Lesson 3-1 Parallel Lines and Transversals

Lesson 3-1 Parallel Lines and Transversals
Lesson 3-2 Angles and Parallel Lines Lesson 3-3 Slopes of Lines Lesson 3-4 Equations of Lines Lesson 3-5 Proving Lines Parallel Lesson 3-6 Perpendiculars and Distance Chapter Menu

Five-Minute Check (over Chapter 2) Main Ideas and Vocabulary
Example 1: Identify Relationships Example 2: Real-World Example: Identify Transversals Key Concept: Transversals and Angles Example 3: Identify Angle Relationships Lesson 3-1 Menu

Lesson 3-1 Ideas/Vocabulary
Identify the relationships between two lines or two planes. Name angles formed by a pair of lines and a transversal. parallel lines consecutive interior angles alternate exterior angles alternate interior angles corresponding angles parallel planes skew lines transversal Lesson 3-1 Ideas/Vocabulary

Identify Relationships
A. Name all planes that are parallel to plane AEF. Answer: plane BHG Lesson 3-1 Example 1a

__ B. Name all segments that intersect AF. Answer: EF, GF, DA, and BA __ Lesson 3-1 Example 1b

__ C. Name all segments that are skew to AD. Answer: FG, GB, EH, EC, and CH __ Lesson 3-1 Example 1c

A. Name a plane that is parallel to plane RST.
B C D plane WTZ plane SYZ plane WXY plane QRX Lesson 3-1 CYP 1a

B. Name a segment that intersects YZ.
XY WX QW RS A. B. C. D. Lesson 3-1 CYP 1b

C. Name a segment that is parallel to RX.
B. C. D. ZW TZ QR ST Lesson 3-1 CYP 1c

D. Name a segment that is skew to TZ. __
RX ST XW QW __ ___ A B C D Lesson 3-1 CYP 1d

Identify Transversals
A. BUS STATION Some of a bus station’s driveways are shown. Identify the sets of lines to which line v is a transversal. Answer: If the lines are extended, line v intersects lines u, w, x, and z. Lesson 3-1 Example 2a

B. BUS STATION Some of a bus station’s driveways are shown. Identify the sets of lines to which line y is a transversal. Answer: lines u, w, x, z Lesson 3-1 Example 2b

C. BUS STATION Some of a bus station’s driveways are shown. Identify the sets of lines to which line u is a transversal. Answer: lines v, x, y, z Lesson 3-1 Example 2c

A. HIKING A group of nature trails is shown
A. HIKING A group of nature trails is shown. Identify the sets of lines to which line a is a transversal. A B C D lines c, f lines c, d, e lines c, d, f lines c, d, e, f Lesson 3-1 CYP 2a

B. HIKING A group of nature trails is shown
B. HIKING A group of nature trails is shown. Identify the sets of lines to which line b is a transversal. A B C D no lines lines c, f lines c, d, e, f lines a, c, d, e, f Lesson 3-1 CYP 2b

C. HIKING A group of nature trails is shown
C. HIKING A group of nature trails is shown. Identify the sets of lines to which line c is a transversal. A B C D lines a, b lines a, b, d, e, f lines a, d, f lines a, b, e Lesson 3-1 CYP 2c

D. HIKING A group of nature trails is shown
D. HIKING A group of nature trails is shown. Identify the sets of lines to which line d is a transversal. A B C D lines e, f lines a, b, f lines a, b, c, e, f lines a, b, c Lesson 3-1 CYP 2d

Lesson 3-1 Key Concept 1

Identify Angle Relationships
A. Identify 7 and 3 as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: corresponding Lesson 3-1 Example 3a

B. Identify 8 and 2 as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: alternate exterior Lesson 3-1 Example 3b

C. Identify 4 and as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: corresponding Lesson 3-1 Example 3c

D. Identify 7 and 1 as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: alternate exterior Lesson 3-1 Example 3d

E. Identify 3 and 9 as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: alternate interior Lesson 3-1 Example 3e

F. Identify 7 and as alternate interior, alternate exterior, corresponding, or consecutive interior angles. Answer: consecutive interior Lesson 3-1 Example 3f

