Triangles

Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Московский государственный индустриальный университет

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

basic mathematics for college students.pdf

Скачиваний:

147

Добавлен:

10.06.2015

Размер:

15.55 Mб

Скачать

☆

<<< < Предыдущая 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 5859 / 6759 60 61 62 63 64 65 66 67 > Следующая >>>

Strategy

EXAMPLE 1 a. a triangle

736	Chapter 9 An Introduction to Geometry

Objectives

1Classify polygons.

2Classify triangles.

3Identify isosceles triangles.

4Find unknown angle measures of triangles.

S E C T I O N 9.3

A point where two sides of a polygon intersect is

A point where two sides of a polygon intersect is

of the polygon.

polygon has sides that are all the same

length and angles that all have the same measure.

We will now discuss geometric ﬁgures called polygons. We see these shapes every day. For example, the walls of most buildings are rectangular in shape. Some tile and vinyl ﬂoor patterns use the shape of a pentagon or a hexagon. Stop signs are in the shape of an octagon.

In this section, we will focus on one speciﬁc type of polygon called a triangle. Triangular shapes are especially important because triangles contribute strength and stability to walls and towers. The gable roofs of houses are triangular, as are the sides of many ramps.

1 Classify polygons.

Polygon

A polygon is a closed geometric ﬁgure with at least three line segments for its sides.

Polygons are formed by ﬁtting together line segments in such a way that

The House of the Seven

Gables, Salem, Massachusetts

•no two of the segments intersect, except at their endpoints, and

•no two line segments with a common endpoint lie on the same line.

The line segments that form a polygon

are called its sides. The point where two sides		Vertex	Side
		Side		Vertex
of a polygon intersect is called a vertex of the
				Side
polygon (plural vertices). The polygon shown	Vertex

to the right has 5 sides and 5 vertices.		Side		Vertex
Polygons are classiﬁed according to the			Side
		Vertex
number of sides that they have. For example,

in the ﬁgure below, we see that a polygon with four sides is called a quadrilateral, and a polygon with eight sides is called an octagon. If a polygon has sides that are all the same length and angles that are the same measure, we call it a regular polygon.

Triangle	Quadrilateral	Pentagon	Hexagon	Heptagon	Octagon	Nonagon	Decagon	Dodecagon
3 sides	4 sides	5 sides	6 sides	7 sides	8 sides	9 sides	10 sides	12 sides

Polygons

Regular polygons

Self Check 1

Give the number of vertices of:

a.a quadrilateral

b.a pentagon

Now Try Problems 25 and 27

Give the number of vertices of: b. a hexagon

We will determine the number of angles that each polygon has.

WHY The number of its vertices is equal to the number of its angles.

9.3 Triangles

737

Solution

a.From the ﬁgure on the previous page, we see that a triangle has three angles and therefore three vertices.

b.From the ﬁgure on the previous page, we see that a hexagon has six angles and

therefore six vertices.

Success Tip From the results of Example 1, we see that the number of vertices of a polygon is equal to the number of its sides.

2 Classify triangles.

A triangle is a polygon with three sides (and three vertices). Recall that in geometry points are labeled with capital letters. We can use the capital letters that denote the vertices of a triangle to name the triangle. For example, when referring to the triangle in the right margin, with vertices A, B, and C, we can use the notation ABC (read as “triangle ABC”).

The Language of Mathematics When naming a triangle,

we may begin with any vertex. Then we move around the	C
we may begin with any vertex. Then we move around the
ﬁgure in a clockwise (or counterclockwise) direction as we
list the remaining vertices. Other ways of naming the triangle A	B
shown here are ACB, BCA, BAC, CAB, and CBA.

The Language of Mathematics The ﬁgures below show how triangles can be classiﬁed according to the lengths of their sides. The single tick marks drawn on each side of the equilateral triangle indicate that the sides are of equal length. The double tick marks drawn on two of the sides of the isosceles triangle indicate that they have the same length. Each side of the scalene triangle has a different number of tick marks to indicate that the sides have different lengths.

Equilateral triangle	Isosceles triangle	Scalene triangle
(all sides equal length)	(at least two sides of	(no sides of equal length)
	equal length)

The Language of Mathematics Since every angle of an equilateral triangle has the same measure, an equilateral triangle is also equiangular.

The Language of Mathematics Since equilateral triangles have at least two sides of equal length, they are also isosceles. However, isosceles triangles are not necessarily equilateral.

738	Chapter 9 An Introduction to Geometry

Triangles may also be classiﬁed by their angles, as shown below.

