Having looked at area as the measure of the interior space of a polygon, we now turn our attention to measuring angles. What is an angle and how do we measure it?
A line is a one-dimensional figure that extends infinitely in opposite directions. In Euclidean geometry, lines are straight, but we will encounter a geometry in which they are curved. One important trait of lines is that any pair of distinct points lie on exactly one line. For this reason, we may denote the line through points \(A\) and \(B\) as \(\overleftrightarrow{AB}\text{.}\)
A line segment is the part of a line bounded by two endpoints. Line segment \(\overline{AB}\) consists of endpoints \(A\) and \(B\) and all points on line \(\overleftrightarrow{AB}\) which lie between \(A\) and \(B\text{.}\)
Unlike a line, a line segment has finite length. We define the length of \(\overline{AB}\) to be the distance from point \(A\) to point \(B\text{.}\) We use the notation \(m(\overline{AB})\) or simply \(AB\) to denote the length of \(\overline{AB}\text{.}\)
Ray \(\overrightarrow{AB}\) consists of point \(A\text{,}\) point \(B\text{,}\) and all points on line \(\overleftrightarrow{AB}\) which are on the same side of \(A\) as point \(B\text{.}\) A ray extends infinitely in one direction, has a single endpoint, and cannot be measured.
An angle is the union of two rays with a common endpoint. In particular, \(\angle BAC\text{,}\) shown in Figure 2.2.4, consists of all points that lie on the rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\text{.}\) The common endpoint \(A\) of the rays is called the vertex of the angle and the two rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are called the sides of the angle.
To think of the measure of an angle as the space between the rays is vague and misleading. The area between the rays is infinite. Note also that the points \(B\) and \(C\) on the sides of \(\angle{BAC}\) do not play a role in defining or measuring the angle. If \(D\) is any other point on ray \(\overrightarrow{AB}\) other than \(A\text{,}\) then \(\angle{DAC}\) is the same angle as \(\angle{BAC}\) and hence has the same measure. Thus it would make no sense to use the linear distance between \(B\) and \(C\) for measuring \(\angle{BAC}\text{.}\) The sides of an angle are infinite and extend beyond what can be drawn.
Instead, we think of angle measurement in terms of rotational movement around the vertex, specifically the amount of rotation needed to move one side of the angle on top of the other.
To begin our study of angle measure, we will adopt the practice of assuming that there are 360 degrees in a full rotation about any point. The choice of 360 can be traced back to the Babylonians who were studying planetary paths around 400 B.C. and used a base 60 (instead of base 10) number system. One advantage of the number 360 is that it has a lot of factors including 2, 3, 4, 5, 6, and 8. This results in more pleasing numbers when we consider fractions of a circle.
Suppose point \(A\) lies between points \(B\) and \(C\) on line \(\overleftrightarrow{BC}\) as shown in Figure 2.2.6. In this case, we say that rays \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are opposite rays. Since \(\frac{360}{2}=180\text{,}\) a rotation of 180 degrees will take the ray \(\overrightarrow{AB}\) to the ray \(\overrightarrow{AC}\) on the opposite side of \(A\text{.}\)
Continuing, we define the measurement of angles according to what fraction of a full (or half) rotation they represent. Notice how the protractor in Figure 2.2.7 divides the semicircular arc into 180 segments, eighteen of which are labeled with numerical values. These multiples of ten are fractions of the half rotation 180 degrees, for example \(\frac{1}{18}(180)=10, \frac{2}{18}(180)=20, \dots, \frac{18}{18}(180)=180\text{.}\) To find the measure of \(\angle{DAB}\text{,}\) we place the center dot of the protractor over the vertex \(A\) of the angle and lined up ray \(\overrightarrow{AB}\) with the mark for 0 degrees. The other side of the angle \(\overrightarrow{AD}\) passes through the mark for 40 degrees.
The measure of \(\angle ABC\), written \(m\angle ABC\text{,}\) is the number of degrees of rotation about vertex \(B\) needed to move ray \(\overrightarrow{BC}\) onto \(\overrightarrow{BA}\text{.}\)
If two angles \(\angle ABC\) and \(\angle DEF\) have the same measure, we say that the angles are congruent and write \(\angle ABC\cong\angle DEF\text{.}\) Thus, \(\angle ABC\cong\angle DEF\) means \(m\angle ABC = m\angle DEF\text{.}\)
The GeoGebra app, Figure 2.2.10 is useful in completing this activity. You may also perform this activity by carefully cutting out a scalene triangle and tracing it on a sheet of paper.
Rotate triangle \(\Delta ABC\) 180 degrees about the midpoint \(D\) of side \(\overline{BC}\) (see hint). Both the original triangle and the rotated version should be visible. What do you notice?
Note that two of the points are double labeled, once for each triangle. If you rightclick on the label, you can move it to make the original label visible.
