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Geometry
UNIT 1
Tools of Geometry
Unit 1 Day 1
Points, Lines, and Planes
Page 8: 13 – 31, 40, 43 - 48, 49, 52, 53
Unit 1 Day 2
Linear Measure
Page 18: 10 – 19, 21, 23, 25, 27 - 32
Unit 1 Day 3
Distance and Midpoints
Page 31: 13 – 31 left column, 33-53 left column
Unit 1 Day 4
Angle measure
Quiz Review Page 45: 1 - 15
Unit 1 Day 5
QUIZ
Unit 1 Day 6
Angle Relationships
Page 51: 22-24, 29-32, 36-47
Unit 1 Day 7
Two-Dimensional Figures
Page 61: 12, 13, 15, 17, 18, 20, 27, 30abc, 40-42
Unit 1 Day 8
Three Dimensional Figures
Page 71: 18 - 23, 51-55
Review
TEST
VOCABULARY
Point
Definition
Congruent Segments
Ray
Interior
Obtuse
Complementary
Convex
Perimeter
Edge
Cone
Line
Defined Term
Construction
Opposite Rays
Exterior
Straight
Supplementary
n-agon
Circumference
Prism
Sphere
Plane
Space
Distance
Angle
Degree
Adjacent Angles
Perpendicular
Equilateral
Area
Base
Surface Area
Collinear
Line Segment
Midpoint
Side
Right Angle
Linear Pair
Polygon
Equiangular
Polyhedron
Pyramid
Volume
Intersection
Betweenness
Bisector
Vertex
Acute
Vertical Angle
Concave
Regular
Face
Cylinder
Points, Lines & Planes
A point is a location. It has
neither shape nor size.
A line is made up of points and
has no thickness or width.
There is exactly one line
through any two points.
A plane is a flat surface made
up of points that extends
infinitely in all directions.
There is exactly one plane
through any three points not
on the same line.
Named by: a capital letter
Named by: the capital letters
representing two points on the
line or a lower case script letter
Named by: a capital script letter
or by the letters naming three
points that are not all on the
same line.
Example:
Example:
Example:
●
 Points, Lines & Planes
On a subway map, the locations of stops are represented by points. The route the train can take is
modeled by a series of connected paths that look like lines. The flat surface of the map on which
these points and lines lie is representative of a plane.
Unlike real-world objects that they model, shapes, points, lines & planes do not have any actual
size. In geometry, point, line, & plane are considered undefined terms because they are only
explained using examples and descriptions.
Collinear points are points that lie on the same line. Noncollinear points do not lie on the same
line. Coplanar points are points that lie in the same plane. Noncoplanar points do not lie in the
same plane.
Example:
Name the geometric term modeled by each object.
a) stripes on a sweater
b) the corner of a box
c) a sheet of notebook paper (sort of)
1.
2.
3.
4.
5.
Recall From Algebra:
Solving Quadratic Equations:
 Set the equation equal to zero
 Factor using GCF, DOPS, or Trinomial
 “Fish” for the answer
Ex: q 2 – 2q + 1 = 0
Ex: 3x2 – 6x + 9 = 0
Ex: n2 + n = 30
Ex: x2 = 12x – 36
If K is between J and L: JK = x2 – 4x, KL = 3x – 2, JL = 28. Find JK and KL.
Distance & Midpoints
 Distance Between Two Points
The distance between two points is the length of the segment with those points as its endpoints.
(REMEMBER: Length cannot be negative!)
Distance Formula (on Number Line):
The distance between two points is the absolute value of the difference between the
coordinates.
Example:
See page 25 in your textbook (SMART BOARD).
1A) _________________ 1B) __________________ 1C) __________________
Example:
Find the distance between (1,0) and (5, 0).
Example:
Find the distance between (5, 7) and (-3, 7).
______________ is another name for x-value; ______________ is another name for y-value
Distance Formula (in the Coordinate plane) – compare to Pythagorean Theorem
d=
Example:
( x2  x1 ) 2  ( y2  y1 ) 2 or d =
(x) 2  (y ) 2
Find the distance between J(4, 3) and K(-3, -7).
