Introductory Geometry

CHAPTER 9

Chapter

Introductory Geometry

9-1 Basic Notions9-2 Polygons9-3 More About Angles9-4 Geometry in Three Dimensions

9

K–2: Children should

recognize, name, build, draw, compare, and sort two- and three-dimensional shapes

describe attributes and parts of two-and three-dimensional shapes

investigate and predict the results of putting together and taking apart two- and three-dimensional shapes.

NCTM Standard: Geometry

3–5: Students should

identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes

classify two- and three-dimensional shapes according to their properties

investigate, describe, and reason about the results of subdividing, combining, and transforming shapes

make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.

NCTM Standard: Geometry (cont.)

6–8: Students should

precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties

use visual tools such as networks to represent and solve problems.

NCTM Standard: Geometry (cont.)

9-1 Basic Notions

Linear Notions Planar Notions Other Planar Notions Angles Angle Measurement Types of Angles Perpendicular Lines A Line Perpendicular to a Plane

Basic Notions

The fundamental building blocks of geometry are:

Points – each end of a line having no thickness and extends forever in two directions is determined by two points.

Lines – have no thickness and extends forever in two directions. Determined by two points.

Planes – have no thickness and extends indefinitely in two directions. Determined by three points that are not all on the same line.

Undefined terms: points, lines, and planes

Linear Notions

Collinear points: are points on the same line. (Any two points are collinear but not every three points have to be collinear.)

Line ℓ contains points A, B, and C.

Point A, B, and C belong to line ℓ .

Points A, B, and C are collinear.

Points A, B, and D are not collinear.

Between:

Point B is between points A and C on line ℓ.

Point D is not between points A and C.

Line segment, or segment:

A subset of a line that contains two points of the line and all points between those two points.

AB BA

Ray:

A subset of a line that contains the endpoint and all points on the line on one side of the point.

AB

Betweeness can be defined in terms of length if the latter is introduced first.

If AB donates the length of , then B is between A and C if A, B, and C are collinear and AB + BC = AC

AB

A B C* * *

Planar Notions

Coplanar points – are points in the same plane.

Non-coplanar points – cannot be placed in a single plane.

Coplanar lines – lines in the same plane.

Skew lines – are lines that do not intersect, and there is no plane that contains them.

Intersecting lines – are two coplanar lines with exactly one point in common.

Concurrent lines – are lines that contain the same point.

Parallel lines – are two distinct coplanar lines m and n that have no points in common.

Planar Notions

Coplanar:

Points D, E, and G are coplanar.

Points D, E, F, and G are non-coplanar.

Lines DE, DF, and FE are coplanar.

Lines DE and EG are coplanar.

Lines DE and GE are intersecting lines; they intersect at point E.

Skew lines:

Lines GF and DE are skew lines. They do not intersect, and there is no plane that contains them.

Concurrent lines:

Lines DE, EG, and EF are concurrent lines; they intersect at point E.

Parallel lines:

Line m is parallel to line n. They have no points in common.

m n

NOW TRY THIS 9-1 Page 575a) How many different lines can be drawn through two points?

Exactly one line can be drawn through any two points

b) Can skew lines be parallel? Why?

No. Skew lines cannot be parallel. By definition, parallel lines are in the same planes. Skew lines are not.

c) On a globe, a “line” is a great circle, that is, a circle the same size as the equator. How many different lines can be drawn through two different points on a globe?

None if they are collinear. Let 0 be the center of the globe. First consider two points A and B on the globe which are

the endpoints of a diameter, that is, A, O, and B are collinear.

There are infinitely many planes containing A and B and hence the center of the globe, O.

Each of these planes intersects the globe in a “great” circle. Consequently if A, O, and B are collinear, there is no unique

“line” through A and B.

Properties of Points, Lines, and Planes There is exactly one line that contains any two distinct

points.

If two points lie in a plane, then the line containing the points lies in the plane.

