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Transcript
Geometry 2205
Unit 1:
Mrs. Bondi
Unit 1 Coordinate Geometry Topics
Lesson 1: Points, Lines, and Planes (PH text 1.2)
Lesson 2: Measuring Segments (PH text 1.3)
Lesson 3: Measuring Angles (PH text 1.4)
Lesson 4: Exploring Angle Pairs (PH text 1.5, 2.6)
Lesson 5: Midpoint and Distance in the Coordinate Plane (PH text 1.7)
Lesson 6: Perimeter, Circumference and Area (PH text 1.8)
Lesson 7: Patterns and Inductive Reasoning (PH text 2.1)
Lesson 8: Conditional Statements (PH text 2.2)
Lesson 9: Biconditionals and Definitions (PH text 2.3)
Lesson 10: Deductive Reasoning (PH text 2.4)
Lesson 11: Reasoning in Algebra and Geometry (PH text 2.5)
Lesson 12: Lines and Angles (PH text 3.1)
Lesson 13: Properties of Parallel Lines (PH text 3.2)
Lesson 14: Proving Lines Parallel (PH text 3.3)
Lesson 15: Parallel and Perpendicular Lines (PH text 3.4)
Lesson 16: Parallel Lines and Triangles (PH text 3.5)
Lesson 17: Equations of Lines in the Coordinate Plane (PH text 3.7)
Lesson 18: Slopes of Parallel and Perpendicular Lines (PH text 3.8)
Big Ideas and Essential Questions:

Some geometric relationships can be described as functional relationships.

Relations and functions are mathematical relationships that can be represented and analyzed by
using words, tables, graphs, and equations.

Numbers, measures, expressions, equations, and inequalities can represent mathematical
situations and structures in many equivalent forms.

Mathematical statements can be justified through deductive and inductive reasoning and proof.
 How can you use coordinates and algebraic techniques to represent, interpret, and verify
geometric relationships?
 How do you use the ideas of direct and indirect proof, and counter-examples to verify valid
conjectures and refute invalid conjectures?
1
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 1: Points, Lines, and Planes (PH text 1.2)
Objective: Student will be able to
Why???
1) Understand basic terms of geometry
2) Understand basic postulates of geometry
To lay the foundation for your study of geometry
Vocabulary:
Definition:
Diagram:
Point:
Line:
Plane:
Space:
Segment:
Ray:
Opposite Rays:
(Yes)
B
A
(No)
E
C
D
2
F
Geometry 2205
Unit 1:
Vocabulary:
Mrs. Bondi
Definition:
Diagram:
Collinear Points:
Non-collinear Points:
Coplanar:
Non-coplanar:
Intersection:
Postulate:
Axiom:
3
Geometry 2205 Mrs. Bondi
Unit 1:
Postulate 1-1:
Through any two points A and B there is
Draw an example:
“exactly” _____________________________
Postulate 1-2:
If two distinct lines intersect, then they intersect
Draw an example:
in “exactly” ____________________________
Line AB intersects line CD at point P.
Postulate 1-3:
If two planes intersect, then they intersect
R
T
in _______________________________________
W
S
The intersection of plane RST and plane WST is _______.
Postulate 1-4:
Through any three non-collinear points
E
H
F
G
there is “exactly” ________________________
D
Points A, B, and C determine plane _______.
A
B
There are three possible ways to determine a specific plane (draw an example of each)
a) 3 non-collinear points b) a line and a point not on the line
4
c) 2 different intersecting lines
C
Geometry 2205 Mrs. Bondi
Unit 1:
Use figure 1 to answer #1 – 3.
P
N
T
H
O
E
1) Point ______ and point _______ are collinear with point R and point H.
R
S
2) The intersection of line PR and line OT is ______.
W
3) Line ______ and line ______ intersect at point O.
Figure 1
Use Figure 2 to answer # 4 – 7.
4) Name a point that is coplanar with points F, B, and C. ______
5) What is the intersection of plane HFDG and plane BCDF? ______
6) What is the intersection of plane AEG and plane BAC? ______
H
G
E
A
F
B
D
Figure 2
7) What is the intersection of plane HACD and line GD? ____________
8) Which of the following is not an acceptable name for the plane shown below?
A) Plane RSZ
D) Plane RSTW
B) Plane RSWZ
C) Plane WSZ
E) Plane STZ
R
Do together: p.16 #1-6, 62-64
HW day 1: p.16 #8-14, 67-80
HW day 2: p.16 #15-26,28-46 even, 50, 54-58 even
5
T
W
S
Z
C
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 2: Measuring Segments (PH text 1.3)
Objective:
Student will be able to determine the length of a segment
Vocabulary:
coordinate –
distance –
congruent segments –
midpoint –
segment bisector –
Postulate 1-5: (Ruler Postulate)
Every point on a line can be paired with a real number, called the coordinate of the point.
