Download Unit 1 Student Notes - Mattawan Consolidated School

Geometry
Chapter 1 Foundations
Lesson 1: Understanding Points, Lines, and Planes
Learning Targets
Success Criteria
LT1-1: Identify, name, draw and solve problems
involving: points, lines, segments, rays and
planes.
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What?
Euclidean Geometry:
Name points, lines & planes (with accurate
notation).
Draw segments and rays (with accurate
notation).
Identify points and lines in a plane (with
accurate notation).
Represent intersections (with accurate notation).
So What?
Plane Geometry
The math system attributed to Euclid of Alexandria, a Greek
mathematician who write the Elements.
Deals with objects that are flat and located somewhere in space.
Coordinate
Geometry
A system of geometry where the position of points are
described using an ordered pair (x, y).
Undefined Terms:
point:
These terms are defined to fit the branch of mathematics to be
studied.
Names a location; has no size; 0-D
line:
A straight path that has no thickness and
extends forever in two directions. (set of points); 1-D
plane:
A flat surface that has no thickness and extends
forever in all directions; 2-D
Collinear:
(3 or more) Points that lie on the same line.
Coplanar:
(4 or more) Points that lie on the same plane.
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Segment:
Endpoint:
A part of a line consisting of two points (endpoints)
and all points in between.
A point at one end of a segment or the starting point of a ray.
Ray:
A part of a line that starts at one endpoint and
extends forever in one direction.
Opposite rays:
Two rays that have a common endpoint
and form a line.
What?
Postulate (axiom):
So What?
A statement accepted true without proof. An assumption.
Point, Line, Plane
Postulate:
1-1-1 Unique Line
Assumption
2 points determine a unique line.
1-1-2 Unique Plane
Assumption
3 points determine a unique plane.
1-1-3 Flat Plane
Assumption
If two points lie in a plane, then the line
containing the points also lies in the plane.
Intersection of Lines
and Planes Postulate:
1-1-4 Line
Intersection
2 lines intersect at 1 point.
1-1-5 Plane
Intersection
2 planes intersect at a line.
Ex#1: Name points, lines & planes (with accurate notation).
Using the figure at the right:
A. Name three points that are collinear.
B. Name three points that are coplanar.
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Ex#2: Identify points and lines in a plane (with accurate notation).
Use the figure at the right to name each of the following:
A. a line containing point Q.
B. a plane containing points P and Q.
C. Name the line three different ways.
Ex#4: Draw segments and rays (with accurate notation).
A. Draw and label a segment with endpoints M and N. B. Draw and label opposite rays with common
endpoint T.
C. Is &
TH the same as &
HT ? Explain.
Ex#5: Represent intersections (with accurate notation).
A. Sketch two lines intersecting in exactly one point.
B. Sketch a line intersecting a plane.
C. Sketch a line that is contained in Plane Q.
D. Sketch three noncollinear points that are contained
in Plane T.
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Lesson 2: Measuring and Constructing Segments
Learning Targets
Success Criteria
LT1-2: Calculate and construct midpoints, segment
bisectors, and segment lengths.
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Find the length of a segment.
Copy a segment.
Use the Segment Addition Postulate.
Apply measurements and constructions to realworld applications.
What?
So What?
Coordinate:
A point that corresponds to one (and only one) number on a ruler.
Ruler Postulate:
The points on a line can
be put into a one-to-one
correspondence with the
real numbers.
Distance:
The absolute value of the difference
of the coordinates.
AB means the distance from A to B
Congruent segments:
Segments that have the same length.
Segment Addition
Postulate:
If A is between M and P, then
MA + AP = MP.
Between:
A point is between two others if its coordinate is greater than one
endpoint and less than the other.
Midpoint:
(of a segment) is a point, M, on the line AB with AM = MB.
Bisect:
To cut into two equal parts.
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Segment bisector:
A ray, segment, or line that intersects a segment at its midpoint.
Perpendicular bisector: A bisector of a segment that is also perpendicular to it.
Construction:
A way of creating a figure using only a compass and an unmarked
straightedge.
The length or measure of a segment always includes a unit of measure, such as inches, centimeter, etc.
Ex#1: Find the length of a segment.
Find the following..
A. KM=
B. JN=
C. IL=
Caution: KM represents a number, while
KM represents a geometric figure.
