Lines, Angles, and Triangles

1.1   Historical Background of Geometry

The word geometry is derived from the Greek words geos (meaning earth) and metron (meaning measure). The ancient Egyptians, Chinese, Babylonians, Romans, and Greeks used geometry for surveying, navigation, astronomy, and other practical occupations.

The Greeks sought to systematize the geometric facts they knew by establishing logical reasons for them and relationships among them. The work of men such as Thales (600 B.C.), Pythagoras (540 B.C.), Plato (390 B.C.), and Aristotle (350 B.C.) in systematizing geometric facts and principles culminated in the geometry text Elements, written in approximately 325 B.C. by Euclid. This most remarkable text has been in use for over 2000 years.

1.2   Undefined Terms of Geometry: Point, Line, and Plane

1.2A   Point, Line, and Plane are Undefined Terms

These undefined terms underlie the definitions of all geometric terms. They can be given meanings by way of descriptions. However, these descriptions, which follow, are not to be thought of as definitions.

1.2B   Point

A point has position only. It has no length, width, or thickness.

A point is represented by a dot. Keep in mind, however, that the dot represents a point but is not a point, just as a dot on a map may represent a locality but is not the locality. A dot, unlike a point, has size.

A point is designated by a capital letter next to the dot, thus point A is represented: A.

1.2C   Line

A line has length but has no width or thickness.

A line may be represented by the path of a piece of chalk on the blackboard or by a stretched rubber band.

A line is designated by the capital letters of any two of its points or by a small letter, thus:

A line may be straight, curved, or a combination of these. To understand how lines differ, think of a line as being generated by a moving point. A straight line, such as is generated by a point moving always in the same direction. A curved line, such as , is generated by a point moving in a continuously changing direction.

Two lines intersect in a point.

A straight line is unlimited in extent. It may be extended in either direction indefinitely.

A ray is the part of a straight line beginning at a given point and extending limitlessly in one direction:

and designate rays.

In this book, the word line will mean straight line unless otherwise stated.

1.2D   Surface

A surface has length and width but no thickness. It may be represented by a blackboard, a side of a box, or the outside of a sphere; remember, however, that these are representations of a surface but are not surfaces.

A plane surface (or plane) is a surface such that a straight line connecting any two of its points lies entirely in it. A plane is a flat surface.

Plane geometry is the geometry of plane figures—those that may be drawn on a plane. Unless otherwise stated, the word figure will mean plane figure in this book.

SOLVED PROBLEMS

1.1   Illustrating undefined terms

Point, line, and plane are undefined terms. State which of these terms is illustrated by (a) the top of a desk; (b) a projection screen; (c) a ruler’s edge; (d) a stretched thread; (e) the tip of a pin.

Solutions

(a)   surface; (b) surface; (c) line; (d) line; (e) point.

1.3   Line Segments

A straight line segment is the part of a straight line between two of its points, including the two points, called endpoints. It is designated by the capital letters of these points with a bar over them or by a small letter. Thus, or r represents the straight line segment between A and B.

The expression straight line segment may be shortened to line segment or to segment, if the meaning is clear. Thus, and segment AB both mean "the straight line segment AB."

1.3A   Dividing a Line Segment into Parts

If a line segment is divided into parts:

1.  The length of the whole line segment equals the sum of the lengths of its parts. Note that the length of is designated AB. A number written beside a line segment designates its length.

2.  The length of the whole line segment is greater than the length of any part.

Suppose is divided into three parts of lengths a, b, and c; thus . Then AB = a + b + c. Also, AB is greater than a; this may be written as AB > a.

If a line segment is divided into two equal parts:

1.  The point of division is the midpoint of the line segment.

2.  A line that crosses at the midpoint is said to bisect the segment.

Because AM = MB in Fig. 1-1, M is the midpoint of and bisects . Equal line segments may be shown by crossing them with the same number of strokes. Note that and are crossed with a single stroke.

Fig. 1.1

3.  If three points A, B, and C lie on a line, then we say they are collinear. If A, B, and C are collinear and AB + BC = AC, then B is between A and C (see Fig. 1-2).

Fig. 1.2

1.3B   Congruent Segments

Two line segments having the same length are said to be congruent. Thus, if AB = CD, then is congruent to , written

SOLVED PROBLEMS

1.2   Naming line segments and points

See Fig. 1-3.

Fig. 1.3

(a)  Name each line segment shown.

(b)  Name the line segments that intersect at A.

(c)  What other line segment can be drawn using points A, B, C, and D?

(d)  Name the point of intersection of and .

(e)  Name the point of intersection of , and .

