angle pairs formed by a transversal - white plains public ... · web viewa pair of parallel planes...

Chapter 3

Parallel and Perpendicular Lines

Mrs. Steptoe

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Unit 3: Vocabulary1) parallel lines

2) perpendicular lines

3) skew lines

4) parallel planes

5) transversal

6) corresponding angles

7) alternate interior angles

8) alternate exterior angles

9) same-side interior angles

10) slope

11) slope-intercept form of the equation

12) point-slope form of the equation

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Day 1: Parallel and Perpendicular Lines

G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Warm Up Identify each of the following:

1. points that lie in the same plane

2. two angles whose sum is 180°

3. the intersection of two distinct intersecting lines

4. a pair of adjacent angles whose non-common sides are opposite rays

Model Problem

Identify each of the following:

A. A pair of parallel segments

B. A pair of skew segments

C. A pair of perpendicular segments

D. A pair of parallel planes

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Exercise Identify each of the following:

1) A pair of parallel segments

2) A pair of skew segments

3) A pair of perpendicular segments

4) A pair of parallel planes

Angle Pairs formed by a Transversal

When two lines are cut by a third line, called a transversal, angle pairs are formed. It is useful to know the names of these angle pairs.

Look for these characteristics when identifying angle pairs:

* Are the angles INTERIOR or EXTERIOR angles? In other words, are they inside or outside the parallel lines?

* Are they on the SAME SIDE or ALTERNATE SIDES of the transversal?

There are four major type of angle pairs.

(1)Corresponding Angleslie on the same side of the transversal, one in the exterior and one in the interior.

Example #1 Example #2 Example #3

(2)Alternate Interior Angleslie between the two lines, on opposite sides of the transversal.


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(3)Alternate Exterior Angleslie outside the two lines, on opposite sides of the transversal.


(4)Same-Side Interior Angles (or Consecutive Interior Angles)lie inside the two lines, on the same side of the transversal.


Model Problem


A. A pair of corresponding angles

B. A pair of alternate interior angles

C. A pair of alternate exterior angles

D. A pair of same-side interior angles

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Exercise


1) A pair of corresponding angles

2) A pair of alternate interior angles

3) A pair of alternate exterior angles

4) A pair of same-side interior angles

Parallel Lines and Angles

Look at the following pairs of lines cut by a transversal l . In the first example, the pair of lines are not parallel, so alternate interior angles ∠1 and ∠2 are NOT congruent. However, when the two lines are parallel, alternate interior angles ∠1 and ∠2 are congruent.

Example #1 Example #2

1

2 l 1

2 l

Example

In the accompanying diagram, AB∥CD and are cut by a transversal EF.

1) What is the relationship between ∠ AGH and ∠GHD?

2) Find m∠ AGH .

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Relationships in Parallel Lines

You can use these relationships to find every angle in a parallel line diagram:

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Homework

Vocabulary

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Day 2: Parallel Lines and Angles


Warm-Up

Identify a pair of:

a) Corresponding angles __________ & _____________

b) Same-side interior angles __________ & _____________

c) Alternate interior angles __________ & _____________

d) Alternate exterior angles __________ & _____________

If m∠ 9=35 °, must m∠7=35 ° also? Under what conditions would this be true?

Relationships in Parallel Lines: Summary

CONGRUENTCorresponding Angles

Alternate Interior AnglesAlternate Exterior Angles

SUPPLEMENTARY Same-side Interior AnglesSame-Side Exterior Angles

Model Problem

In the accompanying diagram, AG∥ DF.

a) What is the relationship between ∠ ABC and ∠DCE?

b) Find m∠ ABC .

Exercise9

1) In the accompanying diagram, AG∥ DF.

a) What is the relationship between ∠ ABD and ∠GDE?

b) Find m∠GDE in the diagram at right.

2) Given: HE∥ AD

a) What is the relationship between ∠FGH and ∠BCD?

b) Find m∠FHG .

3)

a) What is the relationship between ∠BEF and ∠EFD ?

b) Find m∠BEF.

Proving Lines Parallel

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In our last lesson, we learned that certain angle pairs formed by two parallel lines cut by a transversal were congruent or supplementary.

In this lesson, we will show how knowing that these angle pairs are congruent or supplementary will help us to show that lines are parallel.

Model Problem A

Let m∠1=120 ° and m∠ 6=60 ° . Is r ∥s? Explain why or why not.

