**Fundamental Rules**

Geometry is widely used in science and engineering. The ancient Egyptians and Greeks used geometry to calculate the diameter of the earth and the distance to the moon, and scientists have been busy with it ever since. You do not have to memorize all the formulas in this section, but it’s good to be comfortable with this sort of stuff. So roll up your sleeves and make sure your brain is working.

Here are some of the most important principles of *plane geometry* , which involves the behavior of points, lines, and various figures confined to a flat, two-dimensional (2D) surface. There are lots of illustrations to help you envision how these rules and formulas work.

**Rules of Points**

**Two Point Principle**

Suppose that *P* and *Q* are two distinct geometric points. Then the following statements are true, as shown in Fig. 9-1:

*P*and*Q*lie on a common straight line*L*.*L*is the only straight line on which both points lie.

**Fig. 9-1.** Two point principle.

**Three Point Principle**

Let *P* , *Q* , and *R* be three distinct points, not all of which lie on a straight line. Then the following statements are true:

*P, Q*, and*R*all lie in a common Euclidean (flat) plane*S*.*S*is the only Euclidean plane in which all three points lie.

**Principle of ***n* Points

*n*Points

Let *P* _{1} , *P* _{2} , *P* _{3} ,. . ., *P _{n}* be

*n*distinct points, not all of which lie in the same Euclidean (that is, “non-warped”) space of

*n*– 1 dimensions. Then the following statements are true:

*P*_{1},*P*_{2},*P*_{3},. . .,*P*all lie in a common Euclidean space_{n}*U*of*n*dimensions.*U*is the only*n*-dimensional Euclidean space in which all*n*points lie.

**Distance Notation**

The distance between any two distinct points *P* and *Q* , as measured from *P* towards *Q* along the straight line connecting them, is symbolized by writing *PQ* .

**Midpoint Principle**

Suppose there is a straight line segment connecting two distinct points *P* and *R* . Then there is one and only one point *Q* on the line segment, between *P* and *R* , such that *PQ* = *QR* . This is illustrated in Fig. 9-2.

**Fig. 9-2.** Midpoint principle.

**Rules of Angles**

**Angle Notation**

Imagine that *P* , *Q* , and *R* are three distinct points. Let *L* be the straight line segment connecting *P* and *Q* ; let *M* be the straight line segment connecting *R* and *Q* . Then the angle between *L* and *M* , as measured at point *Q* in the plane defined by the three points, can be written as ∠ *PQR* or as ∠ *RQP* . If the rotational sense of measurement is specified, then ∠ *PQR* indicates the angle as measured from *L* to *M* , and ∠ *RQP* indicates the angle as measured from *M* to *L* (Fig. 9-3). These notations can also stand for the measures of angles, expressed either in degrees or in radians.

**Fig. 9-3.** Angle notation and measurement.

**Angle Bisection**

Suppose there is an angle ∠ *PQR* measuring less than 180° and defined by three distinct points *P* , *Q* , and *R* , as shown in Fig. 9-4. Then there is exactly one straight ray *M* that bisects the angle ∠ *PQR* . If *S* is any point on *M* other than the point *Q* , then ∠ *PQS* ∠ *SQR* . Every angle has one and only one ray that divides the angle in half.

**Fig. 9-4.** Angle bisection principle.

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