Download Geometry Chapter 1 Foundations Lesson 1

Geometry
Chapter 1 Foundations
Lesson 1: Understanding Points, Lines, and Planes
Learning Targets
Success Criteria
LT-1: Identify, name, draw and solve problems
involving: points, lines, segments, rays and
planes.
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•
•
•
What?
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?
Euclidean Geometry:
Plane Geometry
Coordinate Geometry
Undefined Terms:
point:
line:
plane:
Collinear:
Coplanar:
Segment:
endpoint:
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Ray:
Opposite rays:
What?
So What?
Postulate (axiom):
Point, Line, Plane
Postulate:
1-1-1 Unique Line
Assumption
1-1-2 Unique Plane
Assumption
1-1-3 Flat Plane
Assumption
Intersection of Lines and
Planes Postulate:
1-1-4 Line Intersection
1-1-5 Plane Intersection
Ex: 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: Identify points and lines in a plane (with accurate notation).
#2. 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.
#3. Name the line three different ways.
Ex: Draw segments and rays (with accurate notation).
#4. Draw and label a segment with endpoints M and N. #5. Draw and label opposite rays with common
endpoint T.
#6. Is &
TH the same as &
HT ? Explain.
Ex: Represent intersections (with accurate notation).
#7. Sketch two lines intersecting in exactly one point. #8. Sketch a line intersecting a plane.
#9. Sketch a line that is contained in Plane Q.
#10. Sketch three noncollinear points that are contained
in Plane T.
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Lesson 2: Measuring and Constructing Segments
Learning Targets
Success Criteria
LT-2: Calculate and construct midpoints, segment
bisectors, and segment lengths.
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•
•
•
What?
Find the length of a segment.
Copy a segment.
Use the Segment Addition Postulate.
Apply measurements and constructions to realworld applications.
So What?
Coordinate:
Ruler Postulate:
Distance:
Congruent segments:
Segment Addition
Postulate:
Between:
Midpoint:
Bisect:
Segment bisector:
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Perpendicular bisector:
Construction:
The length or measure of a segment always includes a unit of measure, such as inches, centimeter, etc.
Ex: Find the length of a segment.
#1. 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
Ex: Use the Segment Addition Postulate.
#2. If Y is between X and Z. If XY = 17 and XZ = 42, #3. Find y and QP if P is between Q and R, QP = 2y,
what is YZ? (Draw a diagram.)
QR = 3y + 1, and PR = 21. (Draw a diagram.)
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#4. K is the midpoint of JL . If JK = 3x – 4
and LK = 5x – 26, find x and JL.
#5. 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
LT-3: Name, measure, classify, and construct angles
and their bisectors.
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•
•
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What?
Name angles using proper notation.
Measure and classify angles.
Use the Angle Addition Postulate.
Find the measure of an angle.
So What?
Angle
Vertex
Interior of an ∠
Exterior of an ∠
T
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.
Measure:
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Degree:
Protractor Postulate:
Congruent Angles:
O
C
G
T
D
A
Angle Addition Postulate:
Angle Bisector:
Types of Angles:
Acute
L
Right
Obtuse
X
Straight
P
M
N
measure is 0 < m < 90
Q
Y
Z
measure is 90
R
measure is 90 < m < 180
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A
B
measure is 180
C
Ex: Measure and classify angles.
#1. Use the diagram to find the measure of each angle.
Then classify each as acute, right, obtuse, or straight.
∠DAB
∠BAE
∠EAD
∠CAD
1.3 Construction: Congruent Angles
1.3 Construction: Angle Bisector
Ex: Use the Angle Addition Postulate.
#2. If m∠LPR = 127, find each measure.
a. Find m∠LPE
b. Find m∠TPR
Ex: Use the Angle Addition Postulate.
IT bisects ∠BIS . If m∠ BIT= 37°
#4. Suppose &
Find m∠ BIS.