Solutions

(a)   , , , , and . These segments may also be named by interchanging the letters; thus,

are also correct.

(b)  

(c)  

(d)  D

(e)  C

1.3   Finding lengths and points of line segments

See Fig. 1-4.

Fig. 1.4

(a)  State the lengths of

(b)  Name two midpoints.

(c)  Name two bisectors.

(d)  Name all congruent segments.

Solutions

1.4   Circles

A circle is the set of all points in a plane that are the same distance from the center. The symbol for circle is ; for circles, . Thus, O stands for the circle whose center is O.

The circumference of a circle is the distance around the circle. It contains 360 degrees (360°).

A radius is a segment joining the center of a circle to a point on the circle (see Fig. 1-5). From the definition of a circle, it follows that the radii of a circle are congruent. Thus, and OC of Fig. 1-5 are radii of O and

Fig. 1.5

A chord is a segment joining any two points on a circle. Thus, and are chords of O.

A diameter is a chord through the center of the circle; it is the longest chord and is twice the length of a radius. is a diameter of O.

An arc is a continuous part of a circle. The symbol for arc is , so that stands for arc AB. An arc of measure 1° is 1/360th of a circumference.

A semicircle is an arc measuring one-half of the circumference of a circle and thus contains 180°. A diameter divides a circle into two semicircles. For example, diameter cuts O of Fig. 1-5 into two semicircles.

A central angle is an angle formed by two radii. Thus, the angle between radii and is a central angle. A central angle measuring 1° cuts off an arc of 1°; thus, if the central angle between and in Fig. 1-6 is 1°, then measures 1°.

Fig. 1.6

Congruent circles are circles having congruent radii. Thus, if then O ≅ O′.

SOLVED PROBLEMS

1.4   Finding lines and arcs in a circle

In Fig. 1-7 find (a) OC and AB; (b) the number of degrees in ; (c) the number of degrees in .

Fig. 1.7

Solutions

(a)  Radius OC = radius OD = 12. Diameter AB = 24.

(b)  Since semicircle ADB contains 180°, contains 180° – 100° = 80°.

(c)  Since semicircle ACB contains 180°, contains 180° – 70° = 110°.

1.5   Angles

An angle is the figure formed by two rays with a common end point. The rays are the sides of the angle, while the end point is its vertex. The symbol for angle is ∠ or the plural is .

Thus, are the sides of the angle shown in Fig. 1-8(a), and A is its vertex.

1.5A   Naming an Angle

An angle may be named in any of the following ways:

1.  With the vertex letter, if there is only one angle having this vertex, as ∠B in Fig. 1-8(b).

2.  With a small letter or a number placed between the sides of the angle and near the vertex, as ∠a or ∠1 in Fig. 1-8(c).

3.  With three capital letters, such that the vertex letter is between two others, one from each side of the angle. In Fig. 1-8(d), ∠E may be named /DEG or /GED.

Fig. 1.8

1.5B   Measuring the Size of an Angle

The size of an angle depends on the extent to which one side of the angle must be rotated, or turned about the vertex, until it meets the other side. We choose degrees to be the unit of measure for angles. The measure of an angle is the number of degrees it contains. We will write m∠A = 60° to denote that angle A measures 608.

The protractor in Fig. 1-9 shows that ∠A measures of 60°. If were rotated about the vertex A until it met , the amount of turn would be 60°.

In using a protractor, be sure that the vertex of the angle is at the center and that one side is along the 0°–180° diameter.

The size of an angle does not depend on the lengths of the sides of the angle.

Fig. 1.9

The size of ∠B in Fig. 1-10 would not be changed if its sides and were made larger or smaller.

Fig. 1.10

No matter how large or small a clock is, the angle formed by its hands at 3 o’clock measures 908, as shown in Figs. 1-11 and 1-12.

Fig. 1.11

Fig. 1.12

Angles that measure less than 1° are usually represented as fractions or decimals. For example, one-thousandth of the way around a circle is either or 0.36°.

In some fields, such as navigation and astronomy, small angles are measured in minutes and seconds. One degree is comprised of 60 minutes, written 1° = 60′. A minute is 60 seconds, written 1′ = 60″. In this notation, one-thousandth of a circle is 21′36″ because

1.5C   Kinds of Angles

1.  Acute angle: An acute angle is an angle whose measure is less than 90°.

Thus, in Fig. 1-13 a° is less than 90°; this is symbolized as a° < 90°.

Fig. 1.13

2.  Right angle: A right angle is an angle that measures 90°.

Thus, in Fig. 1-14, m(rt. ∠A) ∠ 90°. The square corner denotes a right angle.