 Midpoint of a Segment
The midpoint of a segment is the point halfway in between the endpoints of the segment. The
coordinates of the midpoint can be found by finding the average of the abscissas and the average of
the ordinates.
Midpoint Formula
 x  x 2 y1  y 2 
M=  1
,

2 
 2
Example:
Find the coordinates of the midpoint of a segment with the given coordinates.
A(5, 12) and B(-4, 8)
Example:
Find the coordinates of the missing endpoint if P is the midpoint of EG .
E(-8, 6) and P(-5, 10)
Example:
Find the measure of YZ if Y is the midpoint of XZ and XY = 2x – 3 and
YZ = 27 – 4x.
Angle Measure
One of the skills needed for carpentry is how to cut a miter joint. This joint is created when two
boards are cut at an angle to each other. One miscalculation in an angle measure can result in
mitered edges that do not fit together.
 Measure and Classify Angles
A ray is a part of a line. It has one endpoint and extends indefinitely in one direction. Rays are
named by stating the endpoint first and then any other point on the ray.
▪
If you choose a point on a line, that point exactly two rays called opposite rays. Sine both rays
share a common endpoint, opposite rays are collinear.
▪
An angle is formed by two noncollinear rays that have a common endpoint. The rays are called
sides of the angle. The common endpoint is called the vertex.
When naming angles using three letters, the vertex must be the middle of the three letters. You can
name an angle by using a single letter only when there is exactly one angle at that vertex.
Angles are measured in units called degrees. The degree results from dividing the distance around
a circle into 360 parts.
To measure an angle, use a protractor.
Classify Angles
Congruent angles have the same angle measure.
A ray that divides an angle into two congruent angles is called an angle bisector.
CONSTRUCT a congruent angle using the given angle and make a congruence statement.
CONSTRUCT an angle bisector using the given angle and make a congruence statement.
Angle Relationships
 Pairs of Angles
Special Angle Pairs
Adjacent angles are two angles that lie in the same plane and have a common vertex and a
common side, but no common interior points.
Example:
Nonexample:
A linear pair is a pair of adjacent angles with non common sides that are opposite rays.
Example:
Nonexample:
Vertical angles are two nonadjacent angles formed by two intersecting lines.
Example:
Nonexample:
Things that make you go “Hmmm…”
Linear Pair vs. Supplementary Angles
While the angles in a linear pair are always supplementary, some supplementary angles do not form
a linear pair. Why?
_____________________________________________________________________________
Angle Pair Relationships
Vertical angles are congruent.
Example:
Complementary angles are two angles with measures that have a sum of 90º.
Example:
Supplementary angles are two angles with measures that have a sum of 180º.
Example:
The angles in a linear pair are supplementary.
Example:
Page 51 in textbook:
#19 - 21
TWO DIMENSIONAL FIGURES
VOCABULARY:
Polygon:
Vertex:
Examples
Convex Polygons
“Non” Examples
vs.
Concave Polygons
No points of lines are in the interior
Some of the lines pass through the interior
Equilateral Polygon:
Number of
Sides
3
4
5
6
7
8
9
10
11
12
n
Equiangular Polygon:
Regular Polygon:
Polygon
Formulas
Triangle
Square
Rectangle
Circle
P=
P=
P=
C=
A=
A=
A=
A=
P = perimeter of the polygon
b = base
h = height
A = area of figure
l = length
w = width
C = circumference
r = radius
d = diameter
1.
Name each polygon by the number of sides.
Classify as concave or convex and regular or irregular.
2.
Find the perimeter or circumference and the area of the figures. Round to the
nearest tenth.
8m
17 m
d = 12.8 cm
15 m
3.
If I have vertices of A(-1, 2), B(3, 6) and C(3, -2), how might I find the perimeter and
area of the triangle formed?
THREE DIMENTIONAL FIGURES
**Do page 80 #38 – 41 as GUIDED PRACTICE