The distance between any two points on the number line is the ___________________
___________________ of the ____________________ of the coordinates.
A
Postulate 1-6: (Segment Addition Postulate)
C
B
If three points A, B, and C are ____________________ and B is ___________________
A and C, then AC = _______ + ________
Example:
1) Point M is on LN . LM = 23 and MN = 18. What is the measure of LN ?
2)
2) If AC  50 , AB  2x  8 and BC  3x 12 , then solve for x to determine the length
of AB , and BC .
x = _____
A
B
C
AB = _____
BC = _____
6
Geometry 2205 Mrs. Bondi
Unit 1:
Congruent Segments: segments with the ______________ _______________________
If AB  CD then AB  CD.
Midpoint of a segment – divides a segment into _______ ____________ ____________________
Label as CD and use tick marks to
show that M is the midpoint of CD .
Segment bisector – a point, line, or ray that intersects a segment at its __________________
Label CD and midpoint M to show that MN
is the segment bisector of CD .
Example:
2) In the first diagram B is the midpoint of AC . Solve for x and find the
length of AB , BC , and AC if AB  6x  4 and BC  3x  8 .
x = _____
A
B
C
AB = _____
BC = _____
AC = _____
Do together: p.23 #1-7, 19
HW: p.24 #8-28 even, 29, 35-36
(Copy diagrams into your homework. DO NOT WRITE IN THE TEXTBOOK!)
7
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 3: Measuring Angles (PH text 1.4)
Objective:
A
Student will be able to determine the measure of an angle
N
Vocabulary:
G
Angle: formed by two _____________ with the same ______________________
Sides:
Vertex: a point where two or more _________________ meet or where two sides of
a polygon meet. The plural is ________________________
Angle
Postulate 1-7: (Protractor Postulate)
Every ray on an angle can be paired with a real number from 0 to 180.
Find the measure of each of the following angles using the protractor.
4) m CAT = _______
5) m MAF = _______
6) m MAT = _______
7) m FAC = _______
8) m CAM = _______
9) m TAF = _______
Types of angles classified by angle measure:
Zero Angle: an angle with a measure of _______.
Acute Angle: an angle with a measure between _______ and _______.
Right Angle: an angle with a measure of _______.
Obtuse Angle: an angle with a measure between _______ and _______.
Straight Angle: an angle with a measure of _______.
Congruent Angles: angles with _____________________________________.
If mA  mB , then A  B .
8
Vertex
Geometry 2205 Mrs. Bondi
Unit 1:
Postulate 1-8: (Angle Addition Postulate)
A
O
If point B is in the interior of AOC ,
then mAOC = __________ + __________.
C
B
B
A
O
C
Examples:
1) M is in the interior of LON . LOM = 34o. MON = 18o.
What is the measure of LON ?
B
2) In the diagram to the right, mAOB =52 mBOC = _______
A
O
C
Practice:
1) Use letters to name the numbered angles in the given diagrams.
A
a) 1 = ____________ or _____________
B
1
2
b) 2 = ____________ or _____________
D
c) 3 = ____________ or _____________ or _____________
C
F
3
E
2) If the mGDE  2x  3 , mFDE  6x  1 , and mGDF  3x  78 , solve
for x to determine the mGDE , mFDE , and mGDF .
x = _____
G
mGDE = _____
F
E
mFDE = _____
D
mGDF = _____
Do together: p.31 #1-5
HW: p.31 #6-23 even, 40 (Copy diagrams!)
9
G
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 4: Exploring Angle Pairs (PH text 1.5, 2.6)
Objective:
Students will be able to identify special angle pairs
and use their relationships to find angle measures.
Review: Draw a diagram where B is in the interior of AOC . If mAOB  40 , mBOC  3x 15 ,
and mAOC  4x 15 solve for x and justify each step.
Angle bisector - ____________________________________________
Label LOV , and draw point E in the interior of the angle.
Mark the diagram to show that OE bisects LOV .
Practice:
2) In the second diagram DF is the angle bisector of GDE . Solve for x
1
and find the measure of GDF and EDG if mGDF  x  9 and mEDG  3x  42 .
2
x = _____
E
mGDF = _____
F
mEDG = _____
D
G
Important Angle Pairs:
Vertical Angles:
Adjacent Angles:
Example:
Example:
Complementary Angles:
Supplementary Angles:
Example:
Example:
10
Geometry 2205
Unit 1:
Mrs. Bondi
Important Postulate and Theorems:
Postulate 1-9 Linear Pair Postulate
If two angles for a linear pair, then they are supplementary.
Theorem 2-1 Vertical Angles Theorem:
Vertical angles are congruent
Theorem 2-2 Congruent Supplements Theorem:
If two angles are supplements of the same angle (or of congruent angles), then the two angles
are congruent
Theorem 2-3 Congruent Complements Theorem:
If two angles are complements of the same angle (or of congruent angles), then the two
angles are congruent
Theorem 2-4 All right angles are congruent.