If two distinct planes intersect, then their intersection is a line.

There is exactly one plane that contains any three distinct noncollinear points.

Properties of Points, Lines, and Planes (cont.)

A line and a point not on the line determine a plane.

Two parallel lines determine a plane.

Two intersecting lines determine a plane.

Property 7, two intersecting lines determine a plane, can be proved as follows:

Two intersecting lines have exactly one point in common (E). Each line contains at least one other point not on the other line (D and G). Thus there are at least three non-collinear points on the two intersecting lines (D, G, and F). These points determine a plane containing the lines.

NOW TRY THIS 9-2 Page 576

Show that statement 5 and 6 follow logically from the first four statements.

Statement 5 follows from statement 1 and 3. The two points on a line and a third non-collinear point determine a plane.

Statement 6 follows from statement 1 and 3. Two points are from one line and the other are form the line parallel to it.

Example

There are 10 people in a room. How many handshakes take place if each person shakes hands with everyone else in the room?

Solution 3 people 3 handshakes4 people 6 handshakes5 people 10 handshakes

n people handshakes

10 people 45 handshakes

2( 1)

2nn n

C-

=

ExampleThere are 10 people in a room. How many handshakes take place if each person shakes hands with everyone else in the room?

NOW TRY THIS 9-3 Page 577Use a geometric model to find the handshakes that take place at a party of 20 people if each person shakes hands with everyone else at the party. (Hint: Think about people as points and handshakes as lines.)

n people handshakes

20 people 190 handshakes

2( 1)

2nn n

C-

=

1902

380

2

)19)(20(

2

)120)(20(

Other Planar Notions

Two distinct planes either intersect in a line or are parallel.

If two points of a line are in the plane, then the entire line containing the points is contained in the plane, as in the middle above, then the line AB separates plane into two half-planes.

AnglesAngle – formed by two rays with the same endpoint.

Sides of an angle – the two rays that form an angle.

Vertex – the common endpoint of the two rays that form an angle.

It is customary simply to name an angle by its vertex, by a number, or by a lowercase Greek letter.

Adjacent angles – two angles with a common vertex and a common side, but without overlapping interiors.

QPR is adjacent to RPS.

Angle MeasurementAngles are measured by the amount of “opening” between its sides.

Degree – of a rotation about a point

Minute – of a degree

Second – of a minute

1

3601

601

60

Protractor

Symbols:

° = degree, for example 29°

’ = minutes, for example 47’

” = seconds, for example 13”

Example of a Measurements = 29°47’13”

NOW TRY THIS 9-4 Page 579

Convert 8.42° to degrees, minutes, and seconds.

8.42°8° + 0.42(60)’8°25.2’ 8°25’ + (.2)(60)” 8°25’12”

***On the calculator 8.42 then press DMS key***

Example

Find the measure of ABC if mDBC = 52°15′ and mDBA = 27°48′ .

Answers:

15(1/60) = .25

48(1/60) = .80

52.25 – 27.80 = 24.45

.45(60) = 27

24°27′

Example

Express 27°48′ as a number of degrees.

Answers:

48•(1/60) =.8

27.8°

Remember going from degree to (dms) degree, minutes, seconds multiply by 60.

When going from (dms) degree, minutes, seconds to degree multiply by 1/60.

Example #10 Page 585Consider a correctly set clock that starts ticking at noon an answer the following:

Find the measure of the angle swept by the hour hand by the time it reaches:

a) 3 P.M.

The hour hand will be pointed directly at the 3, so it will have moved ¼ of 360° = 90°

b) 12:25 P.M.

The hour hand will be 12/60 of the way from the 12 to the 1.in other words, 25/60 of the 30° between 12 and 1(25/60 x 30 = 12.5) and 12.5° = 12°30’ (.5 x 60 = 30)

c) 6:50 P.M.

