Download Chapter 1: Basics of Geometry

Unit 1: Essentials of Geometry
Lesson 1.1: Points, Lines, and Planes

Objective
Understand and use the basic undefined terms and defined terms in geometry such as point, line,
plane, collinear, coplanar.
USING UNDEFINED TERMS AND DEFINTIONS
Point
Line
________________
Name
Plane
______________________
Names
_____________________
Names
Example 1
a. Name three points that are collinear: ___________________________
b. Name four points that are coplanar: ____________________________
c. Name three points that are not collinear: ________________________
Line
Segment
Ray
Example 2
Give another name for GH . _______________
Name all the rays with endpoint J. ______________________
Which of those rays are opposite rays? ___________________
INTERSECTIONS
The intersection of two lines: _____________________
The intersection of two planes: ____________________
Opposite Rays
Example 3
Sketch a plane and a line
that is in the plane.
Sketch a plane and a line that
does not intersect the plane.
Sketch a plane and a line
that intersects at one point.
Example 4
Name the intersection of PQ and line k. ________________
Name the intersection of plane A and plane B. ________________
Name the intersection of plane A and line k. _________________
Example 5
You are given an equation of a line and a point. Use substitution to determine whether the point is on
the line.
y = x – 4; A (5, 1)
_______________________
y = x + 1; A (1, 0)
_______________________
Example 6
Graph the inequality on a number line. Tell whether the graph is a segment, a ray or rays, a point, or a
line.
x<3
_______________________
-7 < x < 4
_______________________
Unit 1: Essentials of Geometry
Lesson 1.2: Use Segments and Congruence

Objective
Find the distance between two points using the Ruler Postulate and Segment Addition Postulate
 Use the Distance Formula to find the distance between two points in the coordinate plane.
 Understand and apply the definition of congruence (congruent segments).
USING SEGMENT POSTULATES
Ruler Postulate
Example 1
Measure the length of the segment to the nearest millimeter.
When three points are collinear, one point is ____________________ the other two.
For instance, point _____________ is between points A and C.
Segment Addition Postulate
Example 2
Use the map to find the distance between Lubbock and St. Louis.
__________________.
Example 3
Use the diagram to find GH .
GH = ____________
CONGRUENT SEGMENTS
Example 4:
Plot J(-3, 4), K(2, 4), L(1, 3), and M(1, -2) in a
coordinate plane. Then determine whether
JK and LM are congruent.
Example 5:
Use the number line to find the indicated distance.
VW = ___________
XY = ___________
XZ = ___________
VX= ___________
Unit 1: Essentials of Geometry
Lesson 1.3 Apply Pythagorean Theorem
Lesson 7.1 from Geometry Textbook

Objectives
Use the Pythagorean Theorem to find the length of the unknown side of a right triangle and the
area of the triangle.
 Determine which numbers form a Pythagorean Triple
RIGHT TRIANGLES
Pythagorean Theorem
_____________________________
Example 1
Find the length of the sides of each of the given triangles.
x = ___________
x = ___________
Example 2
Find the area of the isosceles triangle with side
lengths 10 meters, 13 meters, and 13 meters.
HINT: A = ½(base of triangle)(height of triangle).
Example 3
A = __________________
A = __________________
PYTHAGOREAN TRIPLES
__________________________________________
Example 3
Complete the table of Pythagorean Triples and their multiples.
Multiple of 2
3, 4, 5
6, 8, 10
5, 12, 13
8, 15, 17
7, 24, 25
14, 28, 50
15, 36, 39
40, 75, 85
Multiple of x
Example 4
Ladder A 20 foot ladder is resting against the side of a house. The base of the ladder is 4 feet away
from the house. Approximately how high above the ground does the ladder touch the house?
Example 5
Real Estate An investor owns a triangular plot of land as shown in the diagram.
Find the perimeter of the plot of land.
One acre of land is equivalent to 43,560 square feet.
How many acres are in this plot of land? Round to
two decimal places.
The investor is planning on selling the land. The
market rate in this area is $5000 per acre. How
much should the investor ask for the land?
Unit 1: Essentials of Geometry
Lesson 1.4 Using the Midpoint and Distance Formulas
Lesson 1.3 from Geometry Textbook

Objectives
Use midpoint and distance formula to find the distance between two points and their midpoint.
MIDPOINTS AND SEGMENT BISECTORS
Example 1
The figure shows a gate with diagonal braces.
MO bisects NP at Q. If PQ is 22.6, find PN.
PN = ___________________
Example 2
Point M is the midpoint of VW . Find x and the length of VM .
VM = _________________________
COORDINATE PLANE
The Midpoint Formula
________________________________
Example 3
Find the coordinates of the midpoint M.
Find the coordinates of the endpoint K.
M = ___________________
K = ____________________
Example 4
Find the length of the segment and then find the coordinate of the midpoint of the segment.
Length = ____________
Midpoint _____________
DISTANCE FORMULA
The Distance Formula
If A(x1, y1) and B(x2, y2) are points in the coordinate plane,
then the distance between A and B is
___________________________________________
Example 5
Find the length of RS .
RS = ______________
Example 6
The coordinates of two segments are given. Find each segment length. Tell whether the segments are
congruent.
AB = ________________
CD = ___________________
AB  CD ? ___________________
Unit 1: Essentials of Geometry
Lesson 1.5: Measure and Classify Angles
Lesson 1.4 from Geometry Textbook

