Chapter 1 Discovering Geometry. 1.1 Basic Geometric Figures I. Point A. Geometric figure with no dimensions B. Used to identify a point in space C. Represented

Chapter 1

Discovering Geometry

1.1 Basic Geometric Figures

I. Point

A. Geometric figure with no dimensions

B. Used to identify a point in space

C. Represented by a dot

•

1.1 Basic Geometric Figures

I. Point

A. Geometric figure with no dimensions

B. Used to identify a point in space

C. Represented by a dot

D. Labeled by a capital letter

•A

II. Line

A. Geometric figure having infinite length

B. No width or height

C. Consists of points

D. Represented by a double pointed arrow

II. Line





• •A B

II. Line





E. Labeled by any two point that it contains

• •A B

II. Line






• •A B

AB

•C

II. Line






• •A B

AB

•C

AC BC

II. Line






F. The intersection of two lines is a _______point

•

••

W

P J

•

••

W

P J

The intersection of WP and PJ is P.

II. Line







G. Through any one point there are infinitely many lines

•

II. Line







G. Through any one point there are infinitely many lines

H. Through any two points there is exactly one line

III. Plane

A. Geometric figure having infinite length and width but no height.

B. Represented by a flat rectangular surface

C. Planes consist of lines

D. Labeled by any three points on the plane

Are Points L, K, and M COPLANAR?

Yes, they are COPLANAR because they LIE ON THE SAME PLANE P.

Is point H, coplanar with points L, K, and M?

P

Q

A

B

LK

M

H

C

No, because it lies on plane Q and points L, K, and M are in different plane, on plane P.NON-COPLANAR points are points that lie in different planes.

D

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On what planes does point D lie?

P

Q

A

B

C

D



On what planes does point D lie? It only lies on plane Q.

P

Q

A

C

B D



III. PlaneA. Geometric figure having infinite length and width but no height. B. Represented by a flat rectangular surfaceC. Planes consist of linesD. Labeled by any three points on the plane

E. Through any two points there are infinitely many planes

F. Through any three points, there is exactly one plane

III. PlaneA. Geometric figure having infinite

length and width but no height. B. Represented by a flat rectangular surfaceC. Planes consist of linesD. Labeled by any three points on the planeE. Through any two points there are

infinitely many planesF. Through any three points, there is exactly

one plane G. The intersection of two planes is a_______lineH. The intersection of three planes is a _____________ or ___________linepoint

IV. Line Segment

A. A piece of a line

B. Has two endpoints

C. Labeled by its endpoints

∙ ∙S T

ST

V. Ray

A. Geometric figure with one endpoint

B. Labeled by it’s endpoint and one other point

∙ ∙P Q

PQ

1.2 Measuring Line Segments

I. “Measure” of a Line Segment

A. The distance between its endpoints

B. Always positive


0-1-2-3-4-5 2 3 4 51•A

•B

coordinates

a b


0-1-2-3-4-5 2 3 4 51•A

•B

coordinates

a b

AB

AB = “the measure of AB”AB = _________7 units


I. “Measure” of a Line Segment

A. The distance between its endpoints

B. Always positive

C. AB = b – a or a - b


0-1-2-3-4-5 2 3 4 51•A

•B

coordinates

a b

AB

AB = 3 – (-4)

AB = 3 + (+4) AB = 7

orAB = -4 – 3 AB = -4 + -3 AB = -7


0-1-2-3-4-5 2 3 4 51•A

•B

coordinates

a b

AB

AB = 3 – (-4)

AB = 3 + (+4) AB = 7 units

orAB = -4 – 3 AB = -4 + -3 AB = -7 = 7 units

Examples

•P

•QPQ

23 95

PQ = ________________95 – 23 = 72 units

Examples

•E

•FEF

-15 46

EF = ________________46 – (-15) = 61 units

OR

EF = ________________-15 – 46 = -61 = 61 units

Examples

•R

•SRS

-92 -18

RS = ________________-18 – (-92) = 74 units

OR

RS = ________________-92 – (-18) = -74 = 74 units

|

|

1.2 Measuring Line SegmentsII. Segment Addition

A. “collinear”= “on the same line”

B. If A, B, & C are collinear and B is between A and C, then

AB + BC = AC

• • •A B C

AB BCAC

Examples of Segment AdditionA carpenter must cut a 54 inch board into two pieces so that one piece is twice as long as the other. What will be the length of the two board after the cut?

• • •A B C

X 2x54 in.

AB + BC = AC x + 2x = 54 3x = 54 3 3

Examples of Segment AdditionA carpenter must cut a 54 inch board into two pieces so that one piece is twice as long as the other. What will be the length of the two board after the cut?

• • •A B C

X 2x

54 in.