Acute triangle	Obtuse triangle	Right triangle
(has three acute angles)	(has an obtuse angle)	(has one right angle)

Right triangles have many real-life applications. For example, in ﬁgure (a) below, we see that a right triangle is formed when a ladder leans against the wall of a building.

The longest side of a right triangle is called the hypotenuse, and the other two sides are called legs. The hypotenuse of a right triangle is always opposite the 90° (right) angle. The legs of a right triangle are adjacent to (next to) the right angle, as shown in ﬁgure (b).

Right triangles

Leg

(a)

Hypotenuse

Leg

(b)

3 Identify isosceles triangles.

In an isosceles triangle, the angles opposite the sides of equal length are called base angles, the sides of equal length form the vertex angle, and the third side is called the base. Two examples of isosceles triangles are shown below.

	Isosceles triangles
	Vertex angle
			Vertex angle
Base angle	Base angle	Base angle	Base angle
	Base		Base

We have seen that isosceles triangles have two sides of equal length.The isosceles triangle theorem states that such triangles have one other important characteristic: Their base angles are congruent.

Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

9.3 Triangles

739

The Language of Mathematics Tick marks can be used to denote the sides of a triangle that have the same length. They can also be used to indicate the angles of a triangle with the same measure. For example, we can show that the base angles of the isosceles triangle below are congruent by using single tick marks.

D is opposite FE, and E is opposite

FD. By the isosceles triangle theorem,

if m(FD) m(FE), then m( D) m( E).

D E

If a mathematical statement is written in the form if p . . . , then q . . . , we call the statement if q . . . , then p . . . its converse. The converses of some statements are true, while the converses of other statements are false. It is interesting to note that the converse of the isosceles triangle theorem is true.

Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite the angles have the same length, and the triangle is isosceles.

EXAMPLE 2 Is the triangle shown here an isosceles triangle?

Strategy We will consider the measures of the angles of the triangle.

WHY If two angles of a triangle are congruent, then the		C
WHY If two angles of a triangle are congruent, then the
sides opposite the angles have the same length, and the		80°
triangle is isosceles.		80°
triangle is isosceles.
Solution A and B have the same measure, 50°.		50° 50°
By the converse of the isosceles triangle theorem, if	A	B
By the converse of the isosceles triangle theorem, if

m( A) m( B), we know that m(BC) m(AC) and that ABC is isosceles.

4 Find unknown angle measures of triangles.

If you draw several triangles and carefully measure each angle with a protractor, you will ﬁnd that the sum of the angle measures of each triangle is 180°. Two examples are shown below.

			33°
	89°
62°	29°	37°	110°
62° + 89° + 29° = 180°		37° + 110° + 33° = 180°

Another way to show this important fact about the sum of the angle measures of a triangle is discussed in Problem 82 of the Study Set at the end of this section.

Angles of a Triangle

Self Check 2

Is the triangle shown below an isosceles triangle?

79°

23°

78°

Now Try Problems 33 and 35

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

EXAMPLE 5

740	Chapter 9 An Introduction to Geometry

Self Check 3

In the ﬁgure, ﬁnd y.

60°

Now Try Problem 37

Self Check 4

In DEF, the measure of D exceeds the measure of E by 5°, and the measure of F is three times the measure of E. Find the measure of each angle of DEF.

Now Try Problem 41

Self Check 5

If one base angle of an isosceles triangle measures 33°, what is the measure of the vertex angle?

Now Try Problem 45

	EXAMPLE 3	In the ﬁgure, ﬁnd x.
	EXAMPLE 3	In the ﬁgure, ﬁnd x.	x
Strategy We will use the fact that the sum of the angle measures			x

of any triangle is 180° to write an equation that models the			40°
			40°
situation.

WHY We can then solve the equation to ﬁnd the unknown angle measure, x.

Solution Since the sum of the angle measures of any triangle is 180°, we have

x 40° 90° 180°	The symbol indicates that the measure of the	90
x 40° 90° 180°	The symbol indicates that the measure of the	90
	angle is 90°.	40
x 130° 180°	Do the addition.	130
x 130° 180°	Do the addition.
x 50°	To isolate x, undo the addition of 130° by
	subtracting 130° from both sides.

Thus, x is 50°.

EXAMPLE 4 In the ﬁgure, ﬁnd the measure of each angle of ABC.

Strategy We will use the fact that the sum of the angle measures of any triangle is 180° to write an equation that models the situation.