A. Identify 4 and 5. A B C D alternate interior alternate exterior
corresponding consecutive interior Lesson 3-1 CYP 3a

B. Identify 7 and 9. A B C D alternate interior alternate exterior
corresponding consecutive interior Lesson 3-1 CYP 3b

C. Identify 4 and 7. A B C D alternate interior alternate exterior
corresponding consecutive interior Lesson 3-1 CYP 3c

D. Identify 2 and 11. A B C D alternate interior alternate exterior
corresponding consecutive interior Lesson 3-1 CYP 3d

E. Identify 3 and 5. A B C D alternate interior alternate exterior
corresponding consecutive interior Lesson 3-1 CYP 3e

F. Identify 6 and 9. A B C D alternate interior alternate exterior
corresponding consecutive interior Lesson 3-1 CYP 3e

Five-Minute Check (over Lesson 3-1) Main Ideas
Postulate 3.1: Corresponding Angles Postulate Example 1: Determine Angle Measures Theorem: Parallel Lines and Angle Pairs Theorem 3.4: Perpendicular Transversal Theorem Example 2: Standardized Test Example: Use an Auxiliary Line Example 3: Find Values of Variables Lesson 3-2 Menu

Use the properties of parallel lines to determine congruent angles. Use algebra to find angle measures. Lesson 3-2 Ideas/Vocabulary

Lesson 3-2 Postulate 3.1

Determine Angle Measures
In the figure, x || y and m 11 = 51. Find m 16. Corresponding Angles Postulate Vertical Angles Theorem Transitive Property Definition of congruent angles Substitution Answer: Lesson 3-2 Example 1

In the figure a || b and m 18 = 42. Find m 25.
C D 42 84 48 138 Lesson 3-2 CYP 1

Lesson 3-2 Theorem

Lesson 3-2 Theorem 3.4

What is the measure of RTV?
Use an Auxiliary Line What is the measure of RTV? Read the Test Item You are asked to find mRTV. Be sure to identify it correctly on the figure. Lesson 3-2 Example 2

Draw through T parallel to and
Use an Auxiliary Line Solve the Test Item Draw through T parallel to and Alternate Interior Angles Theorem Definition of congruent angles Substitution Lesson 3-2 Example 2

Alternate Interior Angles Theorem
Use an Auxiliary Line Alternate Interior Angles Theorem Definition of congruent angles Substitution Angle Addition Postulate Answer: 125 Lesson 3-2 Example 2

What is the measurement of IGE?
B C D 45 48 87 93 Lesson 3-2 CYP 2

Find Values of Variables
ALGEBRA If , and , find x and y. Find x. Since p || q, by the Corresponding Angles Postulate. Lesson 3-2 Example 3

Definition of congruent angles Substitution Subtract x from each side and add 10 to each side. Find y. Since m || n, by the Alternate Exterior Angles Theorem. Definition of congruent angles Substitution Lesson 3-2 Example 3

2(25) – 10 = 4 (y – 25) 50 – 10 = 4y – 100 Simplify. Add 100 to each side. Divide each side by 4. Answer: x = 25 and y = 35 Lesson 3-2 Example 3

ALGEBRA If and find x and y.
C D x = 9, y = 14 x = 12, y = 20 x = 10, y = 16 x = 14, y = 24 Lesson 3-2 CYP 3

Five-Minute Check (over Lesson 3-2) Main Ideas and Vocabulary
Key Concept: Slope Formula Example 1: Find the Slope of a Line Example 2: Real-World Example: Use Rate of Change to Solve a Problem Postulates: Parallel and Perpendicular Lines Example 3: Determine Line Relationships Example 4: Use Slope to Graph a Line Lesson 3-3 Menu

Find slopes of lines. Use slope to identify parallel and perpendicular lines. slope rate of change Lesson 3-3 Ideas/Vocabulary

A. Find the slope of the line.
Find the Slope of a Line A. Find the slope of the line. From (–3, 7) to (–1, –1), go down 8 units and right 2 units. Answer: Lesson 3-3 Example 1

B. Find the slope of the line.
Find the Slope of a Line B. Find the slope of the line. Use the slope formula. Let (0, 4) be (x1, y1) and (0, –3) be (x2, y2). which is undefined. Answer: undefined Lesson 3-3 Example 1

C. Find the slope of the line.
Find the Slope of a Line C. Find the slope of the line. Answer: Lesson 3-3 Example 1

D. Find the slope of the line.
Find the Slope of a Line D. Find the slope of the line. Answer: 0 Lesson 3-3 Example 1

A. Find the slope of the line.
B C D Lesson 3-3 CYP 1a

B. Find the slope of the line.
A B C D undefined 7 Lesson 3-3 CYP 1b

C. Find the slope of the line.
–2 2 A B C D Lesson 3-3 CYP 1c

D. Find the slope of the line.
A B C D undefined 3 Lesson 3-3 CYP 1d

Use Rate of Change to Solve a Problem
RECREATION For one manufacturer of camping equipment, between 1995 and 2005 annual sales increased by $7.4 million per year. In 2005, the total sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2015? Let (x1, y1) = (2005, 85.9) and m = 7.4. Slope formula m = 7.4, y1 = 85.9, x1 = 2005, and x2 = 2015 Lesson 3-3 Example 2