Objective
Find the measure of an angle using different postulates such as the Protractor Postulate and
Angle Addition Postulate.
 Classify angles as acute, right, obtuse, and straight.
 Use a protractor to construct and find the measure of an angle.
USING ANGLE POSTULATES
The angle that has sides AB and AC is denoted _______________________.
The point A is the ____________ of the angle.
Example 1
Name the angles in the figure.
____________________________________________
Protractor Postulate
Words: ____________________________________
Symbols: __________________________________
CLASSIFYING ANGLES
________________
________________
________________
_______________
Example 2
Use the diagram to find the measure of the indicated angle. Then classify the angle.
a) KHJ
b) GHK
_______________
________________
_______________
________________
c) GHJ
d) GHL
________________
________________
_______________
________________
Angle Addition Postulate
Example 3
Given that mLKN  145 , find mLKM and mMKN .
mLKM = _______________
mMKN = ________________
CLASSIFYING ANGLES
Angles that have the same measure are called _____________________.
MEASURES ARE EQUAL
ANGLES ARE CONGRUENT
______________________
______________________
Example 4
The figure shows angles formed by the legs of an ironing board. Identify the
congruent angles. If mHGI  40  , what is mGJK ?
______________________________
________________
Angle Bisectors
MN is the ______________________ of PMQ .
PMN  QMN
Example 5
In the diagram at the right, YW bisects XYZ , and mXYW  18. Find mXYZ .
mXYZ  _________________________
Unit 1: Basics of Geometry
Lesson 1.6 Describe Angle Pair Relationships
Lesson 1.5 from Geometry Textbook
Objective
 Classify pairs of angles as vertical, supplementary, complementary, and a linear pair.
 Apply understanding of angle pair relationship to find the measures of given angles.
COMPLEMENTARY AND SUPPLEMENTARY ANGLES
Example 1
In the figure, name a pair of complementary angles
and supplementary angles, and a pair of adjacent angles.
Example 2
Given that <1 is a complement of <2 and m<1 = 68o, find m<2.
______________
Given that <3 is a supplement of <4 and m<4 = 56o, find m<3.
______________
Example 3
Assume that <A is supplementary to <B and complementary to <C. Determine m<A, m<B, and
m<C.
m<A = x°, m<B = (x – 20)° and m<C = (x + 30)°
______________________________
VERTICAL ANGLES AND LINEAR PAIRS
______________________________
______________________________
Example 4
Identify all of the linear pairs and all of the vertical
angles in the figure at right.
Vertical angles ____________________________
Linear pairs ______________________________
Example 5
Two angles form a linear pair. One angle is 5 times larger than the other. Find the measures of the
two angles.
______________
_______________
Example 6
Solve for x and y.
Then find the angle measures.
x = ________
y = __________
Unit 1: Essentials of Geometry
Lesson 1.7 Classify Polygons
Lesson 1.6 from Geometry Textbook
Objectives
 Classify polygons as concave, convex, and regular.
 Classify polygons by the number of their sides.
IDENTIFY POLYGONS

____________________________________________________________________

____________________________________________________________________

____________________________________________________________________

___________________________________________________________________
CONCAVE AND CONVEX POLYGONS
Convex polygon _________________________________
Concave polygon ________________________________
CLASSIFYING POLYGONS
A polygon is named by its number of sides.
# of sides
3
4
5
6
8
9
10
12
7
Type of polygon
# of sides
Type of polygon
EQUILATERAL AND EQUIANGULAR
Equilateral polygon ______________________
Equiangular polygon _____________________
Regular polygon _________________________
n
Example 1
Classify the polygon by the number of its sides. Tell whether the polygon is equilateral, equiangular,
or regular. Explain your reasoning.
_________________________
________________________
_____________________
Example 2
Use your ruler and draw a concave, regular heptagon.
_________________________
Example 3
A racked of billiard balls is shaped like an equilateral triangle.
The expressions shown represent side lengths of the rack. Find the
lengths of each side.
x = ___________
side length ____________________
Example 4
Three vertices of a regular quadrilateral are A( 0, 4) , B(0, -4) ,
and C (4, 4). What would the other vertex of the figure be?
Example 5
The figure is a regular polygon. Expressions are given
for two side lengths. Find the value of x.
_____________________________
_____________________
Unit 1: Essentials of Geometry
Lesson 1.8 Perimeter, Circumference, and Area
Lesson 1.7 from Geometry Textbook

Objective
Use various formulas to find the perimeter, circumference, and area of closed plane figures
such as a quadrilateral, triangle, and circle.
 Develop and utilize a problem solving plan.
REVIEWING PERIMETER, CIRCUMFERENCE, AND AREA
Formulas for Perimeter, Area, and Circumference
Square
Rectangle
Side length s.
length l and width w
P = ________
P = _____________
A = ________
A = _____________
Triangle
Circle
Side lengths a, b, and c.
Base b and height h.
radius r
P = ________
P = _____________
A = ________
A = _____________
Example 1
Example 2
Find the area and perimeter of a rectangle
length 12 inches and width 5 inches.
Find the radius, circumference, and area of
a circle with a diameter of length 8 cm.
A = ___________
r = ____________
P = ___________
C = ____________
Example 3
Find the area and perimeter of the triangle defined
by D(1,3), E(8, 3) and F(4, 7)
A = _____________
USING A PROBLEM SOLVING PLAN
Example 4
You have a part-time job at a school. You need to buy enough
grass seed to cover the school’s soccer field. The field is 50 yards
wide and 100 yards long. The instructions on the seed bags say that
one bag will cover 5000 square feet. How many bags do you need?
Example 5
You are planning a deck along two sides of a pool. The pool
measures 18 feet by 12 feet. The deck is to be 8 feet wide.
What is the area of the deck?
Example 6
You are making a triangular flag with a base of 24 inches
and an area of 360 square inches. How long should it be?