Fig. 1.14

3.  Obtuse angle: An obtuse angle is an angle whose measure is more than 90° and less than 180°.

Thus, in Fig. 1-15, 90° is less than b° and b° is less than 180°; this is denoted by 90° < b° < 180°.

Fig. 1.15

4.  Straight angle: A straight angle is an angle that measures 180°.

Thus, in Fig. 1-16, m(st. ∠B) = 180°. Note that the sides of a straight angle lie in the same straight line. But do not confuse a straight angle with a straight line!

Fig. 1.16

5.  Reflex angle: A reflex angle is an angle whose measure is more than 180° and less than 360°.

Thus, in Fig. 1-17, 180° is less than c° and c° is less than 360°; this is symbolized as 180° < c°< 360°.

Fig. 1.17

1.5D   Additional Angle Facts

1.  Congruent angles are angles that have the same number of degrees. In other words, if m∠A = m∠B, then ∠A ≅ ∠B.

Thus, in Fig. 1-18, rt. ∠A ≅ rt. ∠B since each measures 90°.

Fig. 1.18

2.  A line that bisects an angle divides it into two congruent parts.

Thus, in Fig. 1-19, if bisects ∠A, then ∠1 ≅ ∠2. (Congruent angles may be shown by crossing their arcs with the same number of strokes. Here the arcs of 1 and 2 are crossed by a single stroke.)

Fig. 1.19

3.  Perpendiculars are lines or rays or segments that meet at right angles.

The symbol for perpendicular is ; for perpendiculars, In Fig. 1-20, so right angles 1 and 2 are formed.

Fig. 1.20

4.  A perpendicular bisector of a given segment is perpendicular to the segment and bisects it.

In Fig. 1-21, is the bisector of ; thus, ∠1 and ∠2 are right angles and M is the midpoint of .

Fig. 1.21

SOLVED PROBLEMS

1.5   Naming an angle

Name the following angles in Fig. 1-22: (a) two obtuse angles; (b) a right angle; (c) a straight angle; (d) an acute angle at D; (e) an acute angle at B.

Fig. 1.22

Solutions

(a)  ∠ABC and ∠ADB (or ∠1). The angles may also be named by reversing the order of the letters: ∠CBA and ∠BDA.

(b)  ∠DBC

(c)  ∠ADC

(d)  ∠2 or ∠BDC

(e)  ∠3 or ∠ABD

1.6   Adding and subtracting angles

In Fig. 1-23, find (a) m∠AOC; (b) m∠BOE; (c) the measure of obtuse ∠AOE.

Fig. 1.23

Solutions

1.7   Finding parts of angles

Find (a) of the measure of a rt. ∠; (b) of the measure of a st. ∠; (c) of 31°; (d) of 70°20′.

Solutions

1.8   Finding rotations

In a half hour, what turn or rotation is made (a) by the minute hand, and (b) by the hour hand of a clock? What rotation is needed to turn (c) from north to southeast in a clockwise direction, and (d) from northwest to southwest in a counterclockwise direction (see Fig. 1-24)?

Fig. 1.24

Solutions

(a)  In 1 hour, a minute hand completes a full circle of 360°. Hence, in a half hour it turns 180°.

(b)  In 1 hour, an hour hand turns of 360° or 30°. Hence, in a half hour it turns 15°.

(c)  Add a turn of 90° from north to east and a turn of 45° from east to southeast to get 90° + 45° = 135°.

(d)  The turn from northwest to southwest is

1.9   Finding angles

Find the measure of the angle formed by the hands of the clock in Fig. 1-25, (a) at 8 o’clock; (b) at 4:30.

Fig. 1.25

Solutions

(a)  At 8 o’clock,

(b)  At 4:30,

1.10   Applying angle facts

In Fig. 1-26, (a) name two pairs of perpendicular segments; (b) find m∠a if m∠b = 42°; (c) find m∠AEB and m∠CED.

Fig. 1.26

Solutions

1.6   Triangles

A polygon is a closed plane figure bounded by straight line segments as sides. Thus, Fig. 1-27 is a polygon of five sides, called a pentagon; it is named pentagon ABCDE, using its letters in order.

Fig. 1.27

A quadrilateral is a polygon having four sides.

A triangle is a polygon having three sides. A vertex of a triangle is a point at which two of the sides meet. (Vertices is the plural of vertex.) The symbol for triangle is Δ; for triangles, .

A triangle may be named with its three letters in any order or with a Roman numeral placed inside of it. Thus, the triangle shown in Fig. 1-28 is ΔABC or ΔI; its sides are , , and ; its vertices are A, B, and C; its angles are ∠A, ∠B, and ∠C.

