identify pairs of lines and angles -...
TRANSCRIPT
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“ L I F E I S W H A T Y O U M A K E I T . ” – M R . H ’ S D A D
Identify Pairs of Lines and Angles
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Goal
Students will learn the different ways lines can interact with each other. This included coplanar and non-coplanar lines.
Students will learn what a transversal is and the different types of angles formed by a transversal.
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Parallel and Skew
Parallel lines-never intersect and are coplanar.
Skew lines-never intersect and are not coplanar.
Parallel planes-planes that never intersect and are always the same distance apart.
Perpendicular lines-lines that intersect at a right angle.
lllk and n⊥m
l
m
n
k
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Postulates
Parallel Postulate If there is a line and a point not on the line, then there is
exactly one line through the point parallel to the given line.
Perpendicular Postulate If there is a line and a point not on the line, then there is
exactly one line through the point perpendicular to the given line.
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Questions
What are the three different names for the interaction of lines? Parallel, intersection, or skew.
If I have a point and a line, how many lines could I draw through the point that would be parallel to my given line? Exactly one
T/F Two lines in a plane either intersect or are parallel.
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Transversals
A transversal is a line that intersects two or more coplanar lines at different points.
transversal
No transversal
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Corresponding Angles
Angles that are on the same side of the transversal and in the same corresponding positions.
2
3
4
1
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Alternate Interior Angles
Angles that are on the opposite sides of the transversal and between the two lines.
Interior
2
1 3
4
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Alternate Exterior Angles
Angles that are on the opposite sides of the transversal and outside the two lines.
Interior
2
1
3
4 Exterior
Exterior
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Consecutive interior Angles
Angles that are on the same side of the transversal and between the two lines.
Interior 2
1
3
4
Exterior
Exterior
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Summary
Students will learn the different ways lines can interact with each other. This included coplanar and non-coplanar lines.
Students will learn what a transversal is and the different types of angles formed by a transversal.
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Example 1
Which lines or planes match the given description. Line(s) parallel to line AD and
containing point C
BC
Line(s) perpendicular to line HD and containing point G
HG
Line(s) skew to line EF and containing point H
HD
A
B C
D
E
F G
H
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Example 2
Name a pair of parallel lines. Line AC ll Line BD
Name a pair of perpendicular lines. Line AC ┴ Line BC
Is Line AC ┴ Line AD? Explain.
Not enough info. It is not marked and no way to determine (yet).
Please look at Ex. 2 part c in book.
A
D
C
B
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Example 3
Identify all corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.
1 2
3 4
5 6
7 8
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Example 4
Determine what each angle pair is called.
Angle 1 and angle 2
Angle 9 and angle 13
Angle 11 and angle 2
Angle 9 and angle 4
Angle 9 and angle 16
Angle 8 and angle 14
Angle 6 and angle 7
Angle 5 and angle 1
1 2
3 4
5 6
7 8
9 10
11 12
13 14
15 16
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“ T H E G R E A T U S E O F L I F E I S T O S P E N D I T F O R S O M E T H I N G T H A T W I L L O U T L A S T I T . ” –
W I L L I A M J A M E S
Use Parallel Lines and Transversals
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Goal
Students will learn how lines being parallel affects the angles formed by a transversal.
Students will be able to justify why angles are congruent or supplementary based on their position in a diagram.
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Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
∠1 ≅ ∠5
∠2 ≅ ∠6
∠3 ≅ ∠7
∠4 ≅ ∠8
2
3 4
1
5 6
7 8
m
n
p
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Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
∠3 ≅ ∠6
∠4 ≅ ∠5
2
3 4
1
5 6
7 8
m
n
p
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Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
∠1 ≅ ∠8
∠2 ≅ ∠7
2
3 4
1
5 6
7 8
m
n
p
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Consecutive interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
∠3 𝑎𝑛𝑑 ∠5 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
∠4 𝑎𝑛𝑑 ∠6 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
2
3 4
1
5 6
7 8
m
n
p
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Summary
At this point, you should be able to: Identify special properties of the four new angle pairs when
two parallel lines are cut by a transversal.
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Example 1
Find the unknown angle measures.
m∠2=105°
m∠5=105°
m∠6=105°
m∠3+m∠5=180°
m∠3=75°
m∠1=75°
m∠4=75°
m∠7=75°
105°
4
1
2 3
5
6 7
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Example 2
Find the value of x.
m∠4=85°
m∠4+2x+5=180
85+2x+5=180
2x+90=180
2x=90
x=45
85°
4
(2x+5)°
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Example 3
Find the value of x.
35°+m∠2=180°
m∠2=145°
m∠2=(7x+5)°
145=7x+5
140=7x
20=x
35° 2
(7x+5)°
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Example 4
Find the value of y.
3y+87=180
3y=93
y=31 (3y)°
87°
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Example 5
Prove the Alternate Exterior Angles Theorem.