Caution: Be sure to use equality for numbers (AB = YZ)
and congruence for figures ( AB ≅ YZ )
1.2 Construction: Congruent Segments
You will need a clean sheet of paper and a compass for this construction.
1.2 Construction: Segment Bisector, Perpendicular Bisector, and Midpoint
You will need a clean sheet of paper and a compass for this construction.
Constructions: http://www.matcmadison.edu/is/as/math/kmirus/
www.whistleralley.com/construction/reference.htm
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Ex#2: Use the Segment Addition Postulate.
A. If Y is between X and Z. If XY = 17 and XZ = 42,
what is YZ? (Draw a diagram.)
B. Find y and QP if P is between Q and R, QP = 2y,
QR = 3y + 1, and PR = 21. (Draw a diagram.)
C. K is the midpoint of JL . If JK = 3x – 4
and LK = 5x – 26, find x and JL.
D. X is the midpoint of ZW . XW = 9 – 2a
and ZW = 6a – 9. Find ZX.
Lesson 3: Measuring and Constructing Angles
Learning Targets
Success Criteria
LT1-3: Name, measure, classify, and construct angles
and their bisectors.
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What?
Angle
Vertex
Interior of an ∠
Exterior of an ∠
Name angles using proper notation.
Measure and classify angles.
Use the Angle Addition Postulate.
Find the measure of an angle.
So What?
Angle = a figure formed by the union
of two rays.
T
Vertex = the common endpoint
O
P
How to name an angle:
Angles are named in various ways:
● You can name an angle by a single letter only when there is one angle
shown at that vertex.
● When there is more than one angle at that vertex you must name the angle
with three letters.
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How to name an angle:
Measure:
Degree: ( º )
Protractor Postulate:
Congruent Angles:
The amount of openness of 2 rays that form an angle. Measured in
degrees ( º ).
The common measure of an angle; 1/360 of a circle.
Given line AB and a point O on line AB, all rays that can be drawn
can be put into a one-to-one correspondence with the real numbers
0 to 180.
Angles that have the same measure.
O
C
G
T
A
Angle Addition
Postulate:
If S is in the interior of ∠PQR, then
m∠PQS + ∠SQR = ∠PQR.
Angle Bisector:
A ray that divides an angle into two
congruent angles.
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D
Types of Angles:
Acute
Right
L
Obtuse
X
Straight
P
M
Q
N
measure is 0 < m < 90
Y
Z
measure is 90
R
measure is 90 < m < 180
A
B
C
measure is 180
Ex#1: Measure and classify angles.
Use the diagram to find the measure of each angle.
Then classify each as acute, right, obtuse, or straight.
A. ∠DAB
B. ∠BAE
C. ∠EAD
D. ∠CAD
1.3 Construction: Congruent Angles
1.3 Construction: Angle Bisector
Ex#2: Use the Angle Addition Postulate.
A. If m∠LPR = 127, find each measure.
Find m∠LPE
Find m∠TPR
B. Suppose m∠ATC = 145, m∠ ATY = 6b + 10, and
m∠CTY = 3b + 9.
Find b.
Find m∠ATY.
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Ex#4: Use the Angle Addition Postulate.
IT bisects ∠BIS . If m∠ BIT= 37°
A. Suppose &
Find m∠ BIS.
IT bisects m∠BIS. m∠BIT = 12x + 3
B. Suppose &
and m∠TIS = 10x + 10. Find x and m∠BIS
IT bisects ∠BIS . If m∠ BIS = 44° and m∠TIS = 10x – 13. Find x,
C. Suppose &
Constructions: www.whistleralley.com/construction/reference.htm
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Lesson 4: Pairs of Angles
Learning Targets
Success Criteria
LT1-4: Classify pairs of angles as adjacent, vertical,
complementary, or supplementary and solve
problems involving them.
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Pairs of Angles
Adjacent Angles:
Two angles in the same plane with a common
vertex and a common side, but no common
interior points.
Linear Pair:
A pair of adjacent angles whose noncommon
sides form opposite rays.
Vertical Angles:
Two nonadjacent angles formed by intersecting
lines.
Complementary Angles:
Two angles whose measures have a sum of 90º.
Supplementary Angles:
Two angles whose measures have a sum of
180º.
Linear Pair Theorem:
If two angles form a linear pair, then they are
supplementary.
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Identify angle pairs and use them to solve
problems.
Find the measures of complements and
supplements.