Theorem 2-5 If two angles are both congruent and supplementary, then each is a right angle.
NEVER ASSUME!
A linear pair can be assumed to be on a straight line. NO other assumptions should be made! You
must see congruent markings, or be told of segment/angle relationships. DO NOT ASSUME!!!
HW day 1: p.38 #7-41 (Copy diagrams!)
HW day 2: p.124 #6-11, 14-17, 19, 20-22 (Copy diagrams!)
Good Idea: p.41 Mid-Chapter Quiz
FUN STUFF! Basic constructions, constructing shapes with a straight edge and compass is quite interesting. North Penn is not including it as a
part of our curriculum, but I will gladly work through this with you outside of class time. If we have time at the end of the year, we will return to
it. There are several lessons on constructions in your textbook, beginning with lesson 1.6.
11
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 5a: Midpoint Formula in the Coordinate Plane (PH text 1.7)
Objectives:
to find the midpoint of a segment.
The midpoint of a segment is the halfway point between two endpoints. The
coordinates of a midpoint are the averages of the coordinates of the endpoints.
The midpoint of two coordinates on a number line can be found by ____________________________
-3
10
The midpoint formula: For endpoints P  x1 , y1  and Q  x2 , y2  on the coordinate plane the midpoint
m can be expressed by:
x x y y 
M = 1 2 , 1 2 
2 
 2
Examples:
1. Find the midpoint of A(2, -1) and B(4, -3)
2. Find the midpoint of P(−1, 6) and Q(5, 0)
We can also use the midpoint formula to find an endpoint if we know the midpoint and one of the
endpoints.
Example:
3. Given the midpoint of AB is M(2, 6), and endpoint A(-8, 9), find the coordinates of endpoint B.
4. Given the midpoint of AB is M(4.5, 0.5), and endpoint C(7, 5), find the coordinates of endpoint D.
12
Geometry 2205
Unit 1:
Mrs. Bondi
HW: p.54 #6-21
Lesson 5b: Distance Formula in the Coordinate Plane (PH text 1.7)
Objectives:
to find the distance between two points in a coordinate plane.
Coordinate Plane Review:
1. Label the following on the coordinate plane below: Quadrants, x-axis, y-axis, origin
2. Graph the points ( –3, 4 ), ( 1, 1 ), ( –3, 1) and connect them to form a triangle.
3. Mark the lengths of the legs of the triangle by counting units.
Use the Pythagorean Theorem to
find the length of the hypotenuse.
a 2  b2  c2
Now use the distance formula to find
the length between (1,1) and ( –3, 4 ).
d=
 x2  x1 
13
2

 y 2  y1

2
Geometry 2205
Unit 1:
Mrs. Bondi
The distance formula: For points P  x1, y1  and Q  x2 , y2  in the coordinate plane, the distance d
between the points is given by:
d=
 x2  x1 
2

 y 2  y1

2
Round answers to the nearest tenth.
4. Find the distance between (1, 4) and (−2, −5).
5. Find the distance between (−3, 2) and (3, −2).
Real life situations can make use of this formula by transforming locations into coordinates.
6. One hiker is 4 miles west and 3 miles north of the campground. Another is 6 miles east and 3
miles south of the campground. How far apart are the hikers? (the camp ground is at (0, 0) )
7. Mickey travels 15 miles west, then 20 miles north. Jamie travels
5 miles east, then 10 miles south. How far apart are they?
14
Geometry 2205
Unit 1:
Mrs. Bondi
8. Quadrilateral KLMN has vertices with coordinates K(-3, -2), L(-5, 6), M(2, 6) and N(4, -2).
Show that LK  MN .
HW: p.54 #22-34
15
Geometry 2205
Unit 1:
Mrs. Bondi
Midpoint and Distance Practice
 x2  x1    y2  y1 
The distance formula:
d
The midpoint formula:
 x  x y  y2 
M 1 2, 1

2 
 2
2
16
2
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 6: Perimeter, Circumference and Area (PH text 1.8)
Objective: Student will be able to: 1) find the area and perimeter of basic polygons
2) find the circumference and area of circles
Vocabulary:
Perimeter (P) – the distance ______________________________________ or the sum of the
lengths of the sides of the polygon
Circumference (C) – the distance ___________________________________
Area (A) – the number of ______________________ enclosed by the polygon
Formulas:
Square:
P = ________
A = ________
Rectangle:
P = ________
A = ________
Triangle:
P = ________
A = ________
Circle:
P = ________
A = ________
Circle Notes:
Name using …
π=
Examples:
1. Find the perimeter/circumference and area of each of the following:
a.
b.
