Well, you found things you can measure: a line segment or an angle. They don’t go on forever.
You can measure them and actually stop somewhere. So how do you do it?
Measuring just means assigning a number to something to give an indication of its size. The number depends on the ruler you’re using. Feet? Inches? Centimeters? Furlongs? They all measure length (although you don’t see furlongs used much outside of horse racing).
A ruler is just a line or line segment that you’ve broken up into smaller segments, all the same size, and numbered. You could even use a number line, and many times we will.
If you place a ruler next to a line segment, each endpoint of the segment will line up with some number on the ruler (even if it’s one of the little fraction lines in between the whole numbers).
The numbers that correspond to the endpoints are called coordinates, and the length of the line segment is the difference between the coordinates. Technically, the length is the absolute value of the difference, because direction doesn’t matter.
DEFINITION
A ruler is a line or segment divided into sections of equal size, labeled with numbers, called coordinates, used to measure the length of a line segment.
A number line like this can be used to measure line segments. For example, the length of line segment AB is equal to the distance between coordinates -7 and -2, or five units.
You can also measure angles. Angles are measured by the amount of rotation from one side to the other. Picture the hands of a clock rotating, creating angles of different sizes. It is important to remember that the lengths of the sides have no effect on the measurement of the angle. The hands of the famous clock known as Big Ben are much longer than the hands of your wrist watch or alarm clock, but they all make the same angle at 9 o’clock.
So how do you put a ruler on an angle? For starters, it’s not a ruler. A ruler is a line you use to measure parts of lines. Angles aren’t parts of lines. They’re more like wedges from a circle. So to measure them you create an instrument called a protractor, a circle broken into 360 little sections, each called a degree.
In geometry, angles are measured in degrees. When you put the protractor over the angle with the center of the circle on the vertex of the angle, the sides fall on numbers, called coordinates.
The measure of the angle is the absolute value of the difference of the coordinates.
DEFINITION
A protractor is a circle whose circumference is divided into 360 units, called degrees, which is used to measure angles.
A protractor can be created using any circle, but most people are familiar with the plastic half-circle tool shown here.
-10 -9 -8
A
-7 -6 -5 -4 -3 -2 -1 0
x
1 2 3 4 5 6 7 8 9 10
B C D E F
Z
W
X Y
When two segments have the same length, they are called congruent segments. In symbols, you could write AB~ⴝXY to say that the segment connecting A to B is the same length as the segment connecting X to Y. You could also write AB = XY to say the measurements—the lengths—are the same. With the little segment above the letters, you’re talking about the segment. Without it, you’re talking about the length, a number. Segments are congruent. Lengths are equal.
DEFINITION
Two segments are congruent if they are the same length. Two angles are congruent if they have the same measure.
The same is true of angles and their measures. The symbol A refers to the actual angle, and the symbol mA denotes the measure of that angle. If you write XYZⴝ~ RST, you’re saying the two angles have the same measure. You could also write m XYZ 5m RST. Angles are congruent;
measures are equal.
A full rotation all the way around the circle is 360r. Half of that, or 180r, is the measure of a straight angle. The straight angle takes its name from the fact that it looks like a line.
An angle of 90r, or a quarter rotation, is called a right angle. If one side of a right angle is on the floor, the other side stands upright. Angles between 0r and 90r are called acute angles.
Angles whose measurement in greater than 90r but less that 180r are obtuse angles.
DEFINITION
A straight angle is an angle that measures 180r. A right angle is an angle that measures 90r.
An angle that measures less than 90r is an acute angle. An obtuse angle is an angle that measures more than 90r but less than 180r.
You can classify angles one by one, according to their size, but you can also label angles based on their relationship to one another. Sometimes the relationship is about position or location or what the angles look like. Other times it’s just about measurements.
Two angles whose measurements total to 90r are called complementary angles. If two angles are complementary, each is the complement of the other.
The complement of an angle of 25r can be found by subtracting the known angle, 25r, from 90r.
90r – 25r = 65r, so an angle of 25r and an angle of 65r are complementary. To find the measure of the complement of an angle of 12r, subtract 90r – 12r = 78r. An angle of 12r and an angle of 78r are complementary angles.
Two angles whose measurements total to 180r are called supplementary angles. If two angles are supplementary, each is the supplement of the other.
DEFINITION
Complementary angles are a pair of angles whose measurements total 90r.
Supplementary angles are a pair of angles whose measurements total 180r.
To find the supplement of an angle of 132r, 180r – 132r = 48r. The measure of the supplement of an angle of 103r is 180 – 103 = 77r.
When two lines intersect, the lines make an X and four angles are formed. Each pair of angles across the X from one another is a pair of vertical angles. Vertical angles are always congruent; they always have the same measurement.
The angles AED and CEB are vertical angles, as are the angles DEC and AEB.
Two angles that have the same vertex and share a side but don’t overlap are called adjacent angles.
Two adjacent angles whose exterior sides (the ones they don’t share) make a line are called a linear pair. Linear pairs are always supplementary.
DEFINITION
Vertical angles are a pair of angles both of which have their vertices at the point where two lines intersect and do not share a side.
Adjacent angles are a pair of angles that have the same vertex and share a side but do not overlap one another.
A linear pair is made up of two adjacent angles whose unshared sides form a straight angle.
D
E
B
C
A
Angles RPQ and SPQ are adjacent. Angles ABD and CBD are both adjacent and linear.
P
R
Q
S
A
C B
D
CHECK POINT
6. If mX = 174r, then X is a(n) ______ angle.
7. If mT = 38r, then T is a(n) ______.
8. If X and Y are supplementary, and mX = 174r, then mY = ______.
9. If R and T are complementary, and mT = 38r, then mR = ______.
10. Lines PA and RT intersect at point Y. If mPYR = 51r, then mRYA = ______
and mTYA = ______.