Fig. 1.28

1.6A   Classifying Triangles

Triangles are classified according to the equality of the lengths of their sides or according to the kind of angles they have.

Triangles According to the Equality of the Lengths of their Sides (Fig. 1-29)

1.  Scalene triangle: A scalene triangle is a triangle having no congruent sides.

Thus in scalene triangle ABC, a # b # c. The small letter used for the length of each side agrees with the capital letter of the angle opposite it. Also, # means ‘‘is not equal to.’’

2.  Isosceles triangle: An isosceles triangle is a triangle having at least two congruent sides.

Fig. 1.29

Thus in isosceles triangle ABC, a = c. These equal sides are called the legs of the isosceles triangle; the remaining side is the base, b. The angles on either side of the base are the base angles; the angle opposite the base is the vertex angle.

3.  Equilateral triangle: An equilateral triangle is a triangle having three congruent sides.

Thus in equilateral triangle ABC, a = b = c. Note that an equilateral triangle is also an isosceles triangle.

Triangles According to the Kind of Angles (Fig. 1-30)

Fig. 1.30

1.  Right triangle: A right triangle is a triangle having a right angle.

Thus in right triangle ABC, ∠C is the right angle. Side c opposite the right angle is the hypotenuse. The perpendicular sides, a and b, are the legs or arms of the right triangle.

2.  Obtuse triangle: An obtuse triangle is a triangle having an obtuse angle.

Thus in obtuse triangle DEF, ∠D is the obtuse angle.

3.  Acute triangle: An acute triangle is a triangle having three acute angles.

Thus in acute triangle HJK, ∠H, ∠J, and ∠K are acute angles.

1.6B   Special Lines in a Triangle

1.  Angle bisector of a triangle: An angle bisector of a triangle is a segment or ray that bisects an angle and extends to the opposite side.

Thus , the angle bisector of ∠B in Fig. 1-31, bisects ∠B, making ∠1 ≅ ∠2.

Fig. 1.31

2.  Median of a triangle: A median of a triangle is a segment from a vertex to the midpoint of the opposite side.

Thus , the median to , in Fig. 1-32, bisects , making AM = MC.

Fig. 1.32

3.  Perpendicular bisector of a side: A perpendicular bisector of a side of a triangle is a line that bisects and is perpendicular to a side.

Thus , the perpendicular bisector of in Fig. 1-32, bisects and is perpendicular to it.

4.  Altitude to a side of a triangle: An altitude of a triangle is a segment from a vertex perpendicular to the opposite side.

Thus , the altitude to in Fig. 1-33, is perpendicular to and forms right angles 1 and 2. Each angle bisector, median, and altitude of a triangle extends from a vertex to the opposite side.

Fig. 1.33

5.  Altitudes of obtuse triangle: In an obtuse triangle, the altitude drawn to either side of the obtuse angle falls outside the triangle.

Thus in obtuse triangle ABC (shaded) in Fig. 1-34, altitudes and fall outside the triangle. In each case, a side of the obtuse angle must be extended.

Fig. 1.34

SOLVED PROBLEMS

1.11   Naming a triangle and its parts

In Fig. 1-35, name (a) an obtuse triangle, and (b) two right triangles and the hypotenuse and legs of each. (c) In Fig. 1-36, name two isosceles triangles; also name the legs, base, and vertex angle of each.

Fig. 1.35

Fig. 1.36

Solutions

(a)  Since ∠ADB is an obtuse angle, ∠ADB or ΔII is obtuse.

(b)  Since ∠C is a right angle, ΔI and ΔABC are right triangles. In ΔI, is the hypotenuse and and are the legs. In ΔABC, AB is the hypotenuse and and are the legs.

(c)  Since AD = AE, ΔADE is an isosceles triangle. In ΔADE, and are the legs, is the base, and ∠A is the vertex angle.

Since AB 5 AC, ΔABC is an isosceles triangle. In ΔABC, and are the legs, is the base, and ∠A is the vertex angle.

1.12   Special lines in a triangle

Name the equal segments and congruent angles in Fig. 1-37, (a) if is the altitude to ; (b) if bisects ∠ACB; (c) if is the perpendicular bisector of ; (d) if is the median to .

Fig. 1.37

Solutions

(a)  Since , ∠1 ≅ ∠2.

(b)  Since bisects ∠ACB, ∠3 ≅ ∠4.

(c)  Since is the bisector of , AL = LD and ∠7 ≅ ∠8.

(d)  Since is median to , AF = FC.

