Angle Relationships & Similar Triangles PDF

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Angle Relationships & Similar Triangles...

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1.2 Angle Relationships and Similar Triangles

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1.2 Angle Relationships and Similar Triangles Geometric Properties

Q

Geometric Properties In Figure 13, we extended the sides of angle NMP to form another angle, RMQ. The pair of angles NMP and RMQ are called vertical angles. Another pair of vertical angles, NMQ and PMR, are formed at the same time. Vertical angles have the following important property.

R M

N

Triangles

P

Vertical Angles Vertical angles have equal measures.

Vertical angles Figure 13

q Transversal 1 3 5 7

2 4

6 8

m Parallel lines n

Parallel lines are lines that lie in the same plane and do not intersect. Figure 14 shows parallel lines m and n. When a line q intersects two parallel lines, q is called a transversal. In Figure 14, the transversal intersecting the parallel lines forms eight angles, indicated by numbers. We learn in geometry that the degree measures of angles 1 through 8 in Figure 14 possess some special properties. The following chart gives the names of these angles and rules about their measures. Name

Figure 14

Sketch

Rule q

Alternate interior angles

m

4

5

Angle measures are equal.

n (also 3 and 6)

Alternate exterior angles

Angle measures are equal.

q 1 m n 8 (also 2 and 7)

(continued)

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10 CHAPTER 1 Trigonometric Functions q

Name

Transversal 1 3 5 7

2

Parallel lines n

6 8

Rule q

Interior angles on same side of transversal

m

4

Sketch

m 6

Angle measures add to 180°.

4 n

(also 3 and 5) Figure 14 (repeated)

q

Corresponding angles 2

m

Angle measures are equal.

6

n (also 1 and 5, 3 and 7, 4 and 8)

EXAMPLE 1 Finding Angle Measures

Find the measure of each marked angle in Figure 15, given that lines m and n are parallel.

(3x + 2)

m

(5x – 40)

n

Solution

The marked angles are alternate exterior angles, which are equal.

Thus, 3x  2 苷 5x  40 Subtract 3x; add 40. (Appendix A) 42 苷 2x Divide by 2. 21 苷 x.

Figure 15

One angle has measure 3x  2 苷 3  21  2 苷 65,

Substitute 21 for x.

and the other has measure 5x  40 苷 5  21  40 苷 65. Substitute 21 for x. Now try Exercise 11.

Triangles An important property of triangles, first proved by Greek geometers, deals with the sum of the measures of the angles of any triangle. 2

Angle Sum of a Triangle 1

3

The sum of the measures of the angles of any triangle is 180°.

(a) 2 1

3 (b)

Figure 16

While it is not an actual proof, we give a rather convincing argument for the truth of this statement, using any size triangle cut from a piece of paper. Tear each corner from the triangle, as suggested in Figure 16(a). You should be able to rearrange the pieces so that the three angles form a straight angle, which has measure 180°, as shown in Figure 16( b).

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1.2 Angle Relationships and Similar Triangles

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EXAMPLE 2 Applying the Angle Sum of a Triangle Property

The measures of two of the angles of a triangle are 48° and 61°. (See the figure.) Find the measure of the third angle, x. Solution

x

48  61  x 苷 180 The sum of the angles is 180°. 109  x 苷 180 x 苷 71

48

61

Subtract 109°.

The third angle of the triangle measures 71°. Now try Exercise 15.

We classify triangles according to angles and sides, as shown in the following chart.

Types of Triangles All acute

One right angle

One obtuse angle

Acute triangle

Right triangle

Obtuse triangle

All sides equal

Two sides equal

No sides equal

Equilateral triangle

Isosceles triangle

Scalene triangle

Angles

(a)

Sides

(b)

(c) Figure 17

Now try Exercises 25, 27, and 31.

Many key ideas of trigonometry depend on similar triangles, triangles of exactly the same shape but not necessarily the same size. Figure 17 shows three pairs of similar triangles. The two triangles in Figure 17(c) not only have the same shape but also the same size. Triangles that are both the same size and the same shape are called congruent triangles. If two triangles are congruent, then it is possible to pick one of them up and place it on top of the other so that they coincide. If two triangles are congruent, then they must be similar. However, two similar triangles need not be congruent. The triangular supports for a child’s swing are congruent (and thus similar) triangles, machine-produced with exactly the same dimensions each time. These supports are just one example of similar triangles. The supports of a long bridge, all the same shape but decreasing in size toward the center of the bridge, are examples of similar (but not congruent) triangles.