10 in
12 inches
2 ft
2. Find the perimeter and area of PET . P (5,6), E (5,-2) and T (-1,-2)
y
x
17
c. d = 8 inches
Geometry 2205 Mrs. Bondi
Unit 1:
3. You are designing a rectangular banner for the Winter Ball.
The banner will be 2 yards long and 4 feet wide.
a) How much material will you need?
b) If the material costs $8.95 per square yard, what will be the total cost?
4. You have 32 yd of fencing, and you want to make a rectangular pen for a calf
that you are raising for a 4-H project.
a) What are the dimensions of the rectangle that will give the maximum area?
b) What is the maximum area?
Area Addition Postulate - Postulate 1-10: The area of a region is ___________ of the areas of the
non-overlapping parts.
5. Find the area of the figure using two different methods.
4yd
4yd
2yd 1 yd
2yd 1 yd
3 yd
4
yd
3 yd
4
yd
3 yd
3 yd
6. Amy is planning to install a new, 4 ft wide sidewalk around her 20 ft x 28 ft rectangular pool. Find
the amount of cement that she will need. If she lines the outer edge of the sidewalk with brick, how
many linear feet of brick will she need? (Draw a diagram before you begin.)
18
Geometry 2205
Unit 1:
Mrs. Bondi
HW: p.64 #9, 10-26 even, 32-35, 42
19
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 7: Patterns and Inductive Reasoning (PH text 2.1)
Objective:
Students will be able to use inductive reasoning to make conjectures.
Inductive Reasoning – reasoning based on patterns you ___________
conjecture – a ________________ you reach using inductive reasoning
(predict and test)
counterexample – an example that shows that a conjecture is ____________
Examples:
Look for a pattern. What are the next three terms in each sequence?
1) 1, 1, 2, 3, 5, …
2) M, T, W, …
Practice:
20
Geometry 2205
Unit 1:
Mrs. Bondi
HW: p.85 #6-48 even (skip 18?)
21
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 8: Conditional Statements (PH text 2.2)
Objective: Students will be able to:
1) recognize conditional statements and their parts
2) write converses, inverses, and contrapositives of conditionals
(TONS of) Vocabulary:
Conditional: another name for an ______-_______ statement. It has ____ parts.
Example – “If two angles are congruent, then they have equal measures.”
If p, then q. or “p implies q”
q
p
Hypothesis: the part of the conditional statement that ____________ the _____.
(Do not include the word “if” as part of the hypothesis.)
Conclusion: the part of the conditional statement that ____________ the ______.
(Do not include the word “then” as part of the conclusion.)
Practice 1) Underline the hypothesis once and the conclusion twice.
Try a Venn Diagram.
a) If a triangle is scalene, then none of the sides are congruent.
b) Jess can go out to play if it is not raining.
Truth Value – a conditional can be either _________ or __________
To be a true conditional statement, the conclusion must be true EVERY time the hypothesis is true.
One counterexample is enough to prove a conditional false.
Counterexample: an example that shows the statement is __________________.
Practice 2) Provide a counterexample to show each statement is false.
a) If a number is divisible by 4, then it is divisible by 6. ________________
b) If AB  BC , then B is the midpoint of AC . _______________________
22
Geometry 2205 Mrs. Bondi
Unit 1:
To determine the truth value of a conditional, evaluate both the hypothesis and conclusion. It may be
helpful to use a Venn Diagram.
Create a Venn Diagram for the conditional:
If it is your birthday, then you will have cake.
Look at the following cases to determine the truth value of the conditional.
Conditional: If it is your birthday, then you will have cake.
What is the statement’s truth value?
Evaluate the truth value for each case.
Case 1: It is your birthday, you have cake.
Case 2: It is your birthday, you do not have cake.
Case 3: It is not your birthday, you have cake.
Case 4: It is not your birthday, you do not have cake.
(NOTE: If the hypothesis is false, then the truth value of the conditional is always _________.)
The counterexample must have a true hypothesis.
Practice 3) Decide whether the statement is true or false. If false, provide a counterexample.
b) If x  0 , then x 2  x . (True or False)
a) If x 2  25 , then x  5 . (True or False)
Counterexample: _____________
Counterexample: _____________
Negation: means the ________________ (verb – “to negate”) “~p” means ________
Practice 4) Negate each statement.
a) It is Saturday.
b) You slept late.
23
Geometry 2205 Mrs. Bondi
Unit 1:
Every conditional has three related conditional statements. We use negation to form the related
statements.
Conditional Example:
Converse:
If it is Saturday, then you slept late.
_____________________________ the hypothesis and conclusion of the conditional.
Example - _________________________________________________
Inverse: ________________ BOTH the hypothesis and conclusion of the conditional.
Example - _________________________________________________
Contrapositive: ______________ BOTH the hypothesis and conclusion of the _________________.
or BOTH ________________ and ________________ the hypothesis and conclusion.