1) Find the measure of the numbered angles.

2) Select a pair of angles and state their relationship.

Angles: _____ and __________ Relationship: ____________________________________

3) Are they congruent or supplementary? ___________________________________

4) Answer the question in a full sentence:

Model Problem B

Let m∠1=120 ° and m∠3=50 ° . Is r ∥s? Explain why or why not.

1) Find the measure of the numbered angles.

2) Select a pair of angles and state their relationship.

Angles: _____ and __________ Relationship: ____________________________________

3) Are they congruent or supplementary? ___________________________________

4) Answer the question in a full sentence:

Exercise11

1) Is showing that ∠2≅∠6 sufficient to prove m∥ n?Explain.

2) Is showing that ∠8≅∠ 4 sufficient to prove m∥ n?Explain.

3) Is showing that ∠3≅∠5 sufficient to prove m∥ n?Explain.

4) If m∠2=36 ° and m∠3=144 °, is m∥ n? Explain.

Summary

Each of these postulates and its converse is true. This means two things:

When lines are parallel, these angle pairs are congruent (or supplementary); AND

When these angle pairs are congruent (or supplementary), the lines are parallel.

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Homework (Day 2)

p. 158-160 #1-4, 13-

19, 35

and

p. 167 #24-36 (evens

only)

Day 3: Parallel Lines, Perpendicular Lines, and Constructions


G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Warm-Up

Part A: More Practice with Parallel Lines

7)

13

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10)

Challenge

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1)

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5)

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Constructions with Parallel and Perpendicular Lines

Task 1: Construct a line parallel to a given line, through a point not on the line.

Model Problem

Reflect

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Practice

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Task 2: Construct the perpendicular bisector of a segment.

Reflect

This construction works because the perpendicular bisector is the set of all points equidistant from the endpoints of a segment. On the diagram below, sketch the segment(s) that illustrate this principle.

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Task 3: Construct a line perpendicular to a given line, through a point not on the line.

Reflect

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The altitude to the side of a triangle is a segment perpendicular to a side and passing through the opposite vertex. Construct an altitude to AB through point C.

Construct the perpendicular bisector of the segment below.

Homework: Chapter 3 Skills Practice

Identify two pairs of each type of angle relationship.

4. corresponding angles _________________________

5. alternate exterior angles _______________________

6. same-side interior angles ______________________

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Use the given information to determine if l ∥m. Explain your answer using one of the theorems covered in class.

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Day 4: The Slope FormulaG-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Warm-Up

In the figure at right, lines l and mare cut by transversal t .If m∠2=(x+18)° and m∠ 6=(2 x−38 ) ° , what value of x would make lines l and m parallel?

The Meaning of the Slope Formula

Line segment has endpoints A (7,8) and B(4,2).

What does it mean to find the slope of this segment?

You have already learned that the slope of a line refers to the ratio of its vertical rise to its horizontal run. You may also recall the slope formula:

Slope of (m) =

Example

Calculate the slope of : A (7, 8) B (4, 2)

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Parallel Line Segments

Parallel line segments have the same slope.

In the diagram at right, AB and CD are parallel because they both have a slope of -4.

To show two segments are parallel, we find their slopes and show they are the same.

Model Problem

has endpoints A(1, 3) and B(5,1). has endpoints C(5,6) and D(9, 4).

Show that ║ .

Slope of =

(1, 3) (5,1)

Slope of =

(5,6) (9, 4)

Write a concluding sentence:

Exercise

has endpoints A(2, 2) and B(4, 6). has endpoints C(6,4) and D(7, 6).

Show that ║ .

Slope of =

(2, 2) (4, 6)

Slope of = (6, 4) (7, 6)

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Write a concluding sentence:Perpendicular Line Segments

Perpendicular line segments are segments that form a 90-degree angle.

We can show that two lines will be perpendicular when their slopes are negative reciprocals. That is, the product of their slopes is -1.

For example, ´JH and FG are perpendicular. The slope of ´JH is -1 and the slope of FG is 1.

Since – 1* 1 = -1, the lines are perpendicular.

We write ´JH ⊥ FG.

Exercise

1) Tell whether each pair of numbers are negative reciprocals.

a. 2 and ½ ___________ d. 34 and

−43 ____________

b. 2 and – ½ ___________ e. 0 and 0 ____________

c. 1 and -1 ___________

Model Problem

has endpoints A(2, 3) and B(5, 1). has endpoints C(5,5) and D(1, -2). Show that .

