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For two triangles to be congruent, they have to be the exact same. Angles, sides, and naturally area. In geometry, you'll see a lot of triangle congruency problems, so this is a page full of postulates and theorems to help you prove that two triangles are congruent. Normally, "S" means side and "A" means angle.
Similar triangles have the same angles but the sides can be different.
How do we prove that two triangles are similar? If they have the same angles, then they're similar, and so when triangles are similar, their corresponding angles are the same and their sides are proportional.
If two triangles are congruent, then everything in them will be the same with the other triangle. For example, if triangle ABC is congruent to triangle DEF, then AB = DE (Picture on the left).
The congruency sign is ~~=, but all on one vertical line as shown left in red, and the similarity sign is ~.
Postulates and theorems to prove congruency:
Postulate SSS:
If all corresponding sides are the same with the two triangles, then they're congruent.
Postulate SAS:
If two sides are the same in a triangle are the same and the angle between the sides is the same as the other triangle's, then the triangle is congruent.
Postulate ASA:
If two angles and the side between them are the same with another triangle, then the two triangles are congruent.
Theorem AAS:
Postulate ASA but the side doesn't have to be between the two angles.
Theorem HL (RHL) states that if there is a right triangle, then if one pair of legs are the same and the hypothenuses are equal, then the triangles are congruent.
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