To perform this rotation in the GeoGebra app, click on the angle rotation icon in the toolbar. Next, click on the interior of the triangle to select the object to be rotated, then click on midpoint \(D\) to identify the center of the rotation. When prompted to input the angle measure, replace 45 with 180. Then click OK.
Does your claim work for all triangles or just the one in the picture? To test this in the GeoGebra applet, select the arrow in the tool menu. Then click and drag one of the vertices in the original triangle to change the shape. Do your claims still hold?
Subsection2.2.4Additivity of Angles and Definitions Associated with Angles
Like additivity of area 1.1.3 and additivity of length 1.2.1, the additivity of angle measure is intuitively obvious and will prove useful. We state this as a principle and add it to our toolbox. First, we must clarify what we mean by the interior of an angle.
A point \(D\) is said to be in the interior of \(\angle ABC\) if \(D\) and \(A\) lie on the same side of line \(\overleftrightarrow{BC}\) and \(D\) and \(C\) lie on the same side of line \(\overleftrightarrow{BA}\text{.}\) Ray \(\overrightarrow{BD}\) lies in the interior of \(\angle ABC\) if \(D\) lies in the interior of \(\angle ABC\text{.}\)
When a pair of lines intersect, two pairs of vertical angles are formed. The point where the lines intersect will be the vertex for each angle in a vertical pair and the sides of one angle will be the opposite rays of the sides of the other angle in the pair.
Ray \(\overrightarrow{FE}\) can also be written as \(\overrightarrow{FH}\) or \(\overrightarrow{FC}\text{.}\) All three names refer to an identical ray because they start at point F and extend in the same direction.
\(\angle{HEB}\) can also be written as \(\angle{HEG}\text{,}\)\(\angle{CEB}\text{,}\) or \(\angle{CEG}\text{.}\) All four names refer to the same angle at vertex E, using different points along its rays.
Several right answers; correct if the angles are vertical angles and four pairs are given. One possibility: \(\angle IEH\) and \(\angle FEJ\text{;}\)\(\angle IEB and \angle AEJ\text{;}\)\(\angle BEF\) and \(\angle HEA\text{;}\) and \(\angle IEA\) and \(\angle BEJ\text{.}\) You may choose whether to allow an angle (with multiple names) to be congruent to itself.
Point E lies on both \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CE}\text{.}\) These two lines intersect at point E, making it their common point.
Points A, B, and G lie on both \(\overrightarrow{GB}\) and \(\overrightarrow{AB}\text{.}\)\(\overrightarrow{GB}\) starts at point G and continues infinitely through point B toward point A. \(\overrightarrow{AB}\) starts at point A and continues infinitely through point B toward point G.
Line \(\overleftrightarrow{IJ}\) lies on rays \(\overrightarrow{IE}\) and \(\overrightarrow{JE}\text{.}\) Ray \(\overrightarrow{IE}\) starts at point I and continues infinitely through points E and J, spanning the line \(\overleftrightarrow{IJ}\text{.}\) Ray \(\overrightarrow{JE}\) starts at point J and continue infinitely through points E and I, also spanning the line \(\overleftrightarrow{IJ}\text{.}\)
\(\overrightarrow{IE}\) and \(\overrightarrow{JE}\) share a common point at vertex E. These two rays intersect to form the angle \(\angle IEF\text{.}\)
Segment \(\overline{BG}\) lies on rays \(\overrightarrow{GE}\) and \(\overrightarrow{BG}\text{.}\) Ray \(\overrightarrow{GE}\) starts at point G and extends infinitely through points B, E, and A. Ray \(\overrightarrow{BG}\) starts at point B and extends infinitely through point G. Because both rays travel through points B and G, they share a common line segment \(\overline{BG}\text{.}\)
Ray \(\overrightarrow{EF}\) lies on both \(\overrightarrow{CH}\) and \(\overrightarrow{EF}\text{.}\)\(\overrightarrow{CH}\) starts at point C and extends infinitely through points H, E, F, and D. \(\overrightarrow{EF}\) starts at point E and extends infinitely through points F and D. Since both rays travel through points E and F, they share a common ray \(\overrightarrow{EF}\text{.}\)