1.7   Pairs of Angles

1.7A   Kinds of Pairs of Angles

1.  Adjacent angles: Adjacent angles are two angles that have the same vertex and a common side between them.

Thus, the entire angle of c° in Fig. 1-38 has been cut into two adjacent angles of a° and b°. These adjacent angles have the same vertex A, and a common side between them. Here, a° + b° = c°.

Fig. 1.38

2.  Vertical angles: Vertical angles are two nonadjacent angles formed by two intersecting lines.

Thus, ∠1 and ∠3 in Fig. 1-39 are vertical angles formed by intersecting lines and . Also, ∠2 and ∠4 are another pair of vertical angles formed by the same lines.

Fig. 1.39

3.  Complementary angles: Complementary angles are two angles whose measures total 90°.

Thus, in Fig. 1-40(a) the angles of a° and b° are adjacent complementary angles. However, in (b) the complementary angles are nonadjacent. In each case, a° + b° = 90°. Either of two complementary angles is said to be the complement of the other.

Fig. 1.40

4.  Supplementary angles: Supplementary angles are two angles whose measures total 180°.

Thus, in Fig. 1-41(a) the angles of a° and b° are adjacent supplementary angles. However, in Fig. 1-41(b) the supplementary angles are nonadjacent. In each case, a° + b° = 180°. Either of two supplementary angles is said to be the supplement of the other.

Fig. 1.41

1.7B   Principles of Pairs of Angles

PRINCIPLE 1:  If an angle of c° is cut into two adjacent angles of a° and b°, then a° + b° = c°.

Thus if a° = 25° and b° = 35° in Fig. 1-42, then c° + 25° + 35° = 60°.

Fig. 1.42

PRINCIPLE 2:  Vertical angles are congruent.

   Thus if and are straight lines in Fig. 1-43, then ∠1 ≅ ∠3 and ∠2 ≅ ∠4. Hence, if m∠1 = 40°, then m∠3 = 40°; in such a case, m∠2 = m∠4 = 140°.

Fig. 1.43

PRINCIPLE 3:  If two complementary angles contain a° and b°, then a° + b° = 90°.

Thus if angles of a° and b° are complementary and a° = 40°, then b° = 50° [Fig. 1-44(a) or (b)].

PRINCIPLE 4:  Adjacent angles are complementary if their exterior sides are perpendicular to each other.

Fig. 1.44

Thus in Fig. 1-44(a), a° and b° are complementary since their exterior sides and are perpendicular to each other.

PRINCIPLE 5:  If two supplementary angles contain a° and b°, then a° + b° = 180°.

Thus if angles of a° and b° are supplementary and a° = 140°, then b° = 40° [Fig. 1-45(a) or (b)].

PRINCIPLE 6:  Adjacent angles are supplementary if their exterior sides lie in the same straight line.

Thus in Fig. 1-45(a) a° and b° are supplementary angles since their exterior sides and lie in the same straight line .

Fig. 1.45

PRINCIPLE 7:  If supplementary angles are congruent, each of them is a right angle. (Equal supplementary angles are right angles.)

Thus if ∠1 and ∠2 in Fig. 1-46 are both congruent and supplementary, then each of them is a right angle.

Fig. 1.46

SOLVED PROBLEMS

1.13   Naming pairs of angles

(a)  In Fig. 1-47(a), name two pairs of supplementary angles.

(b)  In Fig. 1-47(b), name two pairs of complementary angles.

(c)  In Fig. 1-47(c), name two pairs of vertical angles.

Fig. 1.47

Solutions

(a)  Since their sum is 180°, the supplementary angles are (1) ∠1 and ∠BED; (2) ∠3 and ∠AEC.

(b)  Since their sum is 90°, the complementary angles are (1) ∠4 and ∠FJH; (2) ∠6 and ∠EJG.

(c)  Since and are intersecting lines, the vertical angles are (1) ∠8 and ∠10; (2) ∠9 and ∠MOK.

1.14   Finding pairs of angles

Find two angles such that:

(a)  The angles are supplementary and the larger is twice the smaller.

(b)  The angles are complementary and the larger is 20° more than the smaller.

(c) The angles are adjacent and form an angle of 120°. The larger is 20° less than three times the smaller.

(d)  The angles are vertical and complementary.

Solutions

In each solution, x is a number only. This number indicates the number of degrees contained in the angle. Hence, if x = 60, the angle measures 60°.

Fig. 1.48

1.15   Finding a pair of angles using two unknowns

For each of the following, be represented by a and b. Obtain two equations for each case, and then find the angles.

(a)  The angles are adjacent, forming an angle of 88°. One is 36° more than the other.

(b)  The angles are complementary. One is twice as large as