Example - _________________________________________________
Equivalent Statements – two statements with the same truth value
A conditional and its contrapositive are equivalent statements.
A converse and inverse are equivalent statements.
They must either both be true, or both be false.
Practice 5) Write the Conditional, Converse, Inverse, and Contrapositive Statements.
a) “I will take my umbrella if it is raining out.”
(If-Then): ___________________________________________________
(Converse): ___________________________________________________
(Inverse)____________________________________________________
(Contrapositive)_________________________________________________
b) Choosey Mothers choose Jiff.
(If-Then): ___________________________________________________
(Converse): ___________________________________________________
(Inverse)____________________________________________________
(Contrapositive)_________________________________________________
HW: p.93 #6-18 even, 51-58
24
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 9: Biconditionals and Definitions (PH text 2.3)
Objective:
Students will be able to write biconditionals and recognize good definitions.
Biconditional statement: when a ________________ and its ________________
are BOTH true, they can be combined into one statement joined by “if and only if”.
Example - _________________________________________________
Practice 1) Given: “Two segments are congruent if and only if they have the same measure.”
a) Write the biconditional statement as a conditional.
_________________________________________________________
b) Write the converse of the conditional.
_________________________________________________________
Practice 2) If two angles are supplementary, then the sum of their measure is 180.
a) Write the converse of the statement.
_________________________________________________________
b) Is the original statement true? ______
Is the converse true? ______
c) If both statements are true, then write a biconditional statement. If neither is true,
then write a counterexample.
_________________________________________________________
Good Definitions are statements that can help you identify or classify an object. A good definition
has three important components.
 Uses clearly understood terms – commonly used or already defined
 Is precise – avoid words like large, sort of, almost, etc.
 Is reversible – can be written as a true biconditional
25
Geometry 2205 Mrs. Bondi
Unit 1:
Examples/Vocabulary:
 Parallel lines are _________________ lines that do not intersect
Draw and label line AB parallel to line CD.
(Use a geometer for neatness/accuracy.)
The symbol for parallel is
Write the definition as a biconditional:
_____________________________________________________________________________________________________________
_____________________________________________________________________________________________________________

Perpendicular lines are two lines that intersect to form ________________________
Draw and label line EF perpendicular to line GH at
point P. Use appropriate marks to indicate the lines
are perpendicular. (Use a geometer for neatness/accuracy.)
The symbol for perpendicular is
Write the definition as a Biconditional:
_____________________________________________________________________________________________________________
_____________________________________________________________________________________________________________

Skew lines are noncoplanar lines that are not _______________ and do not _______________
Draw and label line JK skew to line LM.
(Use a geometer for neatness/accuracy.)
Write the definition as a Biconditional:
_____________________________________________________________________________________________________________
_____________________________________________________________________________________________________________
 Perpendicular bisector of a segment is a line, segment, or ray that is __________________
to the segment at its _________________.
Label CD with midpoint M and draw MN
so that it is the perpendicular bisector CD .
(Use a geometer for neatness/accuracy.)
Write the definition as a Biconditional:
_____________________________________________________________________________________________________________
_____________________________________________________________________________________________________________
HW: p.101 #10-15, 22-26, 31, 38-41
GOOD IDEA: mid-chapter quiz p.105
26
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 10: Deductive Reasoning (PH text 2.4)
Objective:
Students will be able to use deductive reasoning to form a conjecture.
Deductive Reasoning (also called Logical Reasoning) – the process of reasoning logically from given
statements or facts to a conclusion
Property: Law of Detachment
If the hypothesis of a true conditional is true, then the conclusion is true.
Example 1: If Rebekah maintains distinguished honor roll for seventh grade, she may get a cell
phone.
Rebekah ended the year with distinguished honor roll.
You conclude: _____________________________________________________________
Property: Law of Syllogism
Allows you to draw a conclusion based on two true conditional statements, when the conclusion of one
statement is the hypothesis of the other statement.
Example 2: If it is Monday, Bridget has piano lessons at 4:00. If Bridget has a piano lesson at 4:00, I
pick her up from school. Today is Monday.
You conclude:
_____________________________________________________________
We can use the Law of Detachment and the Law of Syllogism, along with the properties, postulates
and theorems to justify our steps in finding a solution to a problem or proving a statement.
Example 3: Refer to the diagram below.
Name two pairs of vertical angles.
1
4
3
2
Example 4: Using the diagram from example 3, if m1  150 , find the measures of 2 and 3 .
Justify each step.
27
Geometry 2205
Unit 1:
Mrs. Bondi
HW: p.110 #6-16, 36-39
28
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 11: Reasoning in Algebra and Geometry (PH text 2.5)
Objective:
Students will be able to connect reasoning in algebra and geometry.
