Angles and Distances Help

Introduction to Angles and Distance

When two lines intersect, four angles exist at the point of intersection. Unless the two lines are perpendicular, two of the angles are “sharp” and two are “dull.” When the two lines are perpendicular, each of the four angles is a right angle . Angles can also be defined by sets of three points when the points are connected by line segments.

Measuring Angles

The two most common units of angular measure are the degree and the radian .

The Degree

The degree (°) is the unit familiar to lay people. One degree (1°) is 1/360 of a full circle. This means that 90° represents a quarter circle, 180° represents a half circle, 270° represents three-quarters of a circle, and 360° represents a full circle.

A right angle has a measure of 90°, an acute angle has a measure of more than 0° but less than 90°, and an obtuse angle has a measure of more than 90° but less than 180°. A straight angle has a measure of 180°. A reflex angle has a measure of more than 180° but less than 360°.

The Radian

The radian (rad) is defined as follows. Imagine two rays emanating outward from the center point of a circle. Each of the two rays intersects the circle at a point; call these points P and Q . Suppose the distance between P and Q , as measured along the arc of the circle, is equal to the radius of the circle. Then the measure of the angle between the rays is one radian (1 rad). There are 2π radians in a full circle, where π (the lowercase Greek letter pi, pronounced “pie”) stands for the ratio of a circle’s circumference to its diameter. The value of π is approximately 3.14159265359, often rounded off to 3.14159 or 3.14.

A right angle has a measure of π/2 rad, an acute angle has a measure of more than 0 rad but less than π/2 rad, and an obtuse angle has a measure of more than π/2 rad but less than π rad. A straight angle has a measure of π rad, and a reflex angle has a measure larger than π rad but less than 2π rad.

Angle Notation

Imagine that P, Q , and R are three distinct points. Let L be the line segment connecting P and Q ; let M be the 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 , as shown in Fig. 1-3.

Fig. 1-3 . Angle notation.

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 . If rotational sense is important, counterclockwise is usually considered positive, and clockwise is considered negative. In Fig. 1-3, ∠ RQP is positive while ∠ PQR is negative. These notations can also stand for the measures of angles, expressed either in degrees or in radians. If we make an approximate guess as to the measures of the angles in Fig. 1-3, we might say that ∠ RQP = +60° while ∠ PQR = −60°.

Rotational sense is not important in basic geometry, but it does matter when we work in coordinate geometry. We’ll get into that type of geometry, which is also called analytic geometry , later in this book. For now, let’s not worry about the rotational sense in which an angle is measured; we can consider all angles positive.