Use Rate of Change to Solve a Problem
Simplify. Multiply each side by 10. Add 85.9 to each side. The coordinates of the point representing the sales for 2015 are (2015, 159.9). Answer: The total sales in 2015 will be about $159.9 million. Lesson 3-3 Example 2

CELLULAR TELEPHONES Between 1994 and 2000, the number of cellular telephone subscribers increased by an average rate of 14.2 million per year. In 2000, the total subscribers were million. If the number of subscribers increases at the same rate, how many subscribers will there be in 2010? A B C D about million about million about million about million Lesson 3-3 CYP 2

Lesson 3-3 Postulates 1

Determine Line Relationships
A. Determine whether and are parallel, perpendicular, or neither. F(1, –3), G(–2, –1), H(5, 0), J(6, 3) Find the slopes of and Lesson 3-3 Example 3

Answer: The slopes are not the same, so and are not parallel. The product of the slopes is So, and are neither parallel nor perpendicular. Lesson 3-3 Example 3

B. Determine whether and are parallel, perpendicular, or neither. F(4, 2), G(6, –3), H(–1, 5), J(–3, 10) Answer: The slopes are the same, so and are parallel. Lesson 3-3 Example 3

A. Determine whether and are parallel, perpendicular, or neither
A. Determine whether and are parallel, perpendicular, or neither. A(–2, –1), B(4, 5), C(6, 1), D(9, –2) A B C parallel perpendicular neither Lesson 3-3 CYP 3a

B. Determine whether and are parallel, perpendicular, or neither
B. Determine whether and are parallel, perpendicular, or neither. A(7, –3), B(1, –2), C(4, 0), D(–3, 1) A B C parallel perpendicular neither Lesson 3-3 CYP 3b

Use Slope to Graph a Line
Graph the line that contains Q(5, 1) and is parallel to with M(–2, 4) and N(2, 1). First find the slope of Slope formula Substitution Simplify. Lesson 3-3 Example 4

Use Slope to Graph a Line
The slopes of two parallel lines are the same. The slope of the line parallel to through Q(5, 1) is Answer: Graph the line. Start at (5, 1). Move down 3 units and then move right 4 units. Label the point R. Draw Lesson 3-3 Example 4

Graph the line that contains R(2, –1) and is parallel to with O(1, 6) and P(–3, 1).
B. C. D. none of these A B C D Lesson 3-3 CYP 4

Example 1: Slope and y-Intercept Example 2: Slope and a Point Example 3: Two Points Example 4: One Point and an Equation Example 5: Real-World Example: Write Linear Equations Lesson 3-4 Menu

Write an equation of a line given information about its graph. Solve problems by writing equations. slope-intercept form point-slope form Lesson 3-4 Ideas/Vocabulary

y = mx + b Slope-intercept form y = 6x + (–3) m = 6, b = –3
Slope and y-intercept Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of –3. y = mx + b Slope-intercept form y = 6x + (–3) m = 6, b = –3 Answer: The slope-intercept form of the equation of the line is y = 6x – 3. Lesson 3-4 Example 1

Write an equation in slope-intercept form of the line with slope of –1 and y-intercept of 4.
B C D x + y = 4 y = x – 4 y = –x – 4 y = –x + 4 Lesson 3-4 CYP 1

Write an equation in point-slope form of the line
Slope and a Point Write an equation in point-slope form of the line whose slope is that contains (–10, 8). Point-slope form Simplify. Answer: Lesson 3-4 Example 2

Write an equation in point-slope form of the line
whose slope is that contains (6, –3). A B C D Lesson 3-4 CYP 2

First find the slope of the line.
Two Points Write an equation in slope-intercept form for a line containing (4, 9) and (–2, 0). First find the slope of the line. Slope formula x1 = 4, x2 = –2, y1 = 9, y2 = 0 Simplify. Lesson 3-4 Example 3

Now use the point-slope form and either point to write an equation.
Two Points Now use the point-slope form and either point to write an equation. Using (4, 9): Point-slope form Distributive Property Add 9 to each side. Lesson 3-4 Example 3

Distributive Property
Two Points Using (–2, 0): Point-slope form Simplify. Distributive Property Answer: Lesson 3-4 Example 3

