Geometry: Chapter 3 Ch. 3. 4: Prove Lines are Parallel Ch. 3.5 Using Properties of Parallel Lines.
Chapter 3 Perpendicular and Parallel Lines Chapter Objectives Identify parallel lines Define angle...

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Transcript of Chapter 3 Perpendicular and Parallel Lines Chapter Objectives Identify parallel lines Define angle...
Chapter 3
Perpendicular and Parallel Lines
Chapter Objectives
Identify parallel linesDefine angle relationships between parallel linesDevelop a Flow ProofUse Alternate Interior, Alternate Exterior, Corresponding, & Consecutive Interior AnglesCalculate slopes of Parallel and Perpendicular lines
Lesson 3.1
Lines and Angles
Lesson 3.1 Objectives
Identify relationships between lines.Identify angle pairs formed by a transversal.Compare parallel and skew lines.
Lines and Angle Pairs
1
5
2
8
4
6
3
7Corresponding Angles – because they lie in corresponding positions of each intersection.
TransversalAlternate Exterior Angles – because they lie outside the two lines and on opposite sides of the transversal.
Alternate Interior Angles – because they lie inside the two lines and on opposite sides of the transversal.
Consecutive Interior Angles – because they lie inside the two lines and on the same side of the transversal.
Example 1
Determine the relationship between the given angles1) 3 and 9
1) Alternate Interior Angles
2) 13 and 52) Corresponding Angles
3) 4 and 103) Alternate Interior Angles
4) 5 and 154) Alternate Exterior Angles
5) 7 and 145) Consecutive Interior Angles
Parallel versus Skew
Two lines are parallel if they are coplanar and do not intersect.Lines that are not coplanar and do not intersect are called skew lines. These are lines that look like they intersect
but do not lie on the same piece of paper.
Skew lines go in different directions while parallel lines go in the same direction.
Example 2
Complete the following statements using the words parallel, skew, perpendicular.
1) Line WZ and line XY are _________.1) parallel
2) Line WZ and line QW are ________.2) perpendicular
3) Line SY and line WX are _________.3) skew
4) Plane WQR and plane SYT are _________.4) parallel
5) Plane RQT and plane WQR are _________.5) perpendicular
6) Line TS and line ZY are __________.6) skew
7) Line WX and plane SYZ are __________.7) parallel.
Parallel and Perpendicular Postulates:Postulate 13Parallel Postulate
If there is a line and a point not on the line, then there is exactly one line through the point that is parallel to the given line.
Parallel and Perpendicular Postulates:Postulate 14Perpendicular 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.
Homework 3.1
In Class 29
p132135
Homework 1031
Due Tomorrow
Lesson 3.2
Proof and Perpendicular Lines
Lesson 3.2 Objectives
Develop a Flow ProofProve results about perpendicular linesUse Algebra to find angle measure
Flow Proof
A flow proof uses arrows to show the flow of a logical argument.
Each reason is written below the statement it justifies.
1. 5 and 6 are a linear pair 6 and 7 are a linear pair
1. Given
2. 5 and 6 are supplementary 6 and 7 are supplementary
2. Linear Pair Postulate
3. 5 7 3. Congruent Supplements Theorem
PROVE: 5 7
GIVEN: 5 and 6 are a linear pair 6 and 7 are a linear pair
56 7
5 and 6 are a linear pair
6 and 7 are a linear pair
5 and 6 are supplementary
6 and 7 are supplementary
5 7
Given Given
Linear PairPostulate
Linear PairPostulate
Congruent Supplements Theorem
Theorem 3.1:Congruent Angles of a Linear Pair
If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
g
h
So g h
Theorem 3.2:Adjacent Angles Complementary
If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Theorem 3.3: Four Right Angles
If two lines are perpendicular, then they intersect to form four right angles.
