Chapter 1Essentials of Geometry

1.1 Identifying Points, Lines, and Planes•Geometry:•Study of land or Earth measurements•Study of a set of points•Includes undefined terms, definitions,

postulates (axioms), theorems, properties, and logic

Undefined terms

•Point- a location, no dimension, occupies no space

•Name: AA

•Line- one dimension, has no thickness, extends without end, contains an infinite number of points, use 2 points to name a line or a lower case letter

•Name: m, AB, BA, BC, CA•Through any 2 points, there is exactly

one line

m

AB

C

•Plane- 2 dimensions, extends without end in all directions, has length and width but no depth

•Name: ABF, EBC, plane T•Through any 3 points there is exactly one

plane.

m

T

AB

D

F

E

GC

•Collinear- points that lie on the same line

m

(non collinear)

F

E

GA

B C

D

•Coplanar- points that lie in the same plane

( non coplanar)

U

AD

E

C

G

F

B

H

J

I

•Segment- consists of the endpoints and all points between the endpoints.

•Endpoint- a point at the end of a segment or the starting point of a ray.

•Name: AE or EA, BC, BD

A

EB C

D

•Ray- consists of a part of a line that starts at a point and extends infinitely in one direction.

•Name: AB, DC

A D

BC

Opposite rays

•If point C lies on AB between A and B, then CA and CB are opposite rays.

A C B

Intersection

•Geometric figures with one or more points in common

•The set of all points that figures have in common

k

j

t

C

1.2 Use Segments and Congruence•Postulate or Axiom- a rule that is accepted

without proof , “ If - then statements”•Postulates: (p. 96)

The intersection of two lines is a _______________. The intersection of two planes is a ______________. Through any two points there is exactly one _______. Through any three noncollinear points there is exactly one ________________.

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

Ruler Postulate

•The points on a line can be matched one to one with the real numbers.

•Coordinate- the real number that corresponds to a point

•Distance- the absolute value of the difference of the coordinates

Segment Addition Postulate

•If B is between A and C on a line, then AB + BC = AC

• If AB + BC = AC, then B is between A and C.

A BC

A

B

C

D

1.3 Use Midpoint and Distance Formula•Midpoint- the point that divides the

segment into two congruent segments•Segment bisector- a point, ray, line, line

segment or plane that intersects a segment at its midpoint

• Midpoint formula :

A

BM

Other Formulas

•Distance formula:

•Slope formula:

•Equation of a line:

Example•The midpoint of AB is M(5,8). One

endpoint is A(2,-3). Find the coordinate of B. Find the distance of AB. Write the equation of the line segment in slope-intercept form.

1.4 Measure and Classify Angles•Angle- a figure formed by 2 rays with a

common endpoint, the union of 2 rays with a common endpoint

•Side (of an angle)- rays•Vertex- the common endpoint of the rays

that form the angle•Interior/ Exterior of an angle-•Name:

B

A

C

Protractor Postulate

•Consider OB and a point A on one side of OB. The ray that forms OA can be matched one to one with real numbers from 0 to 180.

O

A

B

Classifying Angles

•Acute- 0o < x < 90o

•Right - x = 90o

•Obtuse- 90o < x < 180o

•Straight- x = 180o

Angle Addition Postulate

•If P is in the interior of <RST, then m<RST = m<RSP + m<PST

R

S

T

P

•Congruent angles- angles with the same measure

•Angle bisector- a line, segment, or ray that divides an angle into 2 congruent angles

A

B

C

D

2

1

A

BC

D

E

F

Example

•Find m<ABC and m<CBD.

mABD = 85

2x + 4

3x + 6B

A

C

D

1.5 Describe Angle Pair Relationships•Complementary angles- 2 angles whose

measures sum to 90o

(complement)

•Supplementary angles- 2 angles whose measures sum to 180o

(supplement)

• Adjacent angles- 2 angles that share a common vertex and side but have not common interior points

•Linear Pair- 2 angles formed when the endpoint of a ray falls on a line, non common sides are opposite rays

•Vertical angles- 2 angles formed by 2 intersecting lines

4

3

2

1

O

C

A B

Examples

•Name a pair of complementary angles

supplementary anglesadjacent angles

L

M

K

J

I

HG

Find mWYX and mWYZ.

4x - 20 2x + 14

Y

W

V

X Z

1.6 Classify Polygons• Polygon- a closed plane figure

formed by coplanar segments such that ▫Each segment intersects exactly 2 other

segments one at each endpoint and▫No 2 points with a common endpoint are

collinear• Sides- coplanar segments• Vertex- point where 2 segments’ endpoints

intersect• Diagonal- a segment that joins 2

nonconsecutive vertices of a polygon• Name:

AB

C

DE

F

•Convex or Concave

•Convex- no line that contains a side of the polygon contains a point in the interior of the polygon

•Concave- (or nonconvex) a polygon that is not convex

Classifying

Equilateral, equiangular, or regularEquilateral- all sides are congruent

Equiangular- all angles on the interior are congruent

Regular- a convex polygon that is equilateral and equiangular

Classify by sides

•Triangle Octa-•Quadrilateral Nona-•Pentagon Deca-•Hexa Dodeca-•Hepta n-gon

•N-gon n= the number of sides of the polygon

Examples

•Classify the polygon

a b

•c d

•A billiard rack has the shape of an equilateral triangle. Find the length of a side.

6x - 4 in.

4x + 2 in.

1.7 Find Perimeter, Circumference, and Area

•Perimeter- the distance around a figure, measured in linear units

•Circumference- the distance around a circle

•Area- the amount of surface covered by a figure, measured in units2

Formulas

•Square P = 4s A = s2

•RectangleP = 2l + 2w A = lw

•Triangle P = a + b + c A = ½·bh

•Circle C = ∏d or 2 ∏r A = ∏r2

Converting Units

•100 in2 to ft2

•1.2yd3 to in3

•o.39m2 to mm2

Examples

•Find the perimeter of ∆ABC. Leave in simple radical form. A(2,5) B(4,1) C(8,3)

•The base of a triangle is 14cm. The area is 42cm2. Find the height.

•You can mow 600 sq. ft of grass in 10 min. How long will it take to mow a rectangular field that is 60 ft. wide and 80 ft. long?