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# Angles and Distances Help (page 2)

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By McGraw-Hill Professional
Updated on Oct 3, 2011

## Angle Bisection

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.

Fig. 1-4 . Angle bisection principle.

## Perpendicularity

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 .

Fig. 1-5 . Perpendicular principle.

## Perpendicular Bisector

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.

Fig. 1-6 . Perpendicular bisector principle. Line M is unique.

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

PRPQ = QR

PRQR = PQ

Fig. 1-7 . Distance 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

Fig. 1-8 . Angular addition and subtraction.

## Angels and Distance Problems and Solutions

#### PROBLEM 1

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 ?

Fig. 1-6 . Perpendicular bisector principle. Line M is unique.

#### SOLUTION 1

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.

#### PROBLEM 2

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 ?

Fig. 1-8 . Angular addition and subtraction.

#### SOLUTION 2

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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