1. LESSON 3–3
Slopes of Lines

2. Five-Minute Check (over Lesson 3–2) TEKS Then/Now New Vocabulary
Key Concept: Slope of a Line
Example 1: Find the Slope of a Line
Concept Summary: Classifying Slopes
Example 2: Real-World Example: Use Slope as Rate of Change
Postulates: Parallel and Perpendicular Lines
Example 3: Determine Line Relationships
Example 4: Use Slope to Graph a Line
Lesson Menu

3. In the figure, m4 = 146. Find the measure of 2.
B. 34
C. 146
D. 156
5-Minute Check 1

4. In the figure, m4 = 146. Find the measure of 7.
B. 34
C. 146
D. 156
5-Minute Check 2

5. In the figure, m4 = 146. Find the measure of 10.
B. 146
C. 56
D. 34
5-Minute Check 3

6. In the figure, m4 = 146. Find the measure of 11.
B. 160
C. 52
D. 34
5-Minute Check 4

7. Find m11 + m6.
A. 180
B. 146
C. 68
D. 34
5-Minute Check 5

8. In the map shown, 5th Street and 7th Street are parallel
In the map shown, 5th Street and 7th Street are parallel. At what acute angle do Strait Street and Oak Avenue meet?
A. 76
B. 75
C. 53
D. 52
5-Minute Check 6

9. Mathematical Processes G.1(A), G.1(G)
Targeted TEKS
G.2(B) Derive and use the distance, slope, and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.
Mathematical Processes
G.1(A), G.1(G)
TEKS

10. Use slope to identify parallel and perpendicular lines.
You used the properties of parallel lines to determine congruent angles.
Find slopes of lines.
Use slope to identify parallel and perpendicular lines.
Then/Now

11. slope
rate of change
Vocabulary

12. Concept

13. A. Find the slope of the line.
Find the Slope of a Line
A. Find the slope of the line.
Substitute (–3, 7) for (x1, y1) and (–1, –1) for (x2, y2).
Slope formula
Substitution
Simplify.
Answer: –4
Example 1

14. B. Find the slope of the line.
Find the Slope of a Line
B. Find the slope of the line.
Substitute (–2, –5) for (x1, y1) and (6, 2) for (x2, y2).
Slope formula
Substitution
Simplify.
Answer:
Example 1

15. C. Find the slope of the line.
Find the Slope of a Line
C. Find the slope of the line.
Substitute (0, 4) for (x1, y1) and (0, –3) for (x2, y2).
Slope formula
Substitution
Simplify.
Answer: The slope is undefined.
Example 1

16. D. Find the slope of the line.
Find the Slope of a Line
D. Find the slope of the line.
Substitute (–2, –1) for (x1, y1) and (6, –1) for (x2, y2).
Slope formula
Substitution
Simplify.
Answer: 0
Example 1

17. A. Find the slope of the line.B. C. D.
Example 1a

18. B. Find the slope of the line.
A. 0
B. undefined
C. 7 D.
Example 1b

19. C. Find the slope of the line.A. B.
C. –2
D. 2
Example 1c

20. D. Find the slope of the line.
A. 0
B. undefined
C. 3 D.
Example 1d

21. Concept

22. You want to find the sales in 2015.
Use Slope as Rate of Change
RECREATION In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the annual sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2015?
Analyze
Use the data given to graph the line that models the annual sales y as a function of the years x since The sales increase is constant. Plot the points (0, 48.9) and (5, 85.9) and draw a line through them.
You want to find the sales in 2015.
Example 2

23. Use the slope formula to find the slope of the line.
Use Slope as Rate of Change
Formulate
Find the slope of the line. Use this rate of change to find the amount of sales in 2015.
Determine
Use the slope formula to find the slope of the line.
The sales increased at an average of $7.4 million per year.
Example 2

24. Use Slope as Rate of Change
Use the slope of the line and one known point on the line to calculate the sales y when the years x since 2000 is 15.
Slope formula
m = 7.4, x1 = 0, y1 = 48.9, x2 = 15
Simplify.
Multiply each side by 15.
Add 48.9 to each side.
Example 2

25. Answer: Thus, the sales in 2015 will be about $159.9 million.
Use Slope as Rate of Change
Answer: Thus, the sales in 2015 will be about $ million.
Justify From the graph we can estimate that in 2015, the sales will be a little more than $150 million.
Evalulate Since is close to 150, our answer is reasonable. 
Example 2

26. 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. about million
B. about million
C. about million
D. about million
Example 2

27. Concept

28. Step 1 Find the slopes of and .
Determine Line Relationships
Determine whether and are parallel, perpendicular, or neither for F(1, –3), G(–2, –1), H(5, 0), and J(6, 3). Graph each line to verify your answer.
Step 1 Find the slopes of and
Example 3

29. Step 2 Determine the relationship, if any, between the lines.
Determine Line Relationships
Step 2 Determine the relationship, if any, between the lines.
The slopes are not the same, so and are not parallel. The product of the slopes is So, and are not
perpendicular.
Example 3

30. Answer: The lines are neither parallel nor perpendicular.
Determine Line Relationships
Answer: The lines are neither parallel nor perpendicular.
Check When graphed, you can see that the lines are not parallel and do not intersect in right angles.
Example 3

31. Determine whether AB and CD are parallel, perpendicular, or neither for A(–2, –1), B(4, 5), C(6, 1), and D(9, –2)
A. parallel
B. perpendicular
C. neither
Example 3

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

33. The slopes of two parallel lines are the same.
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 up 3 units and then move left 4 units.
Label the point R.
Draw
Example 4

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

35. LESSON 3–3
Slopes of Lines