Write an equation in slope-intercept form for a line containing (3, 2) and (6, 8).
B C D Lesson 3-4 CYP 3

One Point and an Equation
Write an equation in slope-intercept form for a line containing (1, 7) that is perpendicular to the line Since the slope of the line is the slope of a line perpendicular to it is 2. Lesson 3-4 Example 4

One Point and an Equation
y – y1 = m(x – x1) Point-slope form y – 7 = 2(x – 1) m = 2, (x1, y1) = (1, 7) y – 7 = 2x – 2 Distributive Property y = 2x + 5 Add 7 to each side. Answer: y = 2x + 5 Lesson 3-4 Example 4

Write an equation in slope-intercept form for a line containing (–3, 4) that is perpendicular to the line A B C D Lesson 3-4 CYP 4

Write Linear Equations
RENTAL COSTS An apartment complex charges $525 per month plus a $750 annual maintenance fee. A. Write an equation to represent the total first year’s cost A for r months of rent. For each month of rent, the cost increases by $525. So the rate of change, or slope, is 525. The y-intercept is located where 0 months are rented, or $750. A = mr + b Slope-intercept form A = 525r m = 525, b = 750 Answer: The total annual cost can be represented by the equation A = 525r Lesson 3-4 Example 5a

Write Linear Equations
RENTAL COSTS An apartment complex charges $525 per month plus a $750 annual maintenance fee. B. Compare this rental cost to a complex which charges a $200 annual maintenance fee but $600 per month for rent. If a person expects to stay in an apartment for one year, which complex offers the better rate? Evaluate each equation for r = 12. First complex: Second complex: A = 525r A = 600r + 200 = 525(12) r = 12 = 600(12) + 200 = 7050 Simplify. = 7400 Answer: The first complex offers the better rate: one year costs $7050 instead of $7400. Lesson 3-4 Example 5b

A. Write an equation to represent the total cost C for d days of use.
RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit. A. Write an equation to represent the total cost C for d days of use. A B C D C = 25 + d + 100 C = 125 d C = 100d + 25 C = 25d + 100 Lesson 3-4 CYP 5a

RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit.
B. Compare this rental cost to a company which charges a $50 deposit but $35 per day for use. If a person expects to rent a car for 9 days, which company offers the better rate? A B C D first company second company neither cannot be determined Lesson 3-4 CYP 5b

Five-Minute Check (over Lesson 3-4) Main Ideas Postulates 3.4 and 3.5
Theorems 3.5, 3.6, 3.7, and 3.8 Example 1: Identify Parallel Lines Example 2: Solve Problems with Parallel Lines Example 3: Prove Lines Parallel Example 4: Slope and Parallel Lines Lesson 3-5 Menu

Recognize angle conditions that occur with parallel lines. Prove that two lines are parallel based on given angle relationships. Lesson 3-5 Ideas/Vocabulary

Animation: Construct a Parallel Line Through a Point not on Line
Lesson 3-5 Postulates

Lesson 3-5 Theorems

Identify Parallel Lines
Determine which lines, if any, are parallel. Since RQT and SQP are vertical angles, m SQP = 77. Since m UPQ + m SQP = or 180, consecutive interior angles are supplementary. So, a || b. Since m TQR + m VRQ = or 177, consecutive interior angles are not supplementary. So, c is not parallel to a or b. Answer: a || b Lesson 3-5 Example 1

Determine which lines, if any are parallel. I. e || f II. e || g III
Determine which lines, if any are parallel. I. e || f II. e || g III. f || g A B C D I only II only III only I, II, and III Lesson 3-5 CYP 1

Solve Problems with Parallel Lines
ALGEBRA Find x and m ZYN so that || Explore From the figure, you know that m WXP = 11x – 25 and m ZYN = 7x You also know that WXP and ZYN are alternate exterior angles. Lesson 3-5 Example 2

Plan For line PQ to be parallel to MN, the alternate exterior angles must be congruent. So, m WXP = m ZYN. Substitute the given angle measures into this equation and solve for x. Once you know the value of x, use substitution to find m ZYN. Solve m WXP = m ZYN Alternate exterior angles 11x – 25 = 7x + 35 Substitution 4x – 25 = 35 Subtract 7x from each side. 4x = 60 Add 25 to each side. x = 15 Divide each side by 4. Lesson 3-5 Example 2

Now use the value of x to find m ZYN. m ZYN = 7x + 35 Original equation = 7(15) + 35 x = 15 = 140 Simplify. Examine Verify the angle measure by using the value of x to find m WXP. That is, 11x – 25 = 11(15) – 25 or 140. Since m WXP = m ZYN, m WXP m ZYN and || Answer: x = 15, m ZYN = 140 Lesson 3-5 Example 2