Slope of =

(2, 3) (5,1)

Slope of =

(5, 5) (3, 2)


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Exercise

has endpoints A(2, 2) and B(4, 5). BC has endpoints B(4, 5) and C (7, 3). Show that .

Slope of =

(2, 2) (4, 5)

Slope of BC =

(4, 5) (7, 3)


Vertical and Horizontal Lines

Horizontal Lines Vertical Lines

Horizontal lines are parallel to the x-axis.They have a slope of zero.

Ex. Find the slope of the line containing(-4, 3) and (1, 3).

Vertical lines are parallel to the y-axis.We say their slope is undefined, or that they have no slope.

Ex. Find the slope of the line containing(-3, 2) and (-3, -1).

What do you notice about the y-coordinates What do you notice about the 25

of the points above? x-coordinates of the points above?

Note:

All horizontal lines are perpendicular to all vertical lines.

Model Problem

The coordinates of and RS are given. Find the slope of each line segment. Tell if the line segments are parallel, perpendicular, or neither. Explain your choice.

Parallel, perpendicular, or neither?

1. P(1, 1) and Q(2, 4) R(3, -2) and S (4,1)

2. P(1,5) and Q (2,3) R(2, 3) and S(0,2)

Exercise

The coordinates of and RS are given. Find the slope of each line segment. Tell if the line segments are parallel, perpendicular, or neither.

Parallel, perpendicular, or neither?

1. P(2,0) and Q(2,4) R(8, 6) and S(2,6)

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Slope is undefined

Slope is zero

2. P(8,3) and Q(6,-3) R(0, -6) and S(-2, -3)

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Homework

A. Find the slope of the line containing each set of points.

(5,8) and (4,3) (4, 6) and (-5, 9) (3, 5) and (8, 5)

B. Find the slopes of AB and CD . Then tell whether AB∧CD are parallel, perpendicular, or neither.

1. A(1, 0) and B(5, 3) C(6, -1) and D(0, 2)

2. A(5, 1) and B(-1, -1) C(2, 1) and D(3, -2)

3. A(4, 3) and B(-2, 3) C(-6, 7) and D(-6, 2)

4. A(2, 3) and B(5, 1) C(5, 4) and D(1, - 2)

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Day 5: The Slope-Intercept Form of the EquationG-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Warm-Up

Solve each equation for y.

1. y – 6x = 9 2. 4x – 2y = 8

Every line in the coordinate plane can be described by an equation. The equation gives us enough information about the line to sketch it and describe its behavior.

There are two forms of the equation of a line, the point-slope form and the slope-intercept form. In this lesson, we will learn how to write the equation of a line in slope-intercept form.

The Slope-Intercept Form of the Equation of a Line

The Meaning of the Slope-Intercept Form

The slope-intercept form of the equation tells where the graph hits the y-axis (the y-intercept) and how y changes as compared to x (the slope). Look at this family of functions.

The slope intercept form of the equation is

y=m x+bModel Problems Find the slope and the y-intercept of each line.

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y = 3x y = 3x + 5 y = 3x - 5

1) y=3 x+5 2) 3 x−2 y=6 3) x+6 y=18

4) Write the equation of a line whose slope is ¼ and whose y-intercept is -2.

Exercises Find the slope and the y-intercept of each line.

1) y=2 x−5 2) 3 y+2 x=15 3) x− y=7

4) Write the equation of a line whose slope is -3 and whose y-intercept is 0.

Writing the Equation of a Line in Slope-Intercept Form

Step 1 Find the slope.

Step 2 Write y = mx + b and plug in the slope.

Step 3 Plug in the point (3,2) in for x and y. Solve for b.

Step 4 Rewrite y = mx + b with the slope and y-intercept.

Model Problem A

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Write the equation of the line that is parallel to the line 10x + 5y = 10 and passes through the point (3, 2) in slope-intercept form.

Sketch the graph of this line.

Model Problem B

Write the equation of the line that is perpendicular to the line x + 5y = 10 and passes through the point (1, 0) in slope-intercept form.


Exercise31

If the slope is not given, there will always be a clue!

1) Write an equation of a line parallel to the line 2x + y = 7 and passing through the point (3, -1).

2) Write the equation of the line perpendicular to x + 4y = 8 and passing through the point (3, 0) in slope-intercept form.