					A
WHY We can then solve the equation to ﬁnd the
unknown angle measure x, and use it to evaluate the				x + 32°
expressions 2x and x 32°.
Solution			x		2x
Solution			B			C
x 32° x 2x 180°			The sum of the angle measures			7 10
x 32° x 2x 180°			The sum of the angle measures			7 10
			of any triangle is 180°.			180
			of any triangle is 180°.			32
4x 32° 180°			Combine like terms: x x 2x 4x.			32
4x 32° 180°			Combine like terms: x x 2x 4x.			148
4x 32° 32° 180° 32°			To isolate the variable term, 4x,			37
			subtract 32° from both sides.			4
			subtract 32° from both sides.			4	148
	4x 148°		Do the subtractions.			12
	4x 148°		Do the subtractions.			28
	4x	148°				28
	4x	148°	To isolate x, divide both sides by 4.			28
						0
	4	4
	4	4
	x 37°		Do the divisions. This is the measure of B.

To ﬁnd the measures of A and C, we evaluate the expressions x 32° and 2x for x 37°.

x 32° 37° 32° Substitute 37 for x.		2x 2(37°) Substitute 37 for x.
69°		74°
69°		74°

The measure of B is 37°, the measure of A is 69°, and the measure of C is 74°.

If one base angle of an isosceles triangle measures 70°, what is the measure of the vertex angle?

Strategy We will use the isosceles triangle theorem and the fact that the sum of the angle measures of any triangle is 180° to write an equation that models the situation.

WHY We can then solve the equation to ﬁnd the unknown angle measure.

EXAMPLE 6

Solution By the isosceles triangle theorem, if one of the base angles

measures 70°, so does the other. (See the ﬁgure on the right.) If we			x
let x represent the measure of the vertex angle, we have
x 70° 70° 180°	The sum of the measures of the angles of a	70°	70°
	triangle is 180°.	70°	70°
	triangle is 180°.
x 140° 180°	Combine like terms: 70° 70° 140°.
x 40°	To isolate x, undo the addition of 140° by
	subtracting 140° from both sides.

The vertex angle measures 40°.

If the vertex angle of an isosceles triangle measures 99°, what are the measures of the base angles?

Strategy We will use the fact that the base angles of an isosceles triangle have the same measure and the sum of the angle measures of any triangle is 180° to write an equation that mathematically models the situation.

WHY We can then solve the equation to ﬁnd the unknown angle measures.

Solution The base angles of an isosceles triangle		99°
have the same measure. If we let x represent the
measure of one base angle, the measure of the other	x	x
base angle is also x. (See the ﬁgure to the right.) Since	x	x

the sum of the measures of the angles of any triangle is 180°, the sum of the measures of the base angles and of the vertex angle is 180°. We can use this fact to form an equation.

x x 99° 180°					40.5
x x 99° 180°					40.5
2x 99° 180°			Combine like terms: x x 2x.	2			81.0
2x 99° 180°			Combine like terms: x x 2x.	8
				8
	2x 81°		To isolate the variable term, 2x, undo the addition		01
			of 99° by subtracting 99° from both sides.			0
	2x	81°	of 99° by subtracting 99° from both sides.		1			0
			To isolate x, undo the multiplication by 2 by		1			0
			To isolate x, undo the multiplication by 2 by		1			0
			dividing both sides by 2.					0
	2	2
	2	2
	x 40.5°

The measure of each base angle is 40.5°.

ANSWERS TO SELF CHECKS

1. a. 4 b. 5 2. no 3. 30° 4. m( D) 40°, m( E) 35°, m( F) 105° 5. 114° 6. 61.5°

	9.3 Triangles	741

Self Check 6

If the vertex angle of an isosceles triangle measures 57°, what are the measures of the base angles?

Now Try Problem 49

S E C T I O N 9.3 STUDY SET

VOCABULARY


Fill in the blanks.
1.	A	is a closed geometric
	ﬁgure with at least three line
	segments for its sides.
2.	The polygon shown to the right has
	seven		and seven vertices.

5. A triangle with three sides of equal length is called an

	triangle. An	triangle has at least
two sides of equal length. A			triangle has no
sides of equal length.

742

Chapter 9 An Introduction to Geometry

triangle has three acute angles. An

triangle has one obtuse angle. A

triangle has

one right angle.

The longest side of a right triangle is called the

. The other two sides of a right triangle are

called

The

angles of an isosceles triangle have the

same measure. The sides of equal length of an

isosceles triangle form the

angle.