Angles \(\angle{ACD}\) and \(\angle{DCE}\) form a straight line, making them supplementary angles that add up to \(180^{\circ}\) Given that \(m(\angle{DCE})=67^{\circ}\text{,}\)\(\angle{ACD}\) is determined by subtracting \(67^{\circ}\) from \(180^{\circ}\text{:}\)\(180^{\circ}-67^{\circ}=113^{\circ}\text{.}\) Therefore, \(m(\angle{ACD})=113^{\circ}\text{.}\)
To find \(\angle{ACD}\text{,}\) use the Angle Sum Theorem, which states that a triangles interior angles add up to \(180^{\circ}\text{.}\) Given that \(m(\angle{CAD})=37^{\circ}\) and \(m(\angle{ACD})=113^{\circ}\) (from the previous problem), simply subtract these values from \(180^{\circ}\text{:}\)\(180^{\circ}-(37^{\circ}+113^{\circ})=180^{\circ}-150^{\circ}=30^{\circ}\text{.}\) Therefore, \(m(\angle{ADC})=30^{\circ}\text{.}\)
Angle \(\angle{BCE}\) is vertical to \(\angle{ACD}\text{.}\) According to the Vertical Angles Theorem, vertical angles are always congruent and equal in measure. Since \(m(\angle{ACD})=113^{\circ}\text{,}\) then \(m(\angle{BCE})=113^{\circ}\text{.}\)
Angles \(\angle{ACB}\text{,}\)\(\angle{BCF}\text{,}\) and \(\angle{ECF}\) form a straight line and add up to \(180^{\circ}\) Given that \(m(\angle{ACB})=67^{\circ}\) and \(m(\angle{ECF})=89^{\circ}\text{,}\) simply subtract these values from \(180^{\circ}\text{:}\)\(180^{\circ}-(67^{\circ}+89^{\circ})=180^{\circ}-156^{\circ}=24^{\circ}\text{.}\) Therefore, \(m(\angle{BCF})=24^{\circ}\text{.}\)
Angles \(\angle{BFC}\) and \(\angle{EFC}\) form a straight line, making them supplementary angles that add up to \(180^{\circ}\) Given that \(m(\angle{EFC})=52^{\circ}\text{,}\)\(\angle{BFC}\) is determined by subtracting \(52^{\circ}\) from \(180^{\circ}\text{:}\)\(180^{\circ}-52^{\circ}=128^{\circ}\text{.}\) Therefore, \(m(\angle{BFC})=128^{\circ}\text{.}\)
To find \(\angle{CBF}\text{,}\) use the Angle Sum Theorem, which states that a triangles interior angles add up to \(180^{\circ}\text{.}\) Given that \(m(\angle{BCF})=24^{\circ}\) and \(m(\angle{BFC})=128^{\circ}\) (from the previous problem), simply subtract these values from \(180^{\circ}\text{:}\)\(180^{\circ}-(24^{\circ}+113^{\circ})=128^{\circ}-152^{\circ}=28^{\circ}\text{.}\) Therefore, \(m(\angle{CBF})=28^{\circ}\text{.}\)
To find \(\angle{CEF}\text{,}\) use the Angle Sum Theorem, which states that a triangles interior angles add up to \(180^{\circ}\text{.}\) Given that \(m(\angle{ECF})=89^{\circ}\) and \(m(\angle{EFC})=52^{\circ}\text{,}\) simply subtract these values from \(180^{\circ}\text{:}\)\(180^{\circ}-(89^{\circ}+52^{\circ})=128^{\circ}-141^{\circ}=39^{\circ}\text{.}\) Therefore, \(m(\angle{CEF})=39^{\circ}\text{.}\)
Angle \(\angle{ACB}\) is \(67^{\circ}\) because its vertical angle also measures \(67^{\circ}\text{.}\) Angle \(\angle{BCF}\) is \(24^{\circ}\text{.}\) To find \(\angle{ACF}\text{,}\) combine the angle measures of \(\angle{ACB}\) and \(\angle{BCF}\text{:}\)\(67^{\circ}+24^{\circ}=91^{\circ}\text{.}\) Therefore, \(m(\angle{ACF})=91^{\circ}\text{.}\)
In a previous problem, it was determined that \(m(\angle{CEF})=39^{\circ}\text{.}\) To find \(\angle{CDE}\text{,}\) subtract the given \(39^{\circ}\) from \(89^{\circ}\text{:}\)\(89^{\circ}-39^{\circ}=50^{\circ}\text{.}\) Therefore, \(m(\angle{CDE})=91^{\circ}\text{.}\)
The Angle Sum Theorem states that a triangle’s interior angles always sum to \(180^{\circ}\text{.}\) Angle \(m(\angle{ACB})=67^{\circ}\) because it is vertical to \(\angle{DCE}\) and vertical angles are equal in measure. With \(m(\angle{ACB})=67^{\circ}\text{,}\) the remaining two angles, \(\angle{ABC}\) and \(\angle{BAC}\text{,}\) must sum to \(180^{\circ}-67^{\circ}=113^{\circ}\text{.}\)
What does this tell you about the angle sum of a convex quadrilateral? Explain in words and pictures why this will be true for all convex quadrilaterals.
Experiment with other triangles or by moving vertex \(A\) in the applet. Does your claim still hold true? What happens to the angle measure when the quadrilateral is no longer convex?
Different interpretations of angles in a concave quadrilateral can be made here. In your explanation, you may need to specify whether you are looking only at interior angles or only considering angles less than 180 degrees.