Model Problem C

Write the equation of the line that passes through (-1, 3) and (1, -1) in slope-intercept form. Sketch the graph of this line.

Model Problem D Application to Dilations32

Key Point: The original line and the image are __________________.

Exercise

1) Write the equation of the line containing the points (3, 5) and (4, 7).

2)

Challenge

Line l contains the points (2, 3) and (6, 5). Determine if the point (-8, -2) lies on line l.

Homework

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1. Write the equation of the line, in slope-intercept form, that has the given slope and that passes through the given point.

a. m = 3, (1,5) b. m = 0, (-2, 8)

2. Write the equation of the line, in slope-intercept form, that passes through the given points.

a. (12, -5) and (-4, -1) b. (0, 3) and (2, 9)

3. Write the equation of the line, in slope-intercept form, that passes through the given points.

b. (12, -5) and (-4, -1) b. (0, 3) and (2, 9)

3. Write, in slope-intercept form, the equation of a line that:

a) Is parallel to y – 2x = 4 and whose y-intercept is -1.

b) Is parallel to 2y – 4x = 8 and passes through the point (-2, 1).

c) Is perpendicular to the line y = −23

x+2 and passes through (2, 4)

d) Is perpendicular to the line x = 3 and passes through (3, 4).

e) Represents the line y=2x−5after a dilation of 3.

4. Sketch the graph of:

a. y=3 x+2 b. 2 x+3 y=9 c. x−2 y=−8

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Day 6: The Point-Slope Form of the EquationG-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

Warm-Up

Write the equation of a line parallel to the line given by y = 3x + 5 and passing through the point (1, -2).

The Point-Slope Form of the Equation of a Line

Note: It is not necessary for the given point on the line to be the y-intercept. If the y-intercept is given (i.e. the point is in the form (0, b)), then it might be easier to use slope-intercept form.

Model Problem A

Write the equation of a line in point-slope form of a line that contains the point (-2, 3) and has a slope of-½ . Sketch the graph of this line.

Exercise

Write the equation of a line in point-slope form that passes through the point (-2, 3) and has a slope of 2.Sketch the graph of this line.

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The Equations of Horizontal and Vertical Lines

Model Problem Sketch the graphs of:

x=4 y=3 x=0

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Model Problem BWrite the equation of the line parallel to 6x – 2y = 4 and passing through the point (3, –4) in point-slope form.

E xercise Write the equation of the line parallel to 2y = 8x - 4 and passing through the point (1, -5) in point-slope form. Sketch the graph of this line.

Exercise Sketch the graph of:

y=−3x=4

y=0

Homework

1) Write the equation of the line, in point-slope form, that has the given slope and that passes through the given point. Sketch the graph of each equation.

a. m = 3, (1,5) b. m = 0, (-2, 8) c. m = ½, (-2, -3)

2) Write the equation of the line, in point-slope form, that passes through the given points.

c. (12, -5) and (-4, -1) d. (0, 3) and (2, 9)

3) Write, in point-slope form, the equation of a line that:

a) Is parallel to y = 3x + 5 and passes through the point (0, 7)

b) Is parallel to 2x + 3y = 6 and passes through the origin.

c) Is perpendicular to y = 4x – 3 and passes through the point (8, 1)

d) Is perpendicular to 4x + 3y = 9 and passes through the point (5, -3).

4) Graph each line.

a) x=−2 b) y=0 c) x=1

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Day 7 – Review of Chapter 3

Chapter 3-1: Parallel and Perpendicular Lines and Planes

1) a pair of parallel segments _______________________

2) a pair of perpendicular segments __________________

3) a pair of skew segments _________________________

4) a pair of parallel planes __________________________

Chapter 3-2: Angles formed by Parallel Lines

Identify the relationship between the marked angles. Find the measure of the indicated angle.

1)

2)

3)

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Chapter 3-2 Proving Lines Parallel

State whether or not the given information is sufficient to determine whether c ∥d .

1) m∠ 4=63 °∧m∠6=117 °

2) m∠1=40 °∧m∠4=140 °

3) m∠3=85 °∧m∠7=85°

4) m∠2=90 °

Multiple Choice Practice

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Constructions

Construct a line parallel to m and passing through P. Then construct a line perpendicular to m and passing through P.

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Construct the perpendicular bisector of the segment below. Locate the midpoint and label it M.

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angle pairs formed by a transversal - white plains public ... · web viewa pair of parallel planes...

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