AB + BC = AC x + 2x = 54 3x = 54 3 3

3 541

324

8

240x = 18 in.

18 in.= 2(18)

36 in.

Examples of Segment AdditionA 45 foot piece of pipe must be cut so that the longer piece is 9 feet longer than the shorter. What will be the lengths of the two pieces?

• • •A B C

X X + 945 ft.

AB + BC = AC x + x + 9 = 45 2x + 9 = 45

2 2

- 9 = -9 2x = 36

Examples of Segment AdditionA 45 foot piece of pipe must be cut so that the longer piece is 9 feet longer than the shorter. What will be the lengths of the two pieces?

• • •A B C

X X + 9

45 ft.

AB + BC = AC x + x + 9 = 45

2x = 36 2 2

2 361

216

8

160 x = 18 ft.

18 ft.= 18 + 9

27 ft.

2x + 9 = 45 - 9 = -9


III. Midpoint of a Segment

A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB.


III. Midpoint of a Segment


• ••A BC

1.2 Measuring Line SegmentsIII. Midpoint of a Segment


• ••A BC

12 58a b

B. Midpoint Formula



• ••A BC

12 58a b

B. Midpoint Formula

The midpoint of AB = a + b2

12 + 58 2

=

= 70 2

= 35

35



• ••A BC

-15 35a b

B. Midpoint Formula


-15 + 35 2

=

= 20 2

= 10

10



• ••A BC

-84 -12a b

B. Midpoint Formula


-84 + -12 2

=

= -96 2

= -48

-48

Examples of Segment AdditionA carpenter must cut a 65 inch board into two pieces so that one piece is five inches more than twice the length of the other. What will be the length of the two board after the cut?

• • •A B C

X 2x+565 in.

AB + BC = AC x + 2x+5 = 65

1.3 Measuring Angles

A. Using a Protractor

ACUTE Angle less than 90

60°

°



RIGHT Angle

90°



OBTUSE Angle

Greater than 90 °

140°



°120



•

•

•

• •

•A

BC

D

E

FO


A. Using a ProtractorB. Angle Addition

•

••

•

A

B

C

D


A. Using a ProtractorB. Angle Addition

•

••

•

A

B

C

D

m ABD + m DBC = m ABC

•

••

•

A

B

C

D

m ABD + m DBC = m ABC

A 70 angle is divided into two smaller angles such that the larger angle is two more than three times the smaller.

70°

°

x

3x + 2

x + 3x +2 = 704x + 2 = 70

–2 –24x = 68___ ___

4 4

x = 17

17°

3(17) + 2

53°


A. Using a ProtractorB. Angle AdditionC. Vertical Angle Conjecture

“ the vertical angles formed by intersecting lines have equal measure”

1.4 Special AnglesA. Complementary Angles

A pair of angles whose sum is 90

º

12

1.4 Special AnglesA. Complementary Angles

A pair of angles whose sum is 90

º

12

1.4 Special Angles

B. Supplementary Angles

A pair of angles whose sum is 180°

1 2

1.4 Special Angles

B. Supplementary Angles

1 2

An angle is four times it’s compliment. Find both angles.

x4x

x + 4x = 90

5x = 90

x = 18

18°

4(18) = 72°

1.5 Parallel and Perpendicular Lines

A. Parallel Lines

Lines on the same plane that do not intersect

l

m

l || m


A. Parallel Lines

Lines on the same plane that do not intersect

B. Perpendicular Lines

Two lines that intersect at a right angle

1.5 Parallel and Perpendicular Linesk

j

k j


C. Corresponding Angles

1 2 3

m<1 = m<2 = m<3



1 2

m<1 = m<2 = m<3

(3x+20) (5x -10)

3x + 20 = 5x – 10 -3x -3x 20 = 2x – 10 +10 = + 10 30 = 2x 2 2 15 = x

m<1 = 3(15) +20

45

65º

m<2 = 5(15) – 10

75

65º



1 2

m<1 = m<2 = m<3

(6x+30) (3x +57)

6x + 30 = 3x + 57 -3x -3x3x + 30 = 57 - 30 = - 30 3x = 27 3 3 x = 9

m<1 = 6(9) +30

54

84º

m<2 = 3(9) + 57

27

84º



1 2

m<1 = m<2 = m<3

(9x+50) (4x +39)

9x + 50 + 4x + 39 = 180 13x +89 = 180

- 89 - 89 13x = 91 13 13

x = 7

m<1 = 9(7) +50

63

113º

m<2 = 4(7) + 39

28

67º

Documents

Chapter 1 Discovering Geometry. 1.1 Basic Geometric Figures I. Point A. Geometric figure with no dimensions B. Used to identify a point in space C. Represented