Given: mlln
Prove:∠1≅∠3
1
2
3
m
n
Statements Reasons
1. mlln 1. Given
2. ∠1≅∠2 2. Corresponding angles Postulate
3. ∠3≅∠2 3. Vertical Angles Congruence Theorem
4. ∠1≅∠3 4. Transitive Property
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“ O N L Y T H E P E R S O N W H O H A S F A I T H I N H I M S E L F I S A B L E T O B E F A I T H F U L T O
O T H E R S . ” – E R I C H F R O M M
Prove Lines are Parallel
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Goal
Students will learn how the angles formed by a transversal can be used to determine that lines are parallel.
Students will be able to justify why lines are parallel.
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Corresponding Angles ConversePostulate
If two lines are cut by a transversal and the pairs of corresponding angles are congruent, then the lines are parallel.
∠1 ≅ ∠5
∠2 ≅ ∠6
∠3 ≅ ∠7
∠4 ≅ ∠8
2
3 4
1
5 6
7 8
m
n
p
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Alternate Interior Angles Converse Theorem
If two lines are cut by a transversal and the pairs of alternate interior angles are congruent, then the lines are parallel.
∠3 ≅ ∠6
∠4 ≅ ∠5
2
3 4
1
5 6
7 8
m
n
p
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Alternate Exterior Angles Converse Theorem
If two lines are cut by a transversal and the pairs of Alternate Exterior Angles are congruent, then the lines are parallel.
∠1 ≅ ∠8
∠2 ≅ ∠7
2
3 4
1
5 6
7 8
m
n
p
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Consecutive Interior Angles Converse Theorem
If two lines are cut by a transversal and the pairs of consecutive interior angles are supplementary, then the lines are parallel.
∠3 𝑎𝑛𝑑 ∠5 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
∠4 𝑎𝑛𝑑 ∠6 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
2
3 4
1
5 6
7 8
m
n
p
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Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they are parallel to each other.
If mlln and nllp, then mllp.
m
n
p
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Summary
At this point, you should be able to: Identify special properties of the four new angle pairs formed
when two lines are cut by a transversal that cause lines to be parallel.
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Example 1
Find the value of x that makes mlln.
105=3x
35=x 105°
(3x)°
m
n
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Example 2
Find the value of x that makes mlln.
5x=3x+60
2x=60
x=30 (5x)°
(3x+60)°
m
n
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Example 3
Find the value of x that makes mlln.
2x+x+30=180
3x+30=180
3x=150
x=50
(2x)°
(x+30)°
m
n
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Example 4
Find the value of x that makes mlln.
2x=x+30
x=30 (2x)°
(x+30)°
m
n
p q
(7x)°
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Example 5
For each problem determine if there is enough information to state that pllq.
85°
85°
1)
85°
95°
2)
3) 4)
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Example 6
Prove the Alternate Interior Angles Converse Theorem.
Given: ∠1≅∠2
Prove: mlln
1
2
3
m
n
Statements Reasons
1. ∠1≅∠2 1. Given
2. ∠3≅∠2 2. Vertical Angles Congruence Theorem
3. ∠3≅∠1 3. Transitive Property
4. mlln 4. Corresponding Angles Converse Postulate
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Example 6
Prove the Alternate Interior Angles Converse Theorem. Use a Paragraph Proof.
It is given that ∠1≅∠2. From the diagram ∠3≅∠2 due to the vertical angles congruence theorem. ∠3≅∠1 due to the transitive property. Therefore mlln by the corresponding angles converse postulate.
Statements Reasons
1. ∠1≅∠2 1. Given
2. ∠3≅∠2 2. Vertical Angles Congruence Theorem
3. ∠3≅∠1 3. Transitive Property
4. mlln 4. Corresponding Angles Converse Postulate
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“ A H E R O I S N O B R A V E R T H A N A N O R D I N A R Y M A N ( O R W O M A N ) , B U T H E ( / S H E ) I S B R A V E
F I V E M I N U T E S L O N G E R . ” – R A L P H W A L D O E M E R S O N
Find and Use Slopes of Lines
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Goal
You will learn how to find the slope of the line and how the slope of parallel and perpendicular lines relate.
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Slope of a line
Slope can be thought of as the steepness of a line.
Slope of a nonvertical line is the ratio of vertical change (rise) to horizontal change (run) between any two points on the line.
𝑚 =𝑟𝑖𝑠𝑒
𝑟𝑢𝑛=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥=
𝑦2−𝑦1
𝑥2−𝑥1
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Types of slope
m has negative slope
n has positive slope
p has zero slope
q has undefined slope
m
n
p
q
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Slopes of Parallel Lines
Two nonvertical lines are parallel iff they have the same slope.
Any two vertical lines are parallel
m
n
p q
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Slopes of Perpendicular Lines
Two nonvertical lines are perpendicular iff the product of their slope is -1.
Horizontal and vertical lines are perpendicular.
m
n
p
q
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Summary
You should be able to describe slope and identify slope of parallel and perpendicular lines.
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Example 1
Find the slope of lines m, p, and n.
m
n
p
q
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Example 2
Find the slope of lines m, p, and n. Determine if any of the lines are parallel.
m
n
p
q
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Example 3
Find the slope of a line perpendicular to the line containing the points (5, -4) and (7, 0).
