The ________ of a nonhorizontal line is the positive angle (less than measured counterclockwise from the axis to the line.

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If a nonvertical line has inclination $\theta$ and slope $m,$ then $m=$________.

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If two nonperpendicular lines have slopes $m_{1}$ and $m_{2},$ the angle between the two lines is $\tan \theta=$___________.

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The distance between the point $\left(x_{1}, y_{1}\right)$ and the line $A x+B y+C=0$ is given by $d=$________.

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In Exercises $5-12,$ find the slope of the line with inclination $\theta$

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In Exercises $5-12,$ find the slope of the line with inclination $\theta$

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In Exercises $5-12,$ find the slope of the line with inclination $\theta$

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In Exercises $5-12,$ find the slope of the line with inclination $\theta$

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In Exercises $5-12,$ find the slope of the line with inclination $\theta$

$$\theta=\frac{\pi}{3}\text{radians} $$

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In Exercises $5-12,$ find the slope of the line with inclination $\theta$

$$\theta=\frac{5 \pi}{6}\text{radians} $$

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In Exercises $5-12,$ find the slope of the line with inclination $\theta$

$$\theta=\text{1.27 radians} $$

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In Exercises $5-12,$ find the slope of the line with inclination $\theta$

$$\theta=\text{2.88 radians} $$

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In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$

$$m=-1$$

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In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$

$$m=-2$$

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In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$

$$m=1$$

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In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$

$$m=2$$

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In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$

$$m=\frac{3}{4}$$

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In Exercises $13-18,$ find the inclination $\theta$ (in radians and degrees) of the line with a slope of $m .$

$$m=-\frac{5}{2}$$

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In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$$(\sqrt{3}, 2),(0,1)$$

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In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$$(1,2 \sqrt{3}),(0, \sqrt{3})$$

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In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$$(-\sqrt{3},-1),(0,-2)$$

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In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$$(3, \sqrt{3}),(6,-2 \sqrt{3})$$

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In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$$(6,1),(10,8)$$

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In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$$(12,8),(-4,-3)$$

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In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$$(-2,20),(10,0)$$

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In Exercises $19-26,$ find the inclination $\theta$ (in radians and degrees) of the line passing through the points.

$$(0,100),(50,0)$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$2 x+2 y-5=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$x-\sqrt{3} y+1=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$3 x-3 y+1=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$\sqrt{3} x-y+2=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$x+\sqrt{3} y+2=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$-2 \sqrt{3} x-2 y=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$6 x-2 y+8=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$4 x+5 y-9=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$5 x+3 y=0$$

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In Exercises $27-36,$ find the inclination $\theta$ (in radians and degrees) of the line.

$$2 x-6 y-12=0$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{aligned} 3 x+y &=3 \\ x-y &=2 \end{aligned}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{array}{l}{x+3 y=2} \\ {x-2 y=-3}\end{array}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{aligned} x-y &=0 \\ 3 x-2 y &=-1 \end{aligned}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{array}{l}{2 x-y=2} \\ {4 x+3 y=24}\end{array}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{aligned} x-2 y &=7 \\ 6 x+2 y &=5 \end{aligned}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{array}{l}{5 x+2 y=16} \\ {3 x-5 y=-1}\end{array}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{array}{l}{x+2 y=8} \\ {x-2 y=2}\end{array}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{array}{l}{3 x-5 y=3} \\ {3 x+5 y=12}\end{array}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{array}{l}{0.05 x-0.03 y=0.21} \\ {0.07 x+0.02 y=0.16}\end{array}$$

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In Exercises $37-46,$ find the angle $\theta$ (in radians and degrees) between the lines.

$$\begin{array}{l}{0.02 x-0.05 y=-0.19} \\ {0.03 x+0.04 y=0.52}\end{array}$$

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ANGLE MEASUREMENT In Exercises $47-50$ , find the slope of each side of the triangle and use the slopes to find the measures of the interior angles.

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In Exercises $51-58$ , find the distance between the point and the line.

$$\begin{array}{cc}{\text {Point}} & {\text {Line}} \\ {(0,0)} & {4 x+3 y=0}\end{array}$$

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In Exercises $51-58$ , find the distance between the point and the line.

$$\begin{array}{cc}{\text {Point}} & {\text {Line}} \\ {(0,0)} & {2 x-y=4}\end{array}$$

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In Exercises $51-58$ , find the distance between the point and the line.

