However, I will go over this again in more detail in future geometric proof lessons. I will now show you the basics of proving (showing) that two triangles are congruent. NOTE 2: The Angle Side Side theorem (yes, we all know what it spells) does NOT necessarily work. You can draw 2 equilateral triangles that are the same shape but not the same size. NOTE 1: AAA works fine to show that triangles are the same SHAPE (similar), but does NOT work to show congruence. Remember that if we know two sides of a right triangle we know the third side anyway, so this is really just SSS. HL (hypotenuse leg) = If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. SAS (side angle side) = If two sides and the angle in between are congruent to the corresponding parts of another triangle, the triangles are congruent.ĪSA (angle side angle) = If two angles and the side in between are congruent to the corresponding parts of another triangle, the triangles are congruent.ĪAS (angle angle side) = If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
SSS (side side side) = If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. If we need to prove that two triangles are congruent, we have five different methods: We don't have to worry about proving the sides or angles are congruent. Why? Because now all we have to do is prove that two triangles are congruent. Nonetheless, these are still important facts.
If you have two identical triangles, it should be obvious that their angles are identical. Notice that each one of these properties makes common sense. (3) line segment AC is to line segment DF.(2) line segment BC is to line segment EF.(1) line segment AB is to line segment DE.See picture above.įor those same two triangles, ABC and DEF, we know the following: The corresponding congruent angles are marked with arcs. NOTE: The corresponding congruent sides are marked with small straight line segments called hash marks. (2) there are 3 sets of congruent angles.īelow we have two triangles: triangle ABC and triangle DEF.(1) there are 3 sets of congruent sides and.If two triangles are congruent, then naturally all the sides are angles are also congruent with their corresponding pair. It is important to recognize that in a congruent triangle, each part of it is also obviously congruent. That means that parts that are the same and would match up if you stacked the two figures. Two figures that are congruent have what are called corresponding sides and corresponding angles.
We use the following symbol to indicate congruence: It means not only are the two figures the same shape (~), but they have the same size (=). As written above, it means "identical in form." If you cut two identical triangles from a sheet of paper, and couldn't tell them apart based on size or shape, they would be congruent. Congruent Means "Identical In Form" The Definitionīefore we even start, let me remind you that congruent means "the same" in geometry.
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