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12 CHAPTER 1 Trigonometric Functions

Suppose a correspondence between two triangles ABC and DEF is set up as shown in Figure 18.

B

A

C

Angle A corresponds to angle D. Angle B corresponds to angle E. Angle C corresponds to angle F.

E

D

Side AB corresponds to side DE. Side BC corresponds to side EF. Side AC corresponds to side DF.

For triangle ABC to be similar to triangle DEF, the following conditions must hold.

F Figure 18

Conditions for Similar Triangles 1. Corresponding angles must have the same measure. 2. Corresponding sides must be proportional. (That is, their ratios must be equal.) Now try Exercise 41.

EXAMPLE 3 Finding Angle Measures in Similar Triangles N

45

In Figure 19, triangles ABC and NMP are similar. Find the measures of angles B and C.

A 104 45

M 31

P

Since the triangles are similar, corresponding angles have the same measure. Since C corresponds to P and P measures 104°, angle C also measures 104°. Since angles B and M correspond, B measures 31°.

Solution

B

C Figure 19

Now try Exercise 47.

The small arcs found at the angles in Figures 17–19 denote the corresponding angles in the triangles. NOT E

EXAMPLE 4 Finding Side Lengths in Similar Triangles

Given that triangle ABC and triangle DFE in Figure 20 are similar, find the lengths of the unknown sides of triangle DFE.

C E

32 16 24

B Figure 20

As mentioned before, similar triangles have corresponding sides in proportion. Use this fact to find the unknown side lengths in triangle DFE. Side DF of triangle DFE corresponds to side AB of triangle ABC, and sides DE and AC correspond. This leads to the proportion Solution

8 A

D

F

DF 8 苷 . 24 16 Recall the cross-multiplication property of proportions from algebra. If

c a 苷 , d b

then

ad 苷 bc.

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1.2 Angle Relationships and Similar Triangles

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We use this property to solve the equation for DF. DF 8 苷 24 16 8  24 苷 16  DF Cross multiply. 192 苷 16  DF Multiply. Divide by 16. 12 苷 DF Side DF has length 12. Side EF corresponds to CB. This leads to another proportion. EF 8 苷 32 16 8  32 苷 16  EF 16 苷 EF

Cross multiply. Solve for EF.

Side EF has length 16. Now try Exercise 53.

EXAMPLE 5 Finding the Height of a Flagpole

Firefighters at the Arcade Fire Station need to measure the height of the station flagpole. They find that at the instant when the shadow of the station is 18 ft long, the shadow of the flagpole is 99 ft long. The station is 10 ft high. Find the height of the flagpole. Solution Figure 21 shows the information given in the problem. The two triangles are similar, so corresponding sides are in proportion.

N 10 18

MN 99 苷 18 10 MN 11 苷 Lowest terms 10 2 2  MN 苷 110 Cross multiply. MN 苷 55 Divide by 2.

M

99 Figure 21

The flagpole is 55 ft high. Now try Exercise 57.

EXAMPLE 6 Determining When a Solar Eclipse Can Occur Sun Moon

Not to scale

Umbra

Figure 22

Earth

The sun has a diameter of about 865,000 mi with a maximum distance from Earth’s surface of about 94,500,000 mi. The moon has a smaller diameter of 2159 mi. For a total solar eclipse to occur, the moon must pass between Earth and the sun. The moon must also be close enough to Earth for the moon’s umbra (shadow) to reach the surface of Earth. See Figure 22. (Source: Karttunen, H., P. Kröger, H. Oja, M. Putannen, and K. Donners, Editors, Fundamental Astronomy, Fourth Edition, Springer-Verlag, 2003.)

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14 CHAPTER 1 Trigonometric Functions

(a) Calculate the maximum distance that the moon can be from Earth and still have a total solar eclipse occur. (b) In a recent year, the closest approach of the moon to Earth’s surface was 225,745 mi and the farthest was 251,978 mi. (Source: The World Almanac and Book of Facts.) Can a total solar eclipse occur every time the moon is between Earth and the sun? Explain. Solution Sun Moon

Not to scale

Umbra

Figure 22 (repeated)

Earth

(a) Let Ds be the Earth-sun distance, ds the diameter of the sun,Dm the Earthmoon distance, and dm the diameter of the moon. ds Ds 苷 Similar triangles Dm dm D s d m 94,500,000  2159 苷 ⬇ 236,000 mi Dm 苷 ds 865,000 (b) No, a total solar eclipse cannot occur every time. The moon must be less than 236,000 mi away from Earth for an eclipse to occur, and sometimes it is farther than this. Now try Exercise 73.