Use complements and supplements to solve
problems.
Apply knowledge of angles and congruency to
real-world applications.
Identify vertical angles.
Vertical Angle Theorem:
If two angles are vertical angles, then they are
equal in measure.
Ex#1: Find the measures of complements and supplements.
A. Tell whether the angles are only adjacent, adjacent B. Suppose two angles ∠3 and ∠4 are supplementary.
and linear, or not adjacent.
If m∠3 = 47, what is m∠4?
∠5 and ∠6
∠7 and ∠SPU
∠7 and ∠8
S
R
8
Q
P
6
T
7
5
U
C. Suppose two angles ∠3 and ∠4 are
complementary. If m∠3 = x - 28, what is m∠4?
D. An angle is 10 more than 3 times the measure of its
complement. Find the measure of the complement.
Ex#2: Identify angle pairs and use them to solve problems.
A. ∠1 and ∠2 form a linear pair. Suppose
m∠1 = 11n + 13 and m∠2 = 5n – 9. Find n
and m∠1.
B. Suppose m∠1 = 62, find as many angles as you
can in the figure at the right.
If m∠1 = 10k, find as many angles as you can in the
figure above.
In geometry, figures are used to depict a situation. They are not drawn to reflect total accuracy of the situation.
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From a figure, you can assume:
From a figure you cannot assume:
1. Collinearity and betweenness of points drawn
on lines.
1. Collinearity of three of more points that are not
drawn on lines.
2. Intersection of lines at a given point.
2. Parallel lines.
3. Points in the interior of an angle, on an angle, or in 3. Exact measures of angles and lengths of segments.
the exterior of an angle.
4. Measures of angles or lengths of segments are
equal.
Lesson 5: Using Formulas in Geometry
Learning Targets
Success Criteria
LT1-5: Calculate basic perimeter and area of squares,
rectangles, triangles, and circles.
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Find perimeter and area of figures.
Apply geometric formulas to real-world
applications.
Find the circumference and area of a circle.
What?
So What?
Perimeter:
The sum of the side lengths of a figure.
Area:
Base and Height/
(Altitude)
The number of nonoverlapping square units of a given size that
exactly cover a figure.
Base = any side of a triangle
or polygon.
Height/Altitude = a segment from a
vertex that forms a right angle
with a line containing the base.
Diameter & Radius
Diameter = A segment that passes through the
center of a circle and whose endpoints
are on the circle.
Radius = a segment whose endpoints are
the center of the circle and a points on
the circle.
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d i a m
e t e r
r a d i u s
Circle
The set of points in a plane at a certain distance
(radius) from a given point (center).
Circumference:
The distance around a circle.
C = 2πr or
C = πd
Pi:
π = the ratio of a circles circumference to its distance across
=
C
≈ 3.14
d
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Perimeter and Area Formulas
Rectangle
Square
Triangle
P=
P=
P=
A =
A=
A=
Ex#1: Find perimeter and area of figures.
A.
B.
5x
4 in
x + 4
6
6 in
C. The Queens Quilt block includes 12 blue triangles. D. The base of a rectangle is 5 more than 2 times its
The base and height of each triangle are about 4 in.
height. Find the perimeter and area of the
Find the approximate amount of fabric used to
rectangle.
make the 12 triangles.
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Circumference and Area of a Circle
C=
A=
r a d i u s
Ex#2: Finding the Circumference and Area of a Circle
A. Find the exact area and circumference of a circle
B. Find the area and circumference of a circle with a
whose radius is 14 meters.
diameter of 12cm. Round your answers to the
nearest hundredth.
Lesson 6: Midpoint and Distance in the Coordinate Plane
Learning Targets
Success Criteria
LT1-6: Calculate distance and midpoint between two
points in the coordinate plane.
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What?
Find the coordinates of a midpoint.
Find the coordinates of an endpoint.
Use the distance formula.
Find distances in the coordinate plane.
So What?
Coordinate Plane/
Cartesian Plane:
A plane that is divided into 4 regions
by a horizontal line (x-axis) and
a vertical line (y-axis). Location
of points is (x, y).
Midpoint Formula:
The midpoint M of
found by
Distance Formula:
The distance between two points ' x 1 , y 1( and ' x 2 , y 2 ( is
AB with endpoints A ' x 1 , y 1( and B ' x 2 , y 2 ( is
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Pythagorean Theorem: In a right triangle, the sum of the squares
of the lengths of the legs is equal to the
squares of the length of the hypotenuse.
a and b are called ________
c is called _______________
Ex#1: Find the coordinates of a midpoint or endpoint.