The hour hand will have moved 6 whole spaces at 30° each plus 50/60 of another 30° (50/60 x 30 = 25) so, 180° + 25° = 205°

Find each angle measure between the minute and the hour hands at 1:15 p.m.

Each minute moves the hour hand 1/60 of 30° = 0.5°

1:15 P.M. is 75 minutes (60 + 15 = 75), so the hour hand will have moved 0.5 x 75 = 37.5° past 12

In 15 minutes the minute hand moves 90° past 12

90° - 37.5° = 52.5° = 52°30’ between the hands

Example #10 (cont.) Page 585Consider a correctly set clock that starts ticking at noon an answer the following:

At what time between 12 noon and 1 P.M. will the angle measure between the hands be 180°

At noon the angle between the hands is 0°. There will be a 180° angle between the hands when the minute hand has moved 180° more than the hour hand.

Each minute, the minute hand moves 1/60 of 360° = 6°, the hour hand moves 1/60 of 30° = 0.5°

After x minutes, the minute hand will have moved 6x° and the hour hand will move 0.5x°, therefore,

6x – 0.5x = 180 32 minutes + (.72 x 60 seconds)5.5x = 180 .72 x 60 = 43.2x = 32.727272 minutes 32 minutes + 43.2 seconds

There will be a 180° angle in approximately 32 minutes 44 seconds, or at 12:32:44

Example #10 (cont.) Page 585Consider a correctly set clock that starts ticking at noon an answer the following:

Types of Angles

Perpendicular Lines

When two lines intersect so that the angle formed are right angles the lines are perpendicular lines.

nm

A Line Perpendicular to a Plane

AB�������������� �

AB�������������� �

AB�������������� �

A line perpendicular to a plane is perpendicular to every line in the plane.

TheoremA line perpendicular to two distinct lines in the plane through its intersection with the plane is perpendicular to the plane.

If a line and a plane intersect they are perpendicular.

β and γ represents two walls intersecting along the edge is perpendicular to the floor. Every line in the plane of the floor passing through point A is perpendicular to

AB�������������� �

NOW TRY THIS 9-5 Page 584

a) Is it possible for a line intersecting a plane to be perpendicular to exactly one line in the plane through its intersection with the plane? Explain by making an appropriate drawing.

YES. It is possible for a line intersecting a plane to be perpendicular to only one line in the plane.

NOW TRY THIS 9-5 Page 584

b) Is it possible for a line intersecting a plane to be perpendicular to two distinct lines in a plane going through its point of intersection with the plane, and yet not be perpendicular to the plane?

NO. It is not possible for a line intersecting a plane to be perpendicular to two distinct lines and not be perpendicular to the plane.

NOW TRY THIS 9-5 Page 584

c) Can a line be perpendicular to infinitely many lines?

YES. If a line intersects a plane in point P an is perpendicular to two lines in the plane through P, then it is perpendicular to every line in the plane through P.

Example #4 Page 585Use the following drawing (see page 585) of one of the Great Pyramids of Egypt (with a rectangular base) to find the following:

a) The intersection of and

These lines are parallel, therefore, they do not intersect

b) The intersection of planes ABC, ACE, and BCE.

Point C. Three distinct planes intersecting at one common point.

c) The intersection of and

Point A. The intersection of two non-parallel lines

AD

AD�������������� �

CA�������������� �

CE

Example #4 Page 585Use the following drawing (see page 585) of one of the Great Pyramids of Egypt (with a rectangular base) to find the following:

d) A pair of skew lines: lines that do not intersect, and there is no place that contains them.

and or and

e) A pail of parallel lines: are two distinct coplanar lines that have no points in common

and or and are parallel

They are on opposite sides of a rectangle.

f) A plane not determined by one of the triangular faces or by the base.

Planes BCD or BEA they are planes bisecting (dividing) the pyramid.

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CE�������������� �

CE�������������� �

AC�������������� �

AB�������������� �

AC�������������� �

BE�������������� �

DE�������������� �

HOMEWORK 9-1

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