Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

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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 ¼ QR. This is illustrated in Fig. 9-2.

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 ﬀPQR or as ﬀRQP.If

the rotational sense of measurement is speciﬁed, then ﬀPQR indicates the

angle as measured from L to M, and ﬀRQP 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.

ANGLE BISECTION

Suppose there is an angle ﬀPQR measuring less than 1808 and deﬁned by

three distinct points P, Q, and R, as shown in Fig. 9-4. Then there is exactly

Fig. 9-2. Midpoint principle.

Fig. 9-3. Angle notation and measurement.

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one straight ray M that bisects the angle ﬀPQR.IfS is any point on M other

than the point Q, then ﬀPQS ¼ﬀSQR. Every angle has one and only one ray

that divides the angle in half.

PERPENDICULARITY

Suppose that L is a straight line through two distinct points P and Q. Let R

be a point not on L. Then there is one and only one straight line M through

point R, intersecting line L at some point S, such that M is perpendicular

to L. This is shown in Fig. 9-5.

PERPENDICULAR BISECTOR

Suppose that L is a straight line segment connecting two distinct points P

and R. Then there is one and only one straight line M that intersects line

Fig. 9-4. Angle bisection principle.

Fig. 9-5. Perpendicular principle.

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segment L in a point Q, such that M is perpendicular to L and 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. 9-6.

DISTANCE ADDITION AND SUBTRACTION

Let P, Q, and R be distinct points on a straight line L, such that Q is between

P and R. Then the following equations hold concerning distances as

measured along L (Fig. 9-7):

PQ þ QR ¼ PR

PR  PQ ¼ QR

PR  QR ¼ PQ

ANGLE ADDITION AND SUBTRACTION

Let P, Q, R, and S be four distinct points that lie in a common Euclidean

plane. Let Q be the vertex of three angles ﬀPQR, ﬀPQS, and ﬀSQR,as

Fig. 9-6. Perpendicular bisector principle.

Fig. 9-7. Distance addition and subtraction.

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shown in Fig. 9-8. Then the following equations hold concerning the angular

measures:

ﬀPQS þﬀSQR ¼ﬀPQR

ﬀPQR ﬀPQS ¼ﬀSQR

ﬀPQR ﬀSQR ¼ﬀPQS

VERTICAL ANGLES

Suppose that L and M are two straight lines that intersect at a point P.

Opposing pairs of angles, denoted x and y in Fig. 9-9, are known as vertical

angles and always have equal measure.

Fig. 9-8. Angular addition and subtraction.

Fig. 9-9. Vertical angles have equal measure.

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ALTERNATE INTERIOR ANGLES

Suppose that L and M are parallel, straight lines. Let N be a straight

transversal line that intersects L and M at points P and Q, respectively. In

Fig. 9-10, angles labeled x are alternate interior angles; the same holds true

for angles labeled y. Alternate interior angles always have equal measure.

The transversal line N is perpendicular to lines L and M if and only if x ¼ y.

ALTERNATE EXTERIOR ANGLES

Suppose that L and M are parallel, straight lines. Let N be a straight

transversal line that intersects L and M at points P and Q, respectively.

In Fig. 9-11, angles labeled x are alternate exterior angles; the same

holds true for angles labeled y. Alternate exterior angles always have

Fig. 9-10. Alternate interior angles have equal measure.

Fig. 9-11. Alternate exterior angles have equal measure.

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equal measure. The transversal line N is perpendicular to lines L and M if

and only if x ¼ y.

CORRESPONDING ANGLES

Let L and M be parallel, straight lines. Let N be a straight transversal line

that intersects L and M at points P and Q, respectively. In Fig. 9-12, angles

labeled w are corresponding angles; the same holds true for angles labeled x, y,

and z. Corresponding angles always have equal measure. The transversal

line N is perpendicular to lines L and M if and only if w ¼ x ¼ y ¼ z ¼ 908;

that is, if and only if all four angles are right angles.

PARALLEL PRINCIPLE

Suppose L is a straight line, and P is a point not on L. Then there exists one

and only one straight line M through P, such that line M is parallel to line L

(Fig. 9-13). This is one of the most important postulates in Euclidean

geometry. Its negation can take two forms: either there is no such straight

line M, or there exist two or more such straight lines M

, M

, ....

Either form of the negation of this principle constitutes a cornerstone of

non-Euclidean geometry, which is important to physicists and astronomers

interested in the theories of general relativity and cosmology.

MUTUAL PERPENDICULARITY

Let L and M be straight lines that lie in the same plane. Suppose both L and

M intersect a third straight line N, and both L and M are perpendicular to N.

Then lines L and M are parallel to each other (Fig. 9-14).

Fig. 9-12. Corresponding angles have equal measure.

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PROBLEM 9-1

Suppose a straight, ﬂat, two-lane highway crosses a straight set of railroad

tracks, such that the acute angles (angles measuring more than 08 but less

than 908) between the rails and the side lines of the highway measure 508.

What are the measures of the obtuse angles (angles measuring more than

908 but less than 1808) between the rails and the side lines of the highway?

SOLUTION 9-1

When a straight line is crossed by a straight transversal, the measures of

adjacent angles at the intersection point always add up to 1808. The rails

are straight, as are the side lines of the highway. Therefore, the obtuse angles

measure 1808  508, or 1308. You may wish to draw a picture of this situation

to make it easier to envision.

PROBLEM 9-2

In the situation of the previous problem, what are the measures of the acute

angles between the highway center line and the rails? What are the measures

of the obtuse angles between the highway center line and the rails?

Fig. 9-13. The parallel principle.

Fig. 9-14. Mutual perpendicularity.

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SOLUTION 9-2

The highway center line is straight, and is parallel to both of the side lines.

Thus, the acute angles between the rails and the center line have the same

measures as the acute angles between the rails and the side lines, that is,

508. The measures of the obtuse angles between the rails and the center

line are the same as the measures of the obtuse angles between the rails

and the side lines, that is, 1808  508 ¼ 1308.

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.

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

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

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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).

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.

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.

Fig. 9-16. Side–angle–side triangles.

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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.

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 simi-

lar (that is, they have the same shape, but not necessarily the same size).

SUM OF INTERIOR ANGLES OF PLANE TRIANGLE

Suppose we have a general plane triangle, with no special properties other

than the fact that it is deﬁned by three distinct vertex points. Let the measures

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

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

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