A. Find the coordinates of the midpoint of AB with B. M is the midpoint of XY . X has coordinates
endpoints A(-8, 3) and B(-2, 7).
(2, 7), and M has coordinates (6, 1). Find the
coordinates of Y.
Ex#2: Use the distance formula.
Which segments are congruent? Show your work.
You can also use the Pythagorean Theorem to find the distance between points in the coordinate plane.
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Ex#3: Find distances in the coordinate plane.
A. Graph the following points R(3, 4) and S(-2, -5)
B. Use the distance formula and the Pythagorean
Theorem to find the distance to the nearest
hundredth.
Distance Formula:
Pythagorean Thrm:
Ex#4: Find distances in the coordinate plane or using the distance formula.
The four bases on a baseball field form a square with 90 foot sides. A player throws the ball from first base
to a point located between third base and home plate and 10 feet from third base. What is the distance of the
throw, to the nearest tenth?
Lesson 7: Transformation in the Coordinate Plane
Learning Targets
Success Criteria
LT1-7: Identify and graph reflections, rotations, and
translations in the coordinate plane.
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Identify transformations from a picture and use
arrow notation to describe it.
Draw and identify transformations.
Perform translations in the coordinate plane.
Transformation:
A change in the position, size, or shape of a
figure.
Preimage:
The original figure (or point). A
Image:
The resulting figure (or point). Labeled with A'.
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B'
A transformation maps the preimage to the image.
B
A'
A
C'
C
Transformations
Reflection
Rotation
Translation
A reflection is a transformation
A rotation is a transformation about
across a line, called the line of
a point P, called the center of
reflection. Each point and its image rotation. Each point and its image
are the same distance from P.
T'
T
O'
D'
A'
A
C
E
L
O
E'
C'
S
G'
D
G
are the same distance from the line
of reflection.
L'
A
S'
P
A'
Notation:
Notation:
Notation:
A translation is a transformation in
which all the points of a figure move
the same distance in the same
direction.
Ex#1: Name the transformation. Then use arrow notation to describe the transformation.
M '
A.
B.
C.
T
E
I
A'
K
T'
K'
M
E'
I'
A
T
C
C'
R
E
E'
T'
D'
D
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R'
Ex#2: Drawing and Identifying the Transformations
A. A figure has vertices E(2, 0), F(2, -1), G(5, -1) and B. A figure has vertices at A(1, -1), B(2, 3), and
H(5, 0). After a transformation, the image of the
C(4, -2). After a transformation the image of the
figure has vertices at E'(0, 2), F'(1, 2), G'(1, 5), and
figure has vertices at A'(-1, -1), B'(-2, 3), and
H'(0, 5).
C'(-4, -2).
In the coordinate plane, to find the coordinates for the image, add a to the x-coordinates of the preimage and add
b to the y-coordinate of the preimage.
Translation Rule:
' x , y (%' x$a , y$b (
Ex#3: Perform translations in the coordinate plane.
A. A figure has vertices A(-4, 2), B(-3, 4), and
B. A figure has vertices J(1, 1), K(3, 1), M(1, -4), and
C(-1, 1). Find the coordinates for the image of
L(3, -4). Find the coordinates for the image of
∆ABC after the translation ' x , y (%' x$2, y−1(
JKLM after the translation ' x , y (%' x−2, y$4( .
Draw the preimage and image.
Draw the preimage and image.
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Chapter 1 Homework
Section
Problems
Tools
1.1
p. 9-11 #13-25, 28-42
ruler
1.2
p. 17-19 #11, 12, 14, 15, 17, 18, 20-31, 34-39, 48
ruler
1.3
p. 24-27 #12-27, 29-32, 33, 40-45
protractor
1.4
p. 31-33 #14-24, 27, 30, 34-38even, 40-42, 45
1.5
p. 38-41 #10, 11, 13, 15, 17-25, 28, 29, 33, 35, 39, 4143, 47-51
1.6
p. 47-49 #12-18even, 21-25, 31, 32, 35, 36, 42, 43, 48
graph paper
1.7
p. 53-55 #8-17, 19-23, 25-27, 29-32, 41, 44
graph paper
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