ALGEBRA Find x so that || .
C D x = 60 x = 9 x = 12 Lesson 3-5 CYP 2

Prove Lines Parallel Prove: r || s Given: ℓ || m Lesson 3-5 Example 3

Prove Lines Parallel Proof: Statements Reasons 1. 1. Given
Consecutive Interior Angle Theorem Definition of supplementary angles Definition of congruent angles Substitution Definition of supplementary angles If consecutive interior angles are supplementary, then lines are parallel. Lesson 3-5 Example 3

not enough information to determine
Given x || y and , do you need to use the Corresponding Angles Postulate to prove a || b? A B C yes no not enough information to determine Lesson 3-5 CYP 3

Slope and Parallel Lines
Determine whether p || q. slope of p: slope of q: Answer: Since the slopes are equal, p || q. Lesson 3-5 Example 4

Determine whether r || s.
A B C Yes, r is parallel to s. No, r is not parallel to s. It cannot be determined. Lesson 3-5 CYP 4

Key Concept: Distance Between a Point and a Line Example 1: Distance from a Point to a Line Example 2: Construct a Perpendicular Segment Key Concept: Distance Between Parallel Lines Theorem 3.9 Example 3: Distance Between Lines Lesson 3-6 Menu

Find the distance between a point and a line. Find the distance between parallel lines. equidistant Lesson 3-6 Ideas/Vocabulary

Animation: Shortest Distance From a Point to a Line

Distance from a Point to a Line
Draw the segment that represents the distance from A to Answer: Since the distance from a line to a point not on the line is the length of the segment perpendicular to the line from the point, extend BP and draw AT so that AT ___ Lesson 3-6 Example 1

Which segment in the diagram represents the distance from R to
RY RX MX RM ___ A B C D Lesson 3-6 CYP 1

Construct a Perpendicular Segment
Construct a line perpendicular to line s through V(1, 5) not on s. Then find the distance from V to s. Lesson 3-6 Example 2

Graph line s and point V. Place the compass point at point V. Make the setting wide enough so that when an arc is drawn, it intersects s in two places. Label these points of intersection A and B. Lesson 3-6 Example 2

Put the compass at point A and draw an arc below line s. Lesson 3-6 Example 2

Using the same compass setting, put the compass at point B and draw an arc to intersect the one drawn in step 2. Label the point of intersection Q. Lesson 3-6 Example 2

Label point R at the intersection of and s. Use the slopes of and s to verify that the lines are perpendicular. Lesson 3-6 Example 2

The segment constructed from point V(1, 5) perpendicular to the line s, appears to intersect line s at R(–2, 2). Use the Distance Formula to find the distance between point V and line s. Answer: Lesson 3-6 Example 2

Construct a line perpendicular to line m through Q(–4, –1) not on m
Construct a line perpendicular to line m through Q(–4, –1) not on m. Then find the distance from Q to m. A B C D 2.83 units 3.54 units 2.5 units 5 units Lesson 3-6 CYP 2

Theorem 3.9

Distance Between Lines
Find the distance between the parallel lines a and b whose equations are y = 2x + 3 and y = 2x –3, respectively. You will need to solve a system of equations to find the endpoints of a segment that is perpendicular to both a and b. The slope of lines a and b is 2. Lesson 3-6 Example 3

First, write an equation of a line p perpendicular to a and b. The slope of p is the opposite reciprocal of Use the y-intercept of line a, (0, 3), as one of the endpoints of the perpendicular segment. Point-slope form Simplify. Add 3 to each side. Lesson 3-6 Example 3

Next, use a system of equations to determine the point of intersection of line b and p. Substitute 2x – 3 for y in the second equation. Lesson 3-6 Example 3

Group like terms on each side. Simplify on each side. Divide each side by . Substitute 2.4 for x in the equation for p. The point of intersection is (2.4, 1.8). Lesson 3-6 Example 3

Then, use the Distance Formula to determine the distance between (0, 3) and (2.4, 1.8). Distance Formula x2 = 2.4, x1 = 0, y2 = 1.8, y1 = 3 Answer: The distance between the lines is or about 2.7 units. Lesson 3-6 Example 3

Find the distance between the parallel lines a and b whose equations are and respectively.
2.13 units 3.16 units 2.85 units 3 units Lesson 3-6 CYP 3

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Lesson 3-1 Parallel Lines and Transversals

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Lesson 3-1 Parallel Lines and Transversals

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