#3. Suppose m∠ATC = 145, m∠ ATY = 6b + 10, and
m∠CTY = 3b + 9.
a. Find b.
b. Find m∠ATY.
IT bisects m∠BIS. m∠BIT = 12x + 3
#5. 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,
#6. Suppose &
Constructions: www.whistleralley.com/construction/reference.htm
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Lesson 4: Pairs of Angles
Learning Targets
Success Criteria
LT-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:
Linear Pair:
Vertical Angles:
Complementary Angles:
Supplementary Angles:
Linear Pair Theorem:
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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:
Ex: Find the measures of complements and supplements.
#1. Tell whether the angles are only adjacent, adjacent #2. Suppose two angles ∠3 and ∠4 are supplementary.
and linear, or not adjacent.
If m∠3 = 47, what is m∠4?
a. ∠5 and ∠6
S
R
b. ∠7 and ∠SPU
c. ∠7 and ∠8
8
Q
P
6
T
7
5
U
#3. Suppose two angles ∠3 and ∠4 are
complementary. If m∠3 = x - 28, what is m∠4?
#4. An angle is 10 more than 3 times the measure of its
complement. Find the measure of the complement.
Ex: Identify angle pairs and use them to solve problems.
#5. ∠1 and ∠2 form a linear pair. Suppose
m∠1 = 11n + 13 and m∠2 = 5n – 9. Find n
and m∠1.
#6. 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
LT-5: Calculate basic perimeter and area of squares,
rectangles, triangles, and circles.
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What?
Find perimeter and area of figures.
Apply geometric formulas to real-world
applications.
Find the circumference and area of a circle.
So What?
Perimeter:
Area:
Base and Height/
(altitude)
Diameter:
d i a m e t e r
r a d i u s
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Radius:
Circumference:
Pi:
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Perimeter and Area Formulas
Rectangle
Square
Triangle
P=
P=
P=
A =
A=
A=
1.5 Ex: Find perimeter and area of figures.
#1.
#2.
5x
4 in
x + 4
6
6 in
#3. The Queens Quilt block includes 12 blue triangles. #4. 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
1.5 Ex: Finding the Circumference and Area of a Circle
#5. Find the exact area and circumference of a circle
#6. 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
LT-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:
AB with endpoints A ' x 1 , y 1( and B ' x 2 , y 2 ( is
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
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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 _______________
1.6 Ex: Find the coordinates of a midpoint.
1.6 Ex: Find the coordinates of an endpoint.
#1 Find the coordinates of the midpoint of AB with #2. 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.
1.6 Ex: Use the distance formula.
#3. 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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1.6 Ex: Find distances in the coordinate plane.
#4. Graph the following points R(3, 4) and S(-2, -5)
#5. Use the distance formula and the Pythagorean
Theorem to find the distance to the nearest
hundredth.
Distance Formula:
Pythagorean Thrm:
1.6 Ex: Find distances in the coordinate plane or using the distance formula.
#6. 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
LT-7: Identify and graph reflections, rotations, and
translations in the coordinate plane.
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Transformation:
Preimage:
Image:
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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.
B'
B
A transformation maps the preimage to the image.
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.
1.7 Ex: Name the transformation. Then use arrow notation to describe the transformation.
M '
#1.
#2.
#3.
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'
1.7 Examples: Drawing and Identifying the Transformations
#4. A figure has vertices E(2, 0), F(2, -1), G(5, -1) and #5. 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 (
1.7 Ex: Perform translations in the coordinate plane.
#6. A figure has vertices A(-4, 2), B(-3, 4), and
#7. 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, 14-18, 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, 26-27, 29-30, 34-38even, 40-42, 45
1.5
p. 38-41 #10-26, 28, 29, 33, 34, 36, 39, 41-44, 47-51
1.6
p. 47-49 #12-18even, 21-27, 30-38, 42, 43, 48
graph paper
1.7
p. 53-55 #8-17, 19-23, 25-27, 29-32, 42, 44
graph paper
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