicians use to prove that two triangles are congruent.
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The Five Ways to Prove Triangle Congruence
1. SSS (Side-Side-Side)
If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent. No need to measure angles—when the sides match, the angles automatically match too. This is one of the most straightforward methods. For more practice with geometric proofs, explore our free math worksheets.
2. SAS (Side-Angle-Side)
If two sides and the included angle (the angle between those two sides) of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. The key is that the angle must be between the two sides—not an angle at the end.
3. ASA (Angle-Side-Angle)
If two angles and the included side (the side between those two angles) of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. This method is especially useful when you have parallel lines or transversals creating angle relationships.
4. AAS (Angle-Angle-Side)
If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, the triangles are congruent. Note: AAS works because if you know two angles, you automatically know the third (angles in a triangle sum to 180°), so it effectively becomes ASA.
5. HL (Hypotenuse-Leg) – Right Triangles Only
For right triangles specifically: if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, the triangles are congruent. This is a special case that only applies when you have a 90° angle. HL is sometimes considered the fifth method or a special case of SSS.
How to Choose the Right Method
Start by listing what you know: which sides and angles are marked as equal? If you have three sides, use SSS. If you have two sides with the angle between them, use SAS. If you have two angles with the side between them, use ASA. For right triangles with hypotenuse and leg, use HL. Check out our math resources for more geometry practice.
Common Mistakes to Avoid
Don’t use SSA or AAA—these do not prove congruence. SSA (Side-Side-Angle) can produce two different triangles. AAA (Angle-Angle-Angle) only proves similarity, not congruence. Always ensure you’re using one of the five valid methods above.
Frequently Asked Questions
Recommended Resources
What is the difference between congruence and similarity?
Congruent figures are identical in size and shape. Similar figures have the same shape but may differ in size. Congruence implies similarity, but not vice versa.
Can you use more than one method to prove the same triangles congruent?
Yes! Often multiple methods apply. Choose the one that uses the information given most directly.
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