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Example 4
A skydiver made jumps with 3 different parachutes. The graph of his jumps are below. Which statement is true?
0
500
1000
1500
2000
2500
3000
-3 2 7 12
Height (m)
Time (s)
Height of Each Dive
Dive 1
Dive 2
Dive 3
A) Dive 2 and Dive 3 started at the same height.
B) Dive 1 and Dive 2 lasted the same amount of time.
C) Dive 1 and Dive 3 were the same type of parachute.
D) Dive 2 had the parachute that had the slowest rate of decent.
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“ C E R T A I N S I G N S P R E C E D E C E R T A I N E V E N T S . ”
– C I C E R O
Write and Graph Equations of Lines
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Goal
Students will be able to write and graph equations of lines.
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Slope-intercept Form
y=mx+b
M is the slope of the line.
B is the y-intercept of the line (where the line crosses the y-axis).
Y is y-coordinate of a point on the line.
X is the x-coordinate of a point on the line.
To write equation of line we need to find both b and m before writing the equation of the line.
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𝑦 =1
2𝑥 + 3
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Standard Form
Ax+By=C
X-intercept (where the graph crosses the x-axis) is 𝐶
𝐴.
Y-intercept (where the graph crosses the y-axis) is 𝐶
𝐵.
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𝑥 + (−2)𝑦 = −6
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Summary
You should be able to identify the different ways to write an equation of a line.
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Example 1
Write the equation of each line in slope-intercept form.
m
n
p
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Example 2
Graph the equation of the line.
1) y=2x+7
2) 5x+10y=20
3) y=-3x
4) y=8
5) x=1
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Example 3
Write an equation of the line passing through points (2, 4) and is perpendicular to the line with the equation y=2x+7.
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Example 4
What is the slope and y-intercept of these lines.
1) 5x-10y=-20 1) -10y=-5x-20
2) 𝑦 =1
2𝑥 + 2
2) 3x+y=6 1) y=-3x+6
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Example 5
Write the equation of the line with the given information.
1) m=4, b=-2
y=4x-2
1) m=-1, b=7
y=-x+7
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“ A N X I E T Y I S F E A R O F O N E ’ S S E L F . ” –W I L H E L M S T E K E L
Prove Theorems about Perpendicular Lines
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Goal
You will learn how to prove statements about parallel and perpendicular lines.
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Theorem 3.8
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
If , then .
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Theorem 3.9
If two lines are perpendicular, then they intersect to form four right angles.
a
b
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Theorem 3.10
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
If , then ∠1 and ∠2 are complementary. 1 2
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Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
If , then .
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Lines Perpendicular to a Transversal Theorem
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
If , then .
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Distance from a line.
Distance from a point to a line is the length of the perpendicular segment from the point to the line.
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Summary
You should be able to prove statements parallel and perpendicular lines.
You should be able to determine what the distance from a point to a line is.
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Example 1
What is the value of x?
52=x+7
45=x
52°
(x+7)°
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Example 2
Determine which lines, if any, must be parallel in the diagram. Explain your reasoning.
m n p
q
r
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Example 3
Prove that if two sides of two adjacent angles are perpendicular, then the angles are complementary.
Given: ray BA ⊥ ray BC
Prove: ∠1 and ∠2 are complementary
1 2
A
B C
Statements Reasons
1. Ray BA ⊥ Ray BC 1. Given
2. ∠ABC is a right angle 2. ⊥ form 4 right angles
3. m∠ABC=90° 3. Definition of a right angle
4. m∠ABC=m∠1+m∠2 4. Angle Addition Postulate
5. 90°=m∠1+m∠2 5. Transitive Property
6. ∠1 and ∠2 are complementary
6. Definition of complementary angles
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Example 4
How far apart are lines m and n?
Perpendicular slopes have a product of -1
(-1
2)(_)=-1
(_)=2
Use distance formula
𝑦 = −1
2𝑥+5
m
n
𝑦 = −1
2𝑥
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“ R E M E M B E R T H A T H A P P I N E S S I S A W A Y O F T R A V E L — N O T A D E S T I N A T I O N . ” – R O Y M .
G O O D M A N
Taxicab Geometry
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Goal
To apply geometry to real world problems.
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Taxicab Distance
The distance between two points is the sum of the differences in their coordinates.
AB=|𝑥2 − 𝑥1| + |𝑦2 − 𝑦1|
𝐴 𝑥2, 𝑦2 , 𝐵(𝑥1, 𝑦1)
𝑥2 − 𝑥1
𝑦2 − 𝑦1
A(0, 6)
(4, 1)
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Example 1
Find the taxicab distance from A(-1, 5) to B (4, 2). Draw two different shortest paths from A to B.
|4--1|+|2-5|
|5|+|-3|
5+3
8
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Taxicab Circles
These are circles that are the same distance from the center.
Here is an example of a taxicab circle with radius of 3.
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Summary
You should be able to use taxicab geometry to find the “block” distance between two points.