You must remember your basic properties:
Properties of Equality/Congruence and Real Numbers:
Property
Definition
Addition Property
Subtraction Property
Multiplication Property
Division Property
Substitution Property
Distributive Property
Reflexive Property
Symmetric Property
Transitive Property
Proof – a convincing argument that uses deductive reasoning
– logically shows why a conjecture is true
A proof can be written as a paragraph, two-columns (statements and reasons), or a flow chart.
Each statement must be justified, and follow logically from the statements before it. Usually, the first
statement is the given statement, and the last statement is what you want to prove. The statements that
come between are the statements needed to logically progress from the given statement to the
statement to be proven.
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Geometry 2205
Unit 1:
Examples:
5)
Given:
Prove:
Mrs. Bondi
3x + 5 = x – 7
x = -6
Statement
Reason
3x + 5 = x – 7
6)
Given:
Prove:
6  4
1  3
Statement
1
6
Reason
Practice:
HW: p.116 #4-19, 24, 35-41
30
2
5
3
4
Geometry 2205
Unit 1:
Mrs. Bondi
Extra Practice:
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Geometry 2205
Unit 1:
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Geometry 2205
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Mrs. Bondi
HW: p.124 #12-13, 18, 20-24, 30, 40-42
Before beginning ch.3, do p.137.
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Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 12: Lines and Angles (PH text 3.1)
Objective:
Students will be able to
1) identify relationships between figures in space.
2) identify angles formed by two lines and a transversal.
Types of Lines:
1) Parallel Lines:
a) are _______________________
b) do NOT _____________________
c) The symbol for parallel is
*Algebraic Connection: The slopes of 2 parallel lines are __________________.
2) Perpendicular Lines:
a) are _______________________
b) do ______________________
c) The symbol for perpendicular is
*Algebraic Connection: The slopes of perpendicular lines are ______________ ______________
3) Skew Lines:
a) are not __________________
b) are not __________________
c) do not _________________
d) There is no symbol for skew lines.
Parallel Planes: planes that ______ ______ __________________
H
Examples: Use Figure 1 to answer # 1 – 6.
G
E
A
1) List all lines parallel to line HG ( HG ).
F
B
2) List all lines perpendicular to HG .
Figure 1
D
C
3) List all lines skew to line HG .
6) What is the intersection of HF and line BF ?
4) Which planes contain HG ?
7) Name two ⊥ segments that are both || to plane
HGD?
5) Name 3 pair of parallel planes in the figure.
8) What is the intersection of plane HFA and
plane BCD?
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Geometry 2205
Unit 1:
Mrs. Bondi
1
2
Use colored pencils to identify the described angle pairs in the diagrams.
a
3
4
Transversal –
5
b
7
6
8
Interior –
c
Exterior –
Alternate Interior Angles –
Alternate Interior Angles –
Same-Side Interior Angles –
Same-Side Interior Angles –
Corresponding Angles –
Corresponding Angles –
Alternate Exterior Angles –
Alternate Exterior Angles –
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Geometry 2205
Unit 1:
Mrs. Bondi
Parallel lines indicator
1
a
3
5
b
7
2
4
6
8
c
Notice that the angle relationships exist regardless of the relationship (parallel or not) of the lines
being intersected by the transversal. The angle pairs are named the same here, where the lines are
parallel, and in the previous diagram, where they are not.
4. Name the relationship of each pair of angles in #2.
HW: p.143 #10-16, 21-35, 37-42 (draw a diagram for #25-28)
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Geometry 2205
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Mrs. Bondi
Lesson 13: Properties of Parallel Lines (PH text 3.2)
Objective:
Students will be able to prove theorems about parallel lines.
Parallel Lines and Transversals:
The angle relationships we explored in the last lesson hold special meaning when they are formed by
parallel lines and a transversal. When the transversal intersects parallel lines, the special angle pairs
formed are either congruent, supplementary, or both.
Postulate 3-1
Same-Side Interior Angles Postulate
Theorem 3-1
Alternate Interior Angles Theorem
Theorem 3-2
Corresponding Angles Theorem
Theorem 3-3
Alternate Exterior Angles Theorem
37
Geometry 2205 Mrs. Bondi
Unit 1:
Let’s prove the theorems!
