Module 1
(unit 1)
Foundations of Geometry
Question of the Day (warm-up)
Write this down as your warm-up in your composition book. Some version of this question will be on a quiz.
I will not give you the answers to these. You need to figure them out!
You have 5 minutes and then I am changing the screen.
Housekeeping & Setup
***Log into Chromebooks and Sign up for Google Classroom
Username: brownlau000
Password: Bro12345
Setup
If you have your composition book today, write your name on the front in sharpie and the class/block.
Title the first page: Table of Contents
Make 3 Columns
Date Topic Page Number
1-6-16 Vocab, Segment Length and Midpoint 1
�Topic:�(Basic geometric terminology)�
Module 1-1
Fundamentals of Geometry
Goal(s)
What is Geometry?
Who is the father of Geometry?
Euclid’s 5 postulates also known as axiom: an accepted statement of fact.
Fundamentals of Geometry
Goal(s)
Today’s Vocabulary:
Write these down: Take 5 minutes and try to come up with a definition for each word, compare with a partner, share with the class.
Point- a location in space
All geometric figures contain points. They have no size. In a figure, a point is represented by a dot.
Points are named by capital letters.
A
A
Lines
In a figure, a line is shown with arrows at each end.
Lines are usually named by lower case script letters or by writing capital letters for 2 points on the line, with a double arrow over the pair of letters.
C
B
A
Line l
CA
AC
AB
l
A, B, and C are collinear points.
A
C
refers to the name of the segment
refers to the length of
Line SEGMENTS:
A piece of a line.
AC
Rays
They extend indefinitely in one direction.
A
B
Name by starting with the endpoint.
The arrow always points to the right, even if the ray is going the opposite direction!
A
B
AB
Opposite rays
Opposite rays always form a line.
& are opposite rays
A
B
C
BA
BC
Identify each and use the correct notation:
A
B
C
E
F
Plane
A plane is often represented by a parallelogram. They are usually named by a capital script letter or by a minimum of 3 non-collinear points on the plane.
P
E
A
C
D
Plane P or ACD or ACDE
Why would CDAE be wrong?
B
Coplanar VS. Non Coplanar
Non-Coplanar- points not in the same plane. Must be listed with 4 or more points.
C,D, and E are coplanar points
E
F
D
C
C, D, E, and F are non coplanar points.
What are C, E, and F?
Some other Postulates in Geometry:
Answer the following
1. Name all 6 obvious planes.
2. Name the four non obvious planes:
3. Name 2 points that are coplanar w/ points B,H,and K.
4. Name all the lines that intersect at point E.
5. Which planes intersect at segment CD.
6. Name 3 collinear points:
7. Which point (s) do planes ABC, CDE, and AGE have in common?
Go back to the Question of the Day #1- make sure you know how to answer it.
QOD Warm-up
Review Vocabulary:
Point- a location in space
All geometric figures contain points. They have no size. In a figure, a point is represented by a dot.
Points are named by capital letters.
A
A
Lines
In a figure, a line is shown with arrows at each end.
Lines are usually named by lower case script letters or by writing capital letters for 2 points on the line, with a double arrow over the pair of letters.
C
B
A
Line l
CA
AC
AB
l
A, B, and C are collinear points.
A
C
refers to the name of the segment
refers to the length of
Line SEGMENTS:
A piece of a line.
AC
Rays
They extend indefinitely in one direction.
A
B
Name by starting with the endpoint.
The arrow always points to the right, even if the ray is going the opposite direction!
A
B
AB
Opposite rays
Opposite rays always form a line.
& are opposite rays
A
B
C
BA
BC
Identify each and use the correct notation:
A
B
C
E
F
Plane
A plane is often represented by a parallelogram. They are usually named by a capital script letter or by a minimum of 3 non-collinear points on the plane.
P
E
A
C
D
Plane P or ACD or ACDE
Why would CDAE be wrong?
B
Coplanar VS. Non Coplanar
Non-Coplanar- points not in the same plane. Must be listed with 4 or more points.
