Angles and Distances Help (page 2)
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.
The two most common units of angular measure are the degree and the radian .
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 (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.
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.
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.
Suppose there is an angle ∠ PQR measuring less than 180° and defined by three points P, Q , and R , as shown in Fig. 1-4. Then there is exactly one ray M that bisects (divides in half) the angle ∠ PQR . If S is any point on M other than the point Q , then ∠ PQS = ∠ SQR . That is to say, every angle has one, and only one, ray that bisects it.
Suppose that L is a line through points P and Q . Let R be a point not on L . Then there is exactly one line M through point R , intersecting line L at some point S , such that M is perpendicular to L (that is, such that M and L intersect at a right angle). This is shown in Fig. 1-5. The term orthogonal is sometimes used instead of perpendicular. Another synonym for perpendicular, used especially in theoretical physics, is normal .
Suppose that L is a line segment connecting two points P and R . Then there is one and only one line M that is perpendicular to L and that intersects L at a point Q , such that the distance from P to Q is equal to the distance from Q to R . That is, every line segment has exactly one perpendicular bisector . This is illustrated in Fig. 1-6.
Distance Addition And Subtraction
Let P, Q , and R be points on a line L , such that Q is between P and R . Then the following equations hold concerning distances as measured along L (Fig. 1-7):
PQ + QR = PR
PR − PQ = QR
PR − QR = PQ
Angle Addition And Subtraction
Suppose that P, Q, R , and S are points that all lie in the same plane. That is, they are all on a common, perfectly flat surface. Let Q be the vertex of three angles ∠ PQR , ∠ PQS , and ∠ SQR , with ray QS between rays QP and QR as shown in Fig. 1-8. Then the following equations hold concerning the angular measures:
∠ PQS + ∠ SQR = ∠ PQR
∠ PQR − ∠ PQS = ∠ SQR
∠ PQR − ∠ SQR = ∠ PQS
Angels and Distance Problems and Solutions
Look at Fig. 1-6. Suppose S is some point on line M other than point Q . What can we say about the lengths of line segments PS and SR ?
The solutions to problems like this can be made easier by making your own drawings. The more complicated the language (geometry problems can sometimes read like “legalese”), the more helpful drawings become. With the aid of your own sketch, you should be able to see that for every point S on line M (other than point Q , of course), the distances PS and SR are greater than the distances PQ and QR , respectively.
Look at Fig. 1-8. Suppose that point S is moved perpendicularly with respect to the page (either straight toward you or straight away from you), so S no longer lies in the same plane as points P, Q , and R . What can we say about the measures of ∠ PQR, ∠ PQS , and ∠ SQR ?
In this situation, the sum of the measures of ∠ PQS and ∠ SQR is greater than the measure of ∠ PQR . This is because the measures of both ∠ PQS and ∠ SQR increase if point S departs perpendicularly from the plane containing points P, Q , and R . As point S moves further and further toward or away from you, the measures of ∠ PQS and ∠ SQR increase more and more.
Practice problems for these concepts can be found at: Geometry Basic Rules Practice Test.
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