Theorem 3-1
Given:
Alternate Interior Angles Theorem
a b
Statement
Corresponding Angles Theorem
a b
Statement
Statement
Prove:
1  5
Reason
Theorem 3-3
Given:
3  6
Reason
Theorem 3-2
Given:
Prove:
Alternate Exterior Angles Theorem
a b
Prove:
1  8
Reason
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Geometry 2205
Unit 1:
Mrs. Bondi
Parallel Line Proof Practice
5 1
7 8
4
10
r
6
l
1)
Given:
Prove:
Statement
3)
r s
m1  m2
l m
Prove: 1 and 4 are supplementary
Statement
s
m
2)
Given:
Prove:
Reason
Given:
9
2 3
Statement
4)
Given:
Prove:
Reason
Statement
39
r s
1 and 3 are supplementary
Reason
l m,
1 and 4 are supplementary
Reason
Geometry 2205
Unit 1:
Mrs. Bondi
A
B
1
2
3
C
5)
Given:
AB CD
Prove:
m3  m2  mABD
Statement
D
6)
Given:
BC bisects ABD
AB CD
Reason
HW: p.153 #7-30 even & 23
40
Prove:
m3  m2
Statement
Reason
Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 14: Proving Lines Parallel (PH text 3.3)
Objective:
Students will be able to determine whether two lines are parallel.
The angle relationships we studied in the last lesson are actually biconditional. They can also allow us
to prove lines are parallel. We will treat each of these four theorems as biconditional theorems.
Theorem 3-4
Converse of Corresponding Angles Theorem
Theorem 3-5
Converse of Alternate Interior Angles Theorem
Theorem 3-6
Converse of Same-Side Interior Angles Postulate
Theorem 3-7
Converse of Alternate Exterior Angles Theorem
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Geometry 2205 Mrs. Bondi
Unit 1:
Prove Theorem 3-4: Converse of Alternate Interior Angles Theorem with a flow chart proof.
Given: m3  m6
Prove: a b
m3  m6
Given
m3  m4  180
m4  m6  180
_______________
a
b
_______________
_______________
Prove the Converse of Alternate Exterior Angles Theorem with a
flow chart proof.
Given: m1  m8
Prove: a b
m1  m8
Given
m1  m4
_______________
a
_______________
m5  m8
_______________
What types of quadrilaterals can have angles with the following measures? Why?
1)
120, 60, 150, 30
2)
42
72, 108, 45, 135
b
_______________
Geometry 2205
Unit 1:
Mrs. Bondi
3) What is the value for x that will make a b ?
a
4x + 6
2x – 12
b
4) What is the measure of each angle?
c
Practice:
5)
Given:
l m , m4  m3
Prove:
r s
Statement
HW: p.160 #6-10, 12-28, 31-32
43
Reason
Geometry 2205
Unit 1:
Mrs. Bondi
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Geometry 2205
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Mrs. Bondi
HW: p.160 #11, 29, 40, 41, 47-57
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Geometry 2205
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Lesson 15: Parallel and Perpendicular Lines (PH text 3.4)
Objective:
Theorem 3-8
Students will be able to relate parallel and perpendicular lines.
If two lines are parallel to the same line, then they are parallel to each other.
If a b and b c , then a c .
Theorem 3-9
In a plane, if two lines are perpendicular to the same line, then they are parallel
to each other.
If m  t and n  t , then m n .
Theorem 3-10
In a plane, if a line is perpendicular to one of two parallel lines, then it is also
perpendicular to the other.
If n  l and l m , then m  n .
Example 1) A quilter has cut pieces of fabric that are isosceles trapezoids 2 in wide and 3 inches long on the
longest side. The angles measure 135 and 45. When four pieces are fit together to form a square with short
sides facing inward, will the sides be parallel? Explain.
46
Geometry 2205 Mrs. Bondi
Unit 1:
Prove Theorem 3-9
Given: e  g and f  g
Prove: e f
Statement
Reason
e
f
g
2) Prove Cherry St intersects Apple St perpendicularly. You know that Berry St and Apple St are both
perpendicular to Date St. You also know that Berry St is perpendicular to Cherry St. (Hint: you may benefit
from drawing a diagram first.)
Practice:
HW: p.167 #5-6, pick two proofs 7-10, 16, 19-22, 31-32
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Geometry 2205
Unit 1:
Mrs. Bondi
Lesson 16: Parallel Lines and Triangles (PH text 3.5)
Objective:
Students will be able to use parallel lines to prove a theorem about triangles.
to find measures of angles of triangles.
Classifying Triangles:
Triangle:
Equiangular:
Scalene:
Acute Triangle:
Isosceles:
Right Triangle:
Equilateral Triangle:
Obtuse Triangle:
Postulate 3-2
Parallel Postulate
Through a point not on a line, there is one and only one line parallel to the given
line.
Theorem 3-11
Triangle Angle-Sum Theorem
The sum of the measures of a triangle is 180.
Example 1: Find the values of x, y and z.
P
Q
z
90°
x
S
y
110°
R
40°
48
Geometry 2205 Mrs. Bondi
Unit 1:
Exterior angle of a polygon – an angle formed by a _____________ and an extension
of an __________________ _____________
Remote interior angles – in a triangle, the two non-adjacent angles
Theorem 3-12
Triangle Exterior Angle Theorem
The measure of each exterior angle of a triangle equals the sum of the measures
of its two remote interior angles.
m1  m2  m3
Example 2: If m∠NGL = 30 and m∠GNL = 110, find the m∠1.