C,D, and E are coplanar points
E
F
D
C
C, D, E, and F are non coplanar points.
What are C, E, and F?
Some other Postulates in Geometry:
Answer the following
1. Name all 6 obvious planes.
2. Name the four non obvious planes:
3. Name 2 points that are coplanar w/ points B,H,and K.
4. Name all the lines that intersect at point E.
5. Which planes intersect at segment CD.
6. Name 3 collinear points:
7. Which point (s) do planes ABC, CDE, and AGE have in common?
�Topic:�(Segment Length and Midpoints)�
Module 1-1
Fundamentals of Geometry
Goal(s)
Length of Lines:
Ruler Postulate
The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers.
Distance of 2 points or length of a line on a number line:
A
B
3
-1
Segment Addition Postulate
(Algebraic way)
If B is between A and C but not necessarily in the middle, then
AB + BC = AC
A
C
B
Examples:
A
C
B
AC = 100
Evaluate x, AB, and BC
4x-20
2x+30
X= 15
AB = 40
BC = 60
4x-20+2x+30=100
Objective: apply the midpoint and distance formula to real world problems to solve for length of segments in a coordinate plane. �
Origin
(0,0)
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
Y
X
Assignment: 4 problems on graph paper
Part one: Given the coordinates for a segment; Do each of the following
1. A(2,7) B(-1,3)
2. C(3,4) D(6,2)
3. E(-1,3) F(0,2)
4. G(5,3) H(0,-4)
QOD (Warm-up)
Assignment: 4 problems on graph paper
Part one: Given the coordinates for a segment; Do each of the following
1. A(2,7) B(-1,3)
2. C(3,4) D(6,2)
3. E(-1,3) F(0,2)
4. G(5,3) H(0,-4)
The midpoint M of is the point between P and Q such that
PM ≅ MQ.
P
Q
M
1. On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is
a
b
M
-2
8
Using Midpoints Algebraically
Finding Midpoint In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are
M=
Click on image to play a video on how to find a midpoint
a) What is the midpoint of Segment AB if A(3,-8) and B(5,6)?
b) What is the length of segment AB?
Example:
What is the Length of DE? What is the length of each half?
Add to Assignment: 5 problems on graph paper
Part one: Given the coordinates for a segment; Do each of the following
1. A(2,7) B(-1,3)
2. C(3,4) D(6,2)
3. E(-1,3) F(0,2)
4. G(5,3) H(0,-4)
Part two:
5. There exist a segment such that one of the endpoints is (-12,4) and the midpoint of the segment is (1,-2).
Assignment:
Start working on the Module 1 Assignment in Google Drive. It will be due Tuesday.
When you open it, it will be a google doc that you can edit. it should automatically save to your drive. When the assignment is complete, you may attach your annotated copy and turn the assignment in.
Module 1
1-1 Segment Length and Midpoints
1-2 Measuring Angles
1-3 Skipping until Module 2 Transformations
1-4
TNREADY Warm-up
Evaluate the Distance and the Midpoint between the points A(3,7) and B(-2,5)
QOD #3
Warm-up
T
Q
P
S
R
Questions on Assignment:
5 problems on graph paper
Part one: Given the coordinates for a segment; Do each of the following
1. A(2,7) B(-1,3)
2. C(3,4) D(6,2)
3. E(-1,3) F(0,2)
4. G(5,3) H(0,-4)
Part two:
5. There exist a segment such that one of the endpoints is (-12,4) and the midpoint of the segment is (1,-2).
1.2 Angles, Measuring Angles and identifying angle pair types.�
Objectives:
Angles
1
2
Types of angles
You might see a question written as find the m<ABC = _____
Interactive Angle application:
Measure: An actual number value in degrees for <ABC.
�Protractor Postulate
B
O
A
How to find the measure of an angle using a protractor:
How to find the measure of an angle using a protractor:
Find the measure of the angle.
Measuring and Classifying Angles
m<AOC=______
m<BOD=______
m<EOC = _____
Congruent Angles
Angles with the same measurement.