Homework 3.2
None!Move on to Lesson 3.3
Lesson 3.3
Parallel Lines and Transversals
Lesson 3.3 Objectives
Prove lines are parallel using tranversals.Identify properties of parallel lines.
Postulate 15:Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
You must know the lines are parallel in order to assume the angles are congruent.
1
5
2
8
4
6
3
7
Theorem 3.4:Alternate Interior Angles
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Again, you must know that the lines are parallel. If you know the two lines are parallel, then identify where the
alternate interior angles are. Once you identify them, they should look congruent and they
are.
1
5
2
8
4
6
3
7
Theorem 3.5:Consecutive Interior Angles
If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.
Again be sure that the lines are parallel. They don’t look to be congruent, so they MUST be
supplementary.
5
4
6
31 2
8 7
+=180o + = 180o
Theorem 3.6:Alternate Exterior Angles
If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
Again be sure that the lines are parallel.
1 2
8
5
4
6
3
7
Theorem 3.7:Perpendicular Transversal
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Again you must know the lines are parallel. That also means that you now have 8 right
angles!
Example 3 Find the missing angles for the following:
120o
120o
105o
105o
110o
110o 70o
120o
60o
140o
140o
Homework 3.3
In Class 36
p146149
Homework 826, 3444 even
Due TomorrowQuiz Wednesday Lessons 3.13.3
Emphasis on 3.1 & 3.3
Lesson 3.4
Proving Lines are Parallel
Lesson 3.4 Objectives
Prove that lines are parallelRecall the use of converse statements
Postulate 16:Corresponding Angles Converse
If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
You must know the corresponding angles are congruent. It does not have to be all of them, just one pair to make
the lines parallel.
1
5
2
8
4
6
3
7
Theorem 3.8:Alternate Interior Angles Converse
If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.
Again, you must know that alternate interior angles are congruent.
5
4
6
31 2
8 7
Theorem 3.9:Consecutive Interior Angles Converse
If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel.
Be sure that the consecutive interior angles are supplementary.
5
4
6
31 2
8 7
+=180o + = 180o
Theorem 3.10:Alternate Exterior Angles Converse
If two lines are cut by a transversal so that alternate exterior angles are congruent, then lines are parallel.
Again be sure that the alternate exterior angles are congruent.
1 2
8
5
4
6
3
7
Example 4Is it possible to prove the lines are parallel?
If so, explain how.
Yes they are parallel!
Because Corresponding Angles are congruent.
Corresponding Angles Converse
Yes they are parallel!
Because Alternate Interior Angles are congruent.
Alternate Interior Angles Converse
Yes they are parallel!
Because Alternate Exterior Angles are congruent.
Alternate Exterior Angles Converse
No they are not parallel!
No relationship between those two angles.
Example 5Find the value of x that makes the m n.
x = 2x – 95 (AIA)
x = –95 (SPOE)
x = 95 (DPOE)
100 = 4x – 28 (CA)
128 = 4x (APOE)
x = 32 (DPOE)
(3x + 15) + 75 = 180(CIA)
3x + 90 = 180 (CLT)
3x = 90 (SPOE)
x = 30 (DPOE)
Directions do not ask for reasons, I am showing you them because I am a teacher!!
Homework 3.4
In Class 1, 39
p153156
Homework 1035, 37, 38
Due Tomorrow
Lesson 3.5
Using Properties of Parallel Lines
Lesson 3.5 Objectives
Prove more than two lines are parallel to each other.Identify all possible parallel lines in a figure.
Theorem 3.11:3 Parallel Lines Theorem
If two lines are parallel to the same line, then they are parallel to each other. This looks like the transitive property for
parallel lines.
p q r
If p // q and q // r, then p // r.
Theorem 3.12:Parallel Perpendicular Lines Theorem
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
m n
p
If m p and n p, then m // n.