9.In this section, we discussed the sum of the measures of the angles of a triangle. The word sum indicates the

operation of

10. Complete the table.

Number of Sides Name of Polygon

CONCEPTS

11. Draw an example of each type of regular polygon.

a. hexagon

b. octagon

c. quadrilateral

d. triangle

e. pentagon

f. decagon

12.Refer to the triangle below.

a.What are the names of the vertices of the triangle?

b.How many sides does the triangle have? Name them.

c.Use the vertices to name this triangle in three ways.

H J

13. Draw an example of each type of triangle.

a.	isosceles	b.	equilateral
c.	scalene	d.	obtuse
e.	right	f.	acute

14.Classify each triangle as an acute, an obtuse, or a right triangle.

a.	b.	90°
		90°

c.	d.
	91°
15. Refer to the triangle shown below.
a. What is the measure		B
of B?		B
of B?
b. What type of triangle
is it?	A	C
	A	C

c.What two line segments form the legs?

d.What line segment is the hypotenuse?

e.Which side of the triangle is the longest?

f.Which side is opposite B?

16.Fill in the blanks.

a.The sides of a right triangle that are adjacent to

the right angle are called the

b. The hypotenuse of a right triangle is the side the right angle.

17. Fill in the blanks.

a.	The	triangle theorem states that if two
	sides of a triangle are congruent, then the angles
	opposite those sides are congruent.
b.	The	of the isosceles triangle theorem
	states that if two angles of a triangle are
	congruent, then the sides opposite the angles have
	the same length, and the triangle is isosceles.

18. Refer to the given triangle.

a.What two sides are of equal length?

b. What type of triangle is

XYZ? Z

c.Name the base angles.

d.Which side is opposite X?

e.What is the vertex angle?

f.Which angle is opposite side XY?

g.Which two angles are congruent?

19.Refer to the triangle below.

a.What do we know about EF and GF?

b.What type of triangle is EFG?

E G

57° 57°

66°

20.a. Find the sum of the measures of the angles ofJKL, shown in ﬁgure (a).

b.Find the sum of the measures of the angles ofCDE, shown in ﬁgure (b).

c.What is the sum of the measures of the angles of any triangle?

		C
J		57°
		57°
53°		64° D
95°	32°	59°
95°	32°
K	L	E
K		E
(a)		(b)

NOTATION

Fill in the blanks.
21.	The symbol means	.
22.	The symbol m( A) means the		of angle A.

Refer to the triangle below.

23.What fact about the sides of ABC do the tick marks indicate?

24.What fact about the angles of ABC do the tick marks indicate?

9.3 Triangles

743

GUIDED PRACTICE

For each polygon, give the number of sides it has, give its name, and then give the number of vertices that it has. See Example 1.

25. a.		b.

26. a.

27. a.

28. a.

Classify each triangle as an equilateral triangle, an isosceles triangle, or a scalene triangle. See Objective 2.

29.	a.	3 ft	2 ft	b.
			2 ft
			4 ft	55°
			4 ft
				55°
30.	a.			b.

31.	a.	2 in.	b.		4 in.

	2 in.	2 in.		3 in.	5 in.
	2 in.	2 in.			5 in.
32.	a.	15 cm	b.		1.7 in.
					1.7 in.
	20 cm	20 cm		1.8 in.
					1.4 in.

744	Chapter 9 An Introduction to Geometry

State whether each of the triangles is an isosceles triangle.

See Example 2.

33.

78°

34.

24°

78°

45° 45°

35.	19°	18°	36.
		143°	30°
			30°

60°

Find y. See Example 3.
37.	38.	53°	y
35°	y
39.	40.	y
45°			10°
45°

The degree measures of the angles of a triangle are represented by algebraic expressions. First ﬁnd x.Then determine the measure of each angle of the triangle. See Example 4.

41. 42.

x + 20°

x + 10° x

4x – 5° x + 5°

43. 4x	44.
x	x
4x

x + 15° x + 15°

Find the measure of the vertex angle of each isosceles triangle given the following information. See Example 5.

45.The measure of one base angle is 56°.

46.The measure of one base angle is 68°.

47.The measure of one base angle is 85.5°.

48.The measure of one base angle is 4.75°.

Find the measure of one base angle of each isosceles triangle given the following information. See Example 6.

49.The measure of the vertex angle is 102°.

50.The measure of the vertex angle is 164°.

51.The measure of the vertex angle is 90.5°.

52.The measure of the vertex angle is 2.5°.

TRY IT YOURSELF

Find the measure of each vertex angle.

53.

54.

33°

76°

55. 56.

53.5°

47.5°

The measures of two angles of ABC are given. Find the measure of the third angle.