Angle Bisector
Angle Bisector Problem
�Angle Addition Postulate
Angle Addition
Warm-up QOD
- Evaluate x, y, and each angle’s measure.
Complementary Angles: a pair of angles that sum to 90 degrees.
Evaluate the measure of each angle in the figure below;
�Linear Pair Postulate (adjacent supplementary)
supplementary Angles: a pair of angles that sum to 180 degrees.
<DEF is a straight angle. What are the m<DEC and m<CEF?
D
E
F
C
(2x + 10)
(11x – 12)
<ABD and <DBC are a linear pair. Solve for the measure of each angle.
Evaluate each angle measure in the figure below:
Vertical Angles are opposite each other in an intersection and are Congruent:
Angles created by Perpendicular Lines
Evaluate the measure of <1, <2, and <3
Evaluate each angle in the figure below:
Exit Ticket: line a is perpendicular to b
QOD #6 Warm Up - Evaluate x, y, and each angle’s measure.
Assignment:
Angles Worksheet
Warm-up QOD #7
Identify the construction below and discuss the next steps in completing the construction.
Basic Constructions
Some vocabulary…
Constructions:
Concept # 1 - Duplicate a Congruent Segment
S
T
A
B
How would you double or triple a line segment?
Segments continued:
b. Adding Segments steps:
Construct LM such that it is congruent to given segments AB + CD.
A
B
C
D
L
Segments continued:
C. subtracting Segments steps: Construct PQ = AB-CD.
A
B
C
D
P
Perpendicular Bisector of a segment;
A
B
QOD #8 Warm-up
Constructions:
https://sites.google.com/a/wilsonk12tn.us/brownl/
Chromebooks!!!
Module 1
1-1 Segment Length and Midpoints
1-2 Measuring Angles
1-3 Skipped until Module 2- Transformations
1-4 Reasoning & Proof
Question Of the Day #9
A
O
C
3x-40
2x-10
1-4 Reasoning & Proof
Agenda!
Terms:
Objectives:
2 types of reasoning:
Inductive vs. Deductive
When given patterns:
Use Inductive reasoning – looking at several specific situations or patterns to arrive at a conjecture
Conjecture – an educated guess using inductive reasoning (hypothesis)
counterexample – an example that shows that a conjecture is false (Proof by contradiction)
Conditional Statements
If-then statements are called conditional statements.
The portion of the sentence following if is called the hypothesis.
The part following then is called the conclusion.
If it is a tomato, then it is a fruit.
The converse statement is formed by switching the hypothesis and conclusion.
If it is a tomato, then it is a fruit.
Converse: If it is a fruit, then it is a tomato.
The converse may be true or false.
Multiple choice type question:
Deductive Reasoning:
Honors Logic Extension:
The inverse statement is formed by negating the hypothesis and conclusion.
If it is a tomato, then it is a fruit.
Inverse: If it is not a tomato, then it is not a fruit.
The inverse may be true or false.
The Contrapositive statement is formed by switching and negating the hypothesis and conclusion.
If it is a tomato, then it is a fruit.
Contrapositive: If it is not a fruit, then it is not a tomato.
The contrapositive may be true or false.
Statement 1: If you are in H. Geometry, then you are in Mrs. Brown’s class.
Statement 2: If you are in Mrs. Brown’s class, then you are learning about logic.
Conclusion: If you are in H. Geometry, then you are learning about logic.
Statement 1: If you are in H. Geometry, then you are in Mrs. Brown’s class.
Statement 2: ____(student’s name)______ is in Honors Geometry.
Conclusion: ___(student’s name)____ is in Mrs. Brown’s class.
Truth Tables
p | q | p-->q | q-->p | p←>q | ~p→~q | ~q→~p | p ∩ q | p ⋃ q | ~p ⋃ (q-->p) |
T | T | | | | | | | | |
T | F | | | | | | | | |
F | T | | | | | | | | |
F | F | | | | | | | | |
** T → F IS FALSE **
Intersection “and”
Union “or”
�1-4 Reasoning in Algebra: �The beginning of proof writing ☺
I can use deductive reasoning and algebraic properties to write proofs for angle and segment addition problems.
