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I can find the slope of a line.

I can use slopes to identify 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.

G.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).

CN #5 Slopes of Lines Objectives

rise run slope

Vocabulary

The slope of a line in a coordinate plane is a number that describes the steepness of the line. Any two points on a line can be used to determine the slope.

A fraction with zero in the denominator is undefined because it is impossible to divide by zero.

Remember!

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One interpretation of slope is a rate of change. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour.

Example 2: Transportation Application Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line.

Use the points (4, 260) and (7, 455) to graph the line and find the slope.

Example 2 Continued

The slope is 65, which means Justin is traveling at an average of 65 miles per hour.

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If a line has a slope of , then the slope of a

perpendicular line is .

The ratios and are called opposite reciprocals.

Their product is -1.

Four given points do not always determine two lines.

Graph the lines to make sure the points are not collinear.

Caution!

Example 3A: Determining Whether Lines Are Parallel, Perpendicular, or Neither

Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.

UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3)

The products of the slopes is –1, so the lines are perpendicular.

Example 3B: Determining Whether Lines Are Parallel, Perpendicular, or Neither

Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.

GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4)

The slopes are not the same, so the lines are not parallel. The product of the slopes is not –1, so the lines are not perpendicular.

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Example 3C: Determining Whether Lines Are Parallel, Perpendicular, or Neither

Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.

CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3)

The lines have the same slope, so they are parallel.