57.m( A) 30° and m( B) 60°; ﬁnd m( C).

58.m( A) 45° and m( C) 105°; ﬁnd m( B).

59.m( B) 100° and m( A) 35°; ﬁnd m( C).

60.m( B) 33° and m( C) 77°; ﬁnd m( A).

61.m( A) 25.5° and m( B) 63.8°; ﬁnd m( C).

62.m( B) 67.25° and m( C) 72.5°; ﬁnd m( A).

63.m( A) 29° and m( C) 89.5°; ﬁnd m( B).

64.m( A) 4.5° and m( B) 128°; ﬁnd m( C).

In Problems 65–68, ﬁnd x.

65. °

156

66.	x	67.
		x
	86°	75°
68.		x
	5°

69.One angle of an isosceles triangle has a measure of 39°. What are the possible measures of the other angles?

70.One angle of an isosceles triangle has a measure of 2°. What are the possible measures of the other angles?

71.Find m( C).

	73°	E
	73°
		22°
		49°
	61°
A		B C

72.Find:

a.m( MXZ)

b.m( MYN)

		Y
		49°
X	24°	44°
X		N
		M

83°

73. Find m( NOQ).

	N
79°	Q

	64°
M	O

74. Find m( S).

129°	130°
R	T

9.3 Triangles

745

APPLICATIONS

75.POLYGONS IN NATURE As seen below, a starﬁsh ﬁts the shape of a pentagon. What polygon shape do you see in each of the other objects?

a.lemon

b.chili pepper

c.apple

(a)

(b)

(c)

76.CHEMISTRY Polygons are used to represent the chemical structure of compounds. In the ﬁgure below, what types of polygons are used to represent methylprednisolone, the active ingredient in an antiinﬂammatory medication?

Methylprednisolone

	CH2OH

HO H		CO
	H	3C	OH
H3C

	H

H H

O H

CH3

77.AUTOMOBILE JACK Refer to the ﬁgure below. No matter how high the jack is raised, it always forms two isosceles triangles. Explain why.

746Chapter 9 An Introduction to Geometry

78.EASELS Refer to the ﬁgure below. What type of triangle studied in this section is used in the design of the legs of the easel?

79.POOL The rack shown below is used to set up the billiard balls when beginning a game of pool. Although it does not meet the strict deﬁnition of a polygon, the rack has a shape much like a type of triangle discussed in this section. Which type of triangle?

80.DRAFTING Among the tools used in drafting are the two clear plastic triangles shown below. Classify each according to the lengths of its sides and then according to its angle measures.

45°		30°
45°
90°	45°	90°	60°

WRITING

81.In this section, we discussed the deﬁnition of a pentagon. What is the Pentagon? Why is it named that?

82.A student cut a triangular shape out of blue construction paper and labeled the angles 1, 2, and 3, as shown in ﬁgure (a) below. Then she tore off each of the three corners and arranged them as shown in ﬁgure (b). Explain what important geometric concept this model illustrates.

		2
			2
3	1	1	3
	(a)		(b)

83.Explain why a triangle cannot have two right angles.

84.Explain why a triangle cannot have two obtuse angles.

REVIEW

85.Find 20% of 110.

86.Find 15% of 50.

87.What percent of 200 is 80?

88.20% of what number is 500?

89.Evaluate: 0.85 2(0.25)

90.FIRST AID When checking an accident victim’s pulse, a paramedic counted 13 beats during a 15-second span. How many beats would be expected in 60 seconds?

<<< < Предыдущая 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 5859 / 6759 60 61 62 63 64 65 66 67 > Следующая >>>

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]

#
10.12.2018408.58 Кб6antikrizisnoe-upravlenie.doc
#
18.12.2018108.03 Кб6ARHITYeKTURA.doc
#
10.06.20158.32 Mб302aswath_damodaran-investment_philosophies_2012.pdf
#
10.06.201582.94 Кб7audit_risk.doc
#
10.06.20154.17 Mб5BallotingApplicationDocs-SIMSUsersGuide.pdf
#
10.06.201515.55 Mб147basic mathematics for college students.pdf
#
10.06.2015901.06 Кб40Bazy_dannykh_i_znanii_UP_SHirokov_L.A._2000.pdf
#
29.03.20162.45 Mб35Bernd_Fon_Vittenburg_-_Shakh_planete_Zemlya.pdf
#
10.06.201561.93 Кб44bezopasnost.docx
#
10.06.20152.89 Mб85bibliofond_ru_604241.docx
#
10.06.20152.1 Mб30blind_watchmaker.pdf