ssrs barcode font 1.5B Measuring the Size of an Angle in Objective-C

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1.5B Measuring the Size of an Angle
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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 60 . S The protractor in Fig. 1-9 shows that /A measures of 60 . If AC were rotated about the vertex A until it S met AB , 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.
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Fig. 1-9
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The size of /B in Fig. 1-10 would not be changed if its sides AB and BC were made larger or smaller. No matter how large or small a clock is, the angle formed by its hands at 3 o clock measures 90 , as shown in Figs. 1-11 and 1-12.
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Angles that measure less than 1 are usually represented as fractions or decimals. For example, one360 thousandth of the way around a circle is either 1000 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 21 36 1296 360 notation, one-thousandth of a circle is 21 36 because 60 3600 3600 1000 .
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1.5C Kinds of Angles
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1. Acute angle: An acute angle is an angle whose measure is less than 90 .
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Thus, in Fig. 1-13 a is less than 90 ; this is symbolized as a < 90 .
CHAPTER 1 Lines, Angles, and Triangles
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.
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. 1-13
Fig. 1-14
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
Fig. 1-17
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 .
1.5D Additional Angle Facts
1. Congruent angles are angles that have the same number of degrees. In other words, if m/A /A > /B.
Thus, in Fig. 1-18, rt. /A > rt. /B since each measures 90 .
m/B, then
Fig. 1-18
Fig. 1-19
2. A line that bisects an angle divides it into two congruent parts.
Thus, in Fig. 1-19, if AD 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.)
3. Perpendiculars are lines or rays or segments that meet at right angles.
The symbol for perpendicular is ' ; for perpendiculars, $. In Fig. 1-20, CD ' AB, so right angles 1 and 2 are formed.
4. A perpendicular bisector of a given segment is perpendicular to the segment and bisects it.
In Fig. 1-21, GH is the bisector of EF; thus, /1 and /2 are right angles and M is the midpoint of EF.
Fig. 1-20
Fig. 1-21
CHAPTER 1 Lines, Angles, and Triangles
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
(a) m/AOC (b) m/BOE (c) m/AOE m/a m/b 360 m/b m/c (m/a 90 m/d m/b 35 35 m/c 125 45 50 130 360 220 140
m/d)
1.7 Finding parts of angles 1 2 2 1 Find (a) 5 of the measure of a rt. /; (b) 3 of the measure of a st. /; (c) 2 of 31 ; (d) 10 of 70 20 .
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