N
1
L
G
Practice: Find m1.
1.
2.
3.
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Geometry 2205
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Algebra Find the value of each variable.
4.
5.
6.
7. a. Which of the numbered angles are exterior angles?
b. Name the remote interior angles for each exterior angle.
c. Which two exterior angles share the same remote interior angles? Explain.
HW 1: p.175 #8-21
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Geometry 2205 Mrs. Bondi
Unit 1:
More Practice with angles in Triangles:
Algebra Use the given information to find the unknown angle measures in the triangle.
13. The ratio of the angle measures of the acute angles in a right triangle is 1 : 3.
14. The measure of one angle of a triangle is 61. The other two angles are in a ratio of 2 : 5.
15. The measure of the exterior angle of a triangle is 110. The measures of its remote interior angles
are in a ratio of 2 : 3.
16. Think About a Plan The measure of an exterior angle of DEF is 4x. The measure of one of this
angle’s remote interior angles is x + 23. The measure of the other remote interior angle is 2x + 12.
Find the value of x, the measure of each angle of the triangle, and the measure of the exterior angle.
• How can drawing a picture help you find the answers?
• How are the exterior angle and the third angle of the triangle related?
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Geometry 2205
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Mrs. Bondi
Find the values of the variables and the measures of the angles.
17.
18.
19.
20.
HW: p.176 #22-33
(Mid-Chapter Quiz p.181)
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Geometry 2205
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Mrs. Bondi
Lesson 17: Equations of Lines in the Coordinate Plane (PH text 3.7)
Objective:
Students will be able to graph and write linear equations.
Reminder:
Slope:
A slope will have a predictable “look”.
Positive Slope –
Negative Slope –
Zero Slope –
Undefined Slope –
Class Practice:
Find the slope of each line.
Rise
Run
= ______
Rise
Run
= ______
Find the slope of a line
through ( –2, 1) and (3, 5).
Then graph it to check.
Graphing Lines using slope intercept form:
m = ________________
Rise
Run
= ______
Draw a line through the point (1, 2)
with the slope 5/4.
y = mx + b
b = _________________________
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Geometry 2205
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Mrs. Bondi
To write an equation for a line from a graph, get the slope and y-intercept from the graph and
substitute into the form y = mx +b.
1. Find the y-intercept.
b = _____
2. Find the slope using two points.
m = _____
3. Substitute into y = mx + b to write the equation for the line.
Equations of Lines Reminder:
slope-intercept form
point slope form
standard form
Write an equation in slope-intercept form of the line that passes through points ( 1, 4 ) and ( 3, 10 ).
Write an equation of the horizontal line that passes through ( -2, 5 ).
Write an equation of the vertical line that passes through ( -2, 5 ).
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Geometry 2205
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Practice:
HW: p.193 #7, 8-12 even, 20-30 even, 36-39
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Geometry 2205
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Mrs. Bondi
Lesson 18: Slopes of Parallel and Perpendicular Lines (PH text 3.8)
Objective:
Students will be able to relate slope to parallel and perpendicular lines.
Parallel Lines (||) have slopes that are ____________________________________.
(y-intercepts must be different)
Perpendicular Lines (⊥) have slopes that are ____________________________________.
(lines must intersect) (the product of the slopes of the lines is -1)
Example: Find the slope of the line, and lines perpendicular and parallel to the line, for each equation.
1
y  x 5
y = 3x - 2
y=x
3x + 5y = 7
2
m = _______
m = _______
m = _______
m = _______
m⊥= _______
m⊥= _______
m⊥= _______
m⊥= _______
m|| = _______
m|| = _______
m|| = _______
m|| = _______
1) Write an equation for a line in slope intercept form that contains the point ( 5, 4 ) and is
parallel to y = 2x – 2 .
2) Write an equation for a line in slope intercept form that contains the point ( 1, 6 ) and is
perpendicular to y = 3x – 4 .
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Geometry 2205
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Mrs. Bondi
3) LK is the perpendicular bisector of HJ . Solve for x if
LKJ 
3
x 3.
4
L
K
H
J
x = _____
4. Use slopes to determine whether the opposite sides of quadrilateral WXYZ are parallel.
W(1, 1), X(3, 1), Y(2, 4), Z(2, 3)
5. Are
1
and
2
perpendicular? Explain.
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Geometry 2205
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Practice:
HW: p.201 #5, 8-26 even, 33
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Mrs. Bondi
---------------------------------------------------------------------------------------------------------------------Unit 1 Exam Review: (textbook practice – all answers are in back of book – be prepared to use multiple skills in the same question)
Ch.1 - p.70-74 #1-4, 7-25, 30-41
Ch.2 - p.129-132 #1-38
Ch.3 – p.206-210 #1-29, 34-46
59