Introduction
If it’s been a while since you took a course in plane geometry, perhaps you think of triangles when the subject is brought up. Maybe you recall having to learn all kinds of theoretical proofs concerning triangles using “steps and reasons” tables if your teacher was rigid and less formal methods if your teacher was not so stodgy. Well, here, you don’t have to go through the proofs again, but some of the more important facts about triangles are worth stating.
Pointpointpoint
Let P, Q , and R be three distinct points, not all of which lie on the same straight line. Then the following statements are true (Fig. 415):
 P, Q , and R lie at the vertices of a triangle T .
 T is the only triangle having vertices P, Q , and R .
Sidesideside
Let S, T , and U be 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 (see Fig. 415). Then the following statements hold true:
 Line segments S, T , and U determine a triangle.
 This is the only triangle of its size and shape that has sides S, T , and U .
 All triangles having sides of lengths s, t , and u are congruent (identical in size and shape).
Fig. 415 The threepoint principle; sidesideside triangles.
Sideangleside
Let S and T be two distinct line segments. Let P be a point that lies at the ends of both these line segments. Denote the lengths of S and T by their lowercase counterparts s and t , respectively. Suppose that S and T both subtend an angle x at point P (Fig. 416). 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 at point P .
 All triangles containing two sides of lengths s and t that subtend an angle x are congruent.
Anglesideangle
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. 417). Then the following statements are all true:
 S, x , and y determine a triangle.
 This the only triangle determined by S, x , and y .
 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 are congruent.
Angleangleangle
Let L, M , and N be lines that lie in a common plane and intersect in three points, as illustrated in Fig. 418. Let the angles at these points be x, y , and z . Then the following statements are true:
 There are infinitely many triangles with interior angles x, y , and z in the sense shown.
 All triangles with interior angles x, y , and z in the sense shown are similar (that is, they have the same shape but not necessarily the same size).

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