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20 CHAPTER 1 Trigonometric Functions

1.3 Trigonometric Functions Trigonometric Functions y

P(x, y) r y ␪ Q

x

O

Figure 23

x

Quadrantal Angles

Trigonometric Functions The study of trigonometry covers the six trigonometric functions defined here. To define these functions, we start with an angle  in standard position, and choose any point P having coordinates共x, y兲 on the terminal side of angle . (The point P must not be the vertex of the angle.) See Figure 23. A perpendicular from P to the x-axis at point Q determines a right triangle, having vertices at O, P, and Q. We find the distance r fromP共x, y兲 to the origin, 共0, 0兲, using the distance formula. r 苷 兹共x  0兲2  共y  0兲2 苷 兹x 2  y 2

(Appendix B)

Notice that r  0 since distance is never negative.

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1.3 Trigonometric Functions

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The six trigonometric functions of angle  are sine, cosine, tangent, cotangent, secant, and cosecant. In the following definitions, we use the customary abbreviations for the names of these functions.

Trigonometric Functions Let 共x, y兲 be a point other than the origin on the terminal side of an angle in standard position. The distance from the point to the origin is r 苷 兹x 2  y 2. The six trigonometric functions of  are defined as follows. sin ␪ ⴝ

y r

csc ␪ ⴝ

r y

共 y ⴝ 0兲

cos ␪ ⴝ

x r

sec ␪ ⴝ

r x

y x x cot ␪ ⴝ y tan ␪ ⴝ

共x ⴝ 0兲

共x ⴝ 0兲 共 y ⴝ 0兲

EXAMPLE 1 Finding Function Values of an Angle

The terminal side of an angle  in standard position passes through the point 共8, 15兲. Find the values of the six trigonometric functions of angle . Solution Figure 24 shows angle  and the triangle formed by dropping a perpendicular from the point 共8, 15兲 to the x-axis. The point 共8, 15兲 is 8 units to the right of the y-axis and 15 units above the x-axis, so x 苷 8 and y 苷 15. Since r 苷 兹x 2  y 2,

y

(8, 15) 17

15

x= 8 y = 15 r = 17

r 苷 兹82  152 苷 兹64  225 苷 兹289 苷 17. We can now find the values of the six trigonometric functions of angle.

␪ 0

y 15 苷 r 17 r 17 csc  苷 苷 y 15

x

8

sin  苷

Figure 24

x 8 苷 r 17 r 17 sec  苷 苷 x 8 cos  苷

y 15 苷 x 8 x 8 cot  苷 苷 y 15

tan  苷

Now try Exercise 9.

EXAMPLE 2 Finding Function Values of an Angle

The terminal side of an angle  in standard position passes through the point 共3, 4兲. Find the values of the six trigonometric functions of angle  .

y

␪ –3 0 –4

5

x = –3 y = –4 r = 5

Solution

r 苷 兹共3兲2  共4兲2 苷 兹25 苷 5. Remember that r  0.

x

Then use the definitions of the trigonometric functions. sin  苷

4 4 苷 5 5

cos  苷

3 3 苷 5 5

csc  苷

5 5 苷 4 4

sec  苷

5 5 苷 3 3

(–3, –4) Figure 25

As shown in Figure 25, x 苷 3 and y 苷 4. The value of r is

4 4 苷 3 3 3 3 苷 cot  苷 4 4

tan  苷

Now try Exercise 5.