$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(2,3)} & {3 x+y=1}\end{array}$$

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In Exercises $51-58$ , find the distance between the point and the line.

$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(-2,1)} & {x-y=2}\end{array}$$

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In Exercises $51-58$ , find the distance between the point and the line.

$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(-6,2)} & {x+1=0}\end{array}$$

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In Exercises $51-58$ , find the distance between the point and the line.

$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(2,1)} & {-2 x+y-2=0}\end{array}$$

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In Exercises $51-58$ , find the distance between the point and the line.

$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(0,8)} & {6 x-y=0}\end{array}$$

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In Exercises $51-58$ , find the distance between the point and the line.

$$\begin{array}{ll}{\text { Point }} & {\text { Line }} \\ {(4,2)} & {x-y=20}\end{array}$$

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In Exercises $59-62,$ the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$ , and (c) find the area of the triangle.

$$A=(0,0), B=(1,4), C=(4,0)$$

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In Exercises $59-62,$ the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$ , and (c) find the area of the triangle.

$$A=(0,0), B=(4,5), C=(5,-2)$$

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In Exercises $59-62,$ the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$ , and (c) find the area of the triangle.

$$A=\left(-\frac{1}{2}, \frac{1}{2}\right), B=(2,3), C=\left(\frac{5}{2}, 0\right)$$

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In Exercises $59-62,$ the points represent the vertices of a triangle. (a) Draw triangle $A B C$ in the coordinate plane, (b) find the altitude from vertex $B$ of the triangle to side $A C$ , and (c) find the area of the triangle.

$$A=(-4,-5), B=(3,10), C=(6,12)$$

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In Exercises 63 and $64,$ find the distance between the parallel lines.

$$\begin{array}{l}{x+y=1} \\ {x+y=5}\end{array}$$

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In Exercises 63 and $64,$ find the distance between the parallel lines.

$$\begin{array}{l}{3 x-4 y=1} \\ {3 x-4 y=10}\end{array}$$

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ROAD GRADE A straight road rises with an inclination of 0.10 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road.

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ROAD GRADE A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

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PITCH OF A ROOF A roof has a rise of 3 feet for every horizontal change of 5 feet (see figure). Find the inclination of the roof.

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CONVEYOR DESIGN A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel.

(a) Draw a diagram that gives a visual representation of the problem.

(b) Find the inclination of the conveyor.

(c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor.

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The Falls Incline Railway in Niagara Falls, Ontario, Canada is an inclined railway that was designed to carry people from the City of Niagara Falls to Queen Victoria Park. The railway is approximately 170 feet long with a 36% uphill grade (see figure).

(a) Find the inclination $\theta$ of the railway.

(b) Find the change in elevation from the base to the top of the railway.

(c) Using the origin of a rectangular coordinate system as the base of the inclined plane, find the equation of the line that models the railway track.

(d) Sketch a graph of the equation you found in part (c).

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TRUE OR FALSE? In Exercises 71 and $72,$ determine whether the statement is true or false. Justify your answer.

A line that has an inclination greater than $\pi / 2$ radians has a negative slope.

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TRUE OR FALSE? In Exercises 71 and $72,$ determine whether the statement is true or false. Justify your answer.

To find the angle between two lines whose angles of inclination $\theta_{1}$ and $\theta_{2}$ are known, substitute $\theta_{1}$ and $\theta_{2}$ for $m_{1}$ and $m_{2},$ respectively, in the formula for the angle between two lines.

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Consider a line with slope $m$ and $y$ -intercept $(0,4)$ .

(a) Write the distance $d$ between the origin and the line as a function of $m .$

(b) Graph the function in part (a).

(c) Find the slope that yields the maximum distance between the origin and the line.

(d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

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CAPSTONE Discuss why the inclination of a line can be an angle that is larger than $\pi / 2,$ but the angle between two lines cannot be larger than $\pi / 2 .$ Decide whether the following statement is true or false: "The inclination of a line is the angle between the line and the $x$ -axis." Explain.

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Consider a line with slope $m$ and $y$ -intercept $(0,4)$ .

(a) Write the distance $d$ between the point $(3,1)$ and the line as a function of $m .$

(b) Graph the function in part (a).

(c) Find the slope that yields the maximum distance between the point and the line.

(d) Is it possible for the distance to be 0 ? If so, what is the slope of the line that yields a distance of 0$?$

(e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

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