Triangle Congruence and Similarity Help

Introduction to Triangle Congruence and Similarity

There are four criteria that can be used to define sets of triangles that are directly congruent. These are called the side–side–side (SSS), side–angle–side (SAS), angle–side–angle (ASA), and angle–angle–side (AAS) principles. The last of these can also be called side–angle–angle (SAA). A fifth principle, called angle–angle–angle (AAA), can be used to define sets of triangles that are directly similar.

Side–Side–Side (sss)

Let S, T , and U be defined, specific line segments. Let s, t , and u be the lengths of those three line segments, respectively. Suppose that S, T , and U are joined at their end points P, Q , and R (Fig. 2-4). Then the following statements hold true:

• Line segments S, T , and U determine a triangle
• This is the only triangle that has sides S, T , and U in this order, as you proceed around the triangle in the same rotational sense
• All triangles having sides of lengths s, t , and u in this order, as you proceed around the triangles in the same rotational sense, are directly congruent

Fig. 2-4 . The three-point principle; side-side-side triangles.

Side–Angle–Side (sas)

Let S and T be two distinct line segments. Let P be a point that lies at the ends of both of these line segments. Denote the lengths of S and T by their lowercase counterparts s and t , respectively. Suppose S and T subtend an angle x , expressed in the counterclockwise sense, at point P (Fig. 2-5). Then the following statements are all true:

• S, T , and x determine a triangle
• This is the only triangle having sides S and T that subtend an angle x , measured counterclockwise from S to T , at point P 

Fig. 2-5 . Side-angle-side triangles.

• All triangles containing two sides of lengths s and t that subtend an angle x , measured counterclockwise from the side of length s to the side of length t , are directly congruent

Angle–Side–Angle (asa)

Let S be a line segment having length s , and whose end points are P and Q . Let x and y be the angles subtended relative to S by two lines L and M that run through P and Q , respectively (Fig. 2-6), such that both angles are expressed in the counterclockwise sense. Then the following statements are all true:

• x , S , and y determine a triangle
• This is the only triangle determined by x , S , and y , proceeding from left to right
• All triangles containing one side of length s , and whose other two sides subtend angles of x and y relative to the side whose length is s , with x on the left and y on the right and both angles expressed in the counterclockwise sense, are directly congruent

Fig. 2-6 . Angle-side-angle triangles.

Angle–Angle–Side (aas) Or Side–Angle–Angle (saa)

Let S be a line segment having length s , and whose end points are P and Q . Let x and y be angles, one adjacent to S and one opposite, and both expressed in the counterclockwise sense (Fig. 2-7). Then the following statements are all true:

• S, x , and y determine a triangle
• This is the only triangle determined by S, x and y in the counterclockwise sense
• All triangles containing one side of length s , and two angles x and y expressed and proceeding in the counterclockwise sense, are directly congruent

Fig. 2-7 . Angle-angle-side triangles.