Geometry Help: Conditions of Congruence with Triangles

The geometry student will be expected to be able to understand the concepts of congruence with triangles. Two triangles are considered congruent (that is, geometrically identical) if any of the following four conditions stated below are met. The reasons why having these conditions met proves that two triangles are congruent is beyond the scope of this article. (This material is covered in Saxon Algebra 2 Lesson 124.) When working on these kinds of proofs be sure to line up the corresponding triangle vertices so that you are comparing the correct corresponding sides and angles together. Once a student has proven that two triangles are congruent, they can take one step further and prove that two corresponding segments or corresponding angles have equal lengths/measures using the CPCTC ("Corresponding Parts of Congruent Triangles Are Congruent") theorem.

SSS (Side-Side-Side):
Two triangles are considered congruent (geometrically equal) if we can prove that all three corresponding sides are equal (congruent).

AAAS (Angle-Angle-Angle-Side):
Two triangles are considered congruent (geometrically equal) if we can prove that all three corresponding angles are equal and that one corresponding side is equal (congruent). Because of the nature of triangles (the sum of the angles of a triangle equal 180), we actually only have to prove AA (Angle-Angle) to prove AAA (Angle-Angle-Angle); knowing this, we can rewrite this condition as simply AAS (Angle-Angle-Side).

SAS (Side-Angle-Side):
Two triangles are considered congruent (geometrically equal) if we can provide that two corresponding sides and the included corresponding angle are equal.

HL (Hypotenuse-Leg)
This particular condition only works with two right triangles. If the corresponding hypotenuse (that is, the side opposite the right angle) and a corresponding leg (non-hypotenuse side) are equal, then we've proved that the two right triangles are congruent (geometrically equal).

If we were to prove, for two right triangles, that both corresponding legs (non-hypotenuse sides) are equal, we would have proved that the two triangles are congruent (geometrically equal) by SAS (Side-Angle-Side) since the included corresponding angle would have been equal by definition of "right triangle."

Some Basic Geometric Definitions:
To do triangle proofs, the geometry student will need to know not only the above four congruent triangle conditions, but also the meaning of the following geometric terms:

The "midpoint" is the point half way between the ends of a line segment. The two half segments created by the midpoint, therefore, would be of equal lengths by the definition of midpoint.

An "isosceles" triangle is a triangle that has at least two sides of equal length. The corresponding opposite triangle angles would also be equal.

An "equilateral" or "equiangular" triangle is a triangle with all three sides of equal length. All three angles would also be equal (and be equal to 60 degrees).

A line segment that is "perpendicular" to another line segment creates a right angle where they meet.

Blessings!

Source
John H. Saxon, Jr. Algebra 2