What is a Proof?
Name the Algebra Properties of Equality
1. If a = b, then a + c = b + c.
2. If a = b, then a – c = b – c.
3. If a = b, then a • c = b • c.
4. If a = b and then a ÷ c = b ÷ c
Addition Property
Subtraction Property
Multiplication Property
Division Property
5. If a = b, then either a or b may be _________ for the other in any equation or inequality.
6. a = a
7. If a = b, then b = a
Substituted
Reflexive Property
Symmetric Property
8. If a = b and b = c, then a = c.
Transitive Property
9. a(b + c) = ab + ac
10. “collect like terms” for example:
if 2x + x = 9, then 3x=9.
Distributive Property
Simplify
Recall our “New” Geometry Postulates:
Two angles <LMN and <NMP form a linear pair. The measure of <LMN is twice the measure of <NMP. Find m<LMN.
Remember Today’s Objective:
Types of proofs:
There are 3 main types:
Practice Proof: “Together”
Statements | Justifications |
A
B
C
3x
x+7
Given AC = 23, prove x=4. Justify each step.
AB + BC = AC
Seg. Addition Prop.
3x + x+7 = 23
Substitution prop.
4x+7 = 23
Simplify
4x=16
Subtraction prop.
x=4
Division prop.
Example #1
A
O
C
x
2x+10
B
Statements | Justifications |
1. m<AOB + m<BOC = m<AOC
2. x+2x+10=139
3. 3x + 10 = 139
4. 3x=129
5. x=43
Angle add postulate
Substitution
Simplify
subtraction
division
Example #2
Statements | Justifications |
1. | |
2. | |
3. | |
4. | |
5. | |
6. x=20 | Symmetric Prop. |
A
O
C
B
2x+40
4x
OB bisects <AOC
m<AOB = m<BOC
2x+40 = 4x
40=2x
20=x
given
Def. of angle bisector
substitution
subtraction
division
Start with the given, and if it uses a definition in the given…use that as one of your reasons.
Exit Ticket:
List the properties of equality that would complete this proof.
Proof Writing Activity
Example #3
A
O
C
B
6x-10
6x+10
Statements | Justifications |
| |
| |
| |
| |
| |
<AOB + < BOC = 180
Def. of Linear Pair
6x-10 +6x+10=180
substitution
12x=180
simplify
x=15
division
<AOB = 80°
substitution
Example #4
A
O
C
2x
3x-9
Statements | Justifications |
| |
| |
| |
| |
| |
AO+OC=AC
Seg addition postulate
2x+3x-9=21
Substitution
5x-9=21
Simplify
5x=30
Addition
x=6
Division
Example #5
Statements | Justifications |
| |
| |
| |
| |
| |
A
O
C
B
x+6
2x
<AOB + <BOC = 90
Def. of complementary
2x+x+6=90
Substitution
3x+6=90
simplify
3x=84
subtraction
X=28
division
In a few more weeks, learning a few more geometric postulates, you will be able to write a proof similar to the one below!!!
This is our GOAL!
Interactive Proof Writing
Question Of the Day #9
A
O
C
3x-40
2x-10
Go back and solve QOD
A
O
C
Statements | Justifications |
| |
| |
| |
| |
| |
| |
3x-40
2x-10
Question of the Day #10
Answer each statement with Always, Sometimes, or never; and explain.
In Review:
Fundamentals of Geometry
Goal(s) or objectives
Objectives:
Objectives:
Objectives
REVIEW Module 1
10 minutes to write 3 proofs.
Each person needs to have these on their own paper. You will turn in before quiz tomorrow.
Proof #1
A
O
C
B
6x-10
6x+10
Proof #2
A
O
C
2x
3x-9
Proof #3
A
O
C
B
x+6
2x
REVIEW Module 1
10 Sheets of Problems. These will rotate every 5 minutes. Put these on the same paper….. Be ready!
Due Today: Logic Puzzles