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22 CHAPTER 1 Trigonometric Functions y

We can find the six trigonometric functions using any point other than the origin on the terminal side of an angle. To see why any point may be used, refer to Figure 26, which shows an angle  and two distinct points on its terminal side. Point P has coordinates 共x, y兲, and point P (read “P-prime”) has coordinates 共x, y兲. Let r be the length of the hypotenuse of triangle OPQ, and letr be the length of the hypotenuse of triangle OPQ. Since corresponding sides of similar triangles are proportional,

(x , y ) OP = r OP = r

P

(x, y) P

␪ O

Q

x

Q

Figure 26

y y 苷 , r r

(Section 1.2)

y

so sin  苷 r is the same no matter which point is used to find it. A similar result holds for the other five trigonometric functions. We can also find the trigonometric function values of an angle if we know the equation of the line coinciding with the terminal ray. Recall from algebra that the graph of the equation

y

Ax  By 苷 0 x

0

x + 2y = 0, x

0

(Appendix B)

is a line that passes through the origin. If we restrict x to have only nonpositive or only nonnegative values, we obtain as the graph a ray with endpoint at the origin. For example, the graph of x  2y 苷 0, x  0, shown in Figure 27, is a ray that can serve as the terminal side of an angle in standard position. By choosing a point on the ray, we can find the trigonometric function values of the angle.

Figure 27

EXAMPLE 3 Finding Function Values of an Angle

Find the six trigonometric function values of the angle  in standard position, if the terminal side of  is defined by x  2y 苷 0, x  0. The angle is shown in Figure 28. We can use any point except共0, 0兲 on the terminal side of  to find the trigonometric function values. We choose x 苷 2 and find the corresponding y-value.

y

Solution

x=2 y = –1 r = 5

x  2y 苷 0, x  0

␪ x

0

x + 2y = 0, x

Figure 28

(2, –1) 0

2  2y 苷 0 2y 苷 2 y 苷 1

Let x 苷 2. Subtract 2. (Appendix A) Divide by 2.

The point 共2, 1兲 lies on the terminal side, and the corresponding value of r is r 苷 兹22  共1兲2 苷 兹5. Now we use the definitions of the trigonometric functions. 1 1 兹5 y 兹5 苷  苷 sin  苷 苷 5 Multiply the numerators and r 兹5 兹5 兹5 denominators by 兹5 to rationalize

x 2 2 兹5 2兹5 the denominators. 苷 苷  cos  苷 苷 r 5 兹5 兹5 兹5 1 y 1 tan  苷 苷 苷 x 2 2 r r x 2 兹5 兹5 苷 sec  苷 苷 2 苷 兹5 cot  苷 苷 csc  苷 苷 y x y 2 1 1 Now try Exercise 25.

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1.3 Trigonometric Functions

23

Recall that when the equation of a line is written in the formy 苷 mx  b, the coefficient of x is the slope of the line. In Example 3, x  2y 苷 0 can be written as y 苷  12 x, so the slope is 21. Notice thattan  苷  12 . In general, it is true that m ⴝ tan ␪. NOT E The trigonometric function values we found in Examples 1– 3 are exact. If we were to use a calculator to approximate these values, the decimal results would not be acceptable if exact values were required.

Quadrantal Angles If the terminal side of an angle in standard position lies along the y-axis, any point on this terminal side has x-coordinate 0. Similarly, an angle with terminal side on the x-axis has y-coordinate 0 for any point on the terminal side. Since the values of x and y appear in the denominators of some trigonometric functions, and since a fraction is undefined if its denominator is 0, some trigonometric function values of quadrantal angles (i.e., those with terminal side on an axis) are undefined. EXAMPLE 4 Finding Function Values of Quadrantal Angles

Find the values of the six trigonometric functions for each angle. (a) an angle of 90° (b) an angle  in standard position with terminal side through共3, 0兲 Solution

(a) First, we select any point on the terminal side of a 90° angle. We choose the point 共0, 1兲, as shown in Figure 29. Here x 苷 0 and y 苷 1, so r 苷 1. Then, sin 90 苷

1 苷1 1

cos 90 苷

0 苷0 1

tan 90 苷

1 0

csc 90 苷

1 苷1 1

sec 90 苷

1 0

cot 90 苷

0 苷 0. 1

共undefined兲

y

y

A calculator in degree mode returns the correct values for sin 90 and cos 90 . The second screen shows an ERROR message for tan 90 , because 90 is not in the domain of the tangent function.

共undefined兲

(0, 1) 90 0

x

(–3, 0)

Figure 29

x

0

Figure 30

(b) Figure 30 shows the angle. Here, x 苷 3, y 苷 0, and r 苷 3, so the trigonometric functions have the following values. sin  苷

0 苷0 3

cos  苷

3 苷 1 3

tan  苷

0 苷0 3

csc  苷

3 0

sec  苷

3 苷 1 3

cot  苷

3 共undefined兲 0

共undefined兲

...