Finding Parallel Lines
Find any lines that are parallel and explain why.
a b c
x
y
z
125o
55o
Results
x // y Corresponding Angles Converse
Postulate 16
y // z Consecutive Interior Angels Converse
Theorem 3.9
x // z 3 Parallel Lines Theorem
Theorem 3.11
b // c Alternate Exterior Angles Converse
Theorem 3.10
Homework 3.5
In Class 16, 19
p160163
Homework 824, 3336, 4351
Due Tomorrow
Lesson 3.6
Parallel Lines in the Coordinate Plane
Lesson 3.6 Objectives
Review the slope of a lineIdentify parallel lines based on their slopesWrite equations of parallel lines in a coordinate plane
Slope
Recall that slope of a nonvertical line is a ratio of the vertical change divided by the horizontal change.
It is a measure of how steep a line is. The larger the slope, the steeper the line is.
Slope can be negative or positive whether or not the lines slants up or down.
Remember that slope is often referred to asrise
run
Which really meansy2 – y1
x2 – x1
Which we find it in an equation by looking for m. y = mx + b
= m
Example of SlopeYou are given two points:
Now label each point as 1 and 2.1 2
Then substitute as the formula for slope tells you.
3 )B( , 81 )A( , 2
8=
– 2
3 – 1
y1y2– x2– x1
=6
2= 3
Postulate 17: Slopes of Parallel Lines Postulate
In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.
m = 1
m = 1 m = undefined
Writing an Equation inSlope Intercept Form
You will be given Slope
Or at least two points so you can calculate slope
yintercept y = mx + b
slope yintercept
This is the point at which the line touches the yaxis.
Your final answer should always appear in this form.
Writing an Equation GivenSlope and 1 Point
For this, you will be given the slope Or have to determine it from and equation Or determine it from a set of two points
You will also be given 1 point through which the line passesTo solve, use your slopeintercept form to find b
y = mx+b
Plug in Slope for m. The xvalue from your point for x. The yvalue from your point for y.
Solve for b using algebraWhen finished, be sure to rewrite in slope intercept form using your new m and b.
Leave x and y as x and y in your final equation.
Example
Example 5 p167
Write an equation of the line through the point (2,3) that has a slope of 5.
y = mx + b
y = 5x + b
3 = 5(2) + b
3 = 10 + b
b = 7
So, your final answer is:
y = 5x  7
Homework 3.6
WSDue Tomorrow
Lesson 3.7
Perpendicular Lines in the Coordinate Plane
Lesson 3.7 Objectives
Use slope to identify perpendicular linesWrite equations of perpendicular lines
Postulate 18:Slopes of Perpendicular Lines Postulate
In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is –1. Vertical and horizontal lines are
perpendicular.
Identifying Perpendicular Lines
There are two ways to identify perpendicular lines The product of the slopes equal –1
(3/2)(2/3) = 1 Or to get from one slope to the other,
you find the negative reciprocal Remember that reciprocal flips the
number. Well now you flip it and make it negative! 3/2 2/3
Determining Perpendicular Lines
Remember there are two ways to identify perpendicular lines
Multiply the slopes together If the answer is –1, they are perpendicular
Verify that the slopes are negative reciprocals of each other Take one of the slopes, flip it over, and make it negative. If the
answer matches the other slope, they are perpendicular.
Example (#25 p176) y = 3x y = 1/3x – 2
(3)(1/3) = 1 Perpendicular
Or 3 1/3 1/3
Check!
Tougher Example
Example 4 P173Decide whether the lines are perpendicular
Line r: 4x + 5y = 2 Line s: 5x + 4y = 3
Need to change into slopeintercept form.
y = mx + b
Subtract xterm from both sides5y = 4x + 2 4y = 5x + 3
Get y to be alone by dividing off the coefficienty = 4/5x + 2/5 y = 5/4x + 3/4
Now multiply their slopes
(4/5)(5/4) =20/20 = 1
NOT Perpendicular
Homework 3.7
WSDue TomorrowTest Thursday January 29th