**Introduction to Triangles**

If it’s been a while since you took a course in plane geometry, you probably think of triangles when the subject is brought up. Maybe you recall having to learn all kinds of theoretical proofs concerning triangles, using “steps/reasons” tables if your teacher was rigid, and less formal methods if your teacher was not so stodgy. You won’t have to go through the proofs again here, but some of the more important facts about triangles are worth stating.

**Point-Point-Point**

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. 9-15):

*P*,*Q*, and*R*lie at the vertices of a triangle.- This is the only triangle having vertices
*P*,*Q*, and*R*.

**Fig. 9-15.** The three point principle; side-side-side triangles.

**Side-Side-Side**

Let *S* , *T* , and *U* be three distinct, straight 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. 9-15). 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. 9-15.** The three point principle; side-side-side triangles.

**Side-Angle-Side**

Let *S* and *T* be two distinct, straight 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* relative to each other at point *P* (Fig. 9-16). 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.

**Fig. 9-16.** Side–angle–side triangles.

**Angle-Side-Angle**

Let *S* be a straight 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 straight lines *L* and *M* that run through *P* and *Q* , respectively (Fig. 9-17). 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*. - 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.

**Fig. 9-17.** Angle–side–angle triangles.

**Angle-Angle-Angle**

Let *L, M* , and *N* be straight lines that lie in a common plane and intersect in three distinct points, as illustrated in Fig. 9-18. 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).

**Fig. 9-18.** Angle–angle–angle triangles.

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