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Polyominoes, Polygons, and Polyhedra: Discovering Geometry through Exploration

Teresa D. Magnus

Section 2.2 Exploring Angle Measure

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?

Subsection 2.2.1 Linear Objects

To fully grasp the definition of angle, we need to first understand the difference between lines, segments, and rays.

Definition 2.2.1.

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 \(\overline{AB}\text{.}\)
Whereas a line continues infinitely in two directions, a ray continues infinitely in just one direction and has an endpoint.

Definition 2.2.2.

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.

Subsection 2.2.2 Defining and Measuring Angles

We now define an angle and describe how to measure angles.

Definition 2.2.3.

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.
angle BAC
Figure 2.2.4. Angle \(\angle{BAC}\)
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.
A circle centered around point A. An arrow has been added to the circle to indicate rotation in a counterclockwise direction.
Figure 2.2.5. Full rotation about point \(A\text{.}\)
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 \(A\text{.}\)
Line containing C A and B from left to right. Semicircular arc drawn with center A from point B to the line on the opposite side of A.
Figure 2.2.6. 180-degree rotation about point \(A\)
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.
A protractor aligned with line containing C A B, centered at point A. Labels for every multiple of 10 degrees from zero degrees up through 180 degrees are shown, both clockwise and counterclockwise. Ray AD passes through the indicator for 40 degrees.
Figure 2.2.7. Using a protractor to measure angles.

Exploration 2.2.1. Reading a Protractor.

(a)
Use the picture above to determine the following:
  1. The measure of angle \(\angle{EAB}\text{.}\)
  2. The measure of angle \(\angle{CAD}\text{.}\)
  3. The measure of angle \(\angle{CAE}\text{.}\)
  4. The measure of angle \(\angle{EAC}\text{.}\)
(b)
  1. What fraction of a full rotation is 40 degrees?
  2. What fraction of a full rotation is 110 degrees?
(c)
Use a physical protractor and a ruler to draw an angle with measure 78 degrees. Describe your technique.
(d)
It is possible to have angle measures greater than 180 degrees. Use words and/or pictures to describe an angle that measures 240 degrees.
We give a more concise definition of angle measure and introduce the notion of angle congruence.

Definition 2.2.8.

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 \(\overrightarrow{BA}\text{.}\)

Definition 2.2.9.

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 \(m\angle ABC = m\angle DEF\text{.}\)

Subsection 2.2.3 Angles of a Triangle

Exploration 2.2.2.

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.
Figure 2.2.10. An interactive GeoGebra applet for Exploration 2.2.2.
(a)
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.
Hint.
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.
(b)
Also rotate \(\Delta ABC\) 180 degrees about the midpoint \(E\) of side \(\overline{AB}\text{.}\)
  1. Which angle of \(\Delta ABC\) is congruent to \(\angle A'C'B'\text{?}\)
  2. Which angle of \(\Delta ABC\) is congruent to \(\angle B_{1}'A_{1}'C_{1}'\text{?}\)
(c)
What does this tell you about \(m\angle ABC+m\angle BCA + m\angle CAB\text{?}\) Explain how the picture shows this.
(d)
Make a claim about the sum of the interior angles of a triangle.
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?

Subsection 2.2.4 Additivity 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.

Definition 2.2.11.

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 \(\angle ABC\text{.}\)
Angle BAC with interior ray AD
Figure 2.2.12. Additivity of angle measure

Subsection 2.2.5 Vertical Angles

To conclude this section, we will use what we have learned to make a claim about vertical angles.

Definition 2.2.14.

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.

Checkpoint 2.2.15.

Determine two pairs of vertical angles in the figure to the right.
Two lines meet at point E. One line contains points A, E, and B in order and the other contains points C, E, and D in that order.
Figure 2.2.16. Lines intersecting to form vertical angles
Hint.
Each pair of vertical angles consists of two angles.
Answer.
One pair consists of \(\angle AEC\) and \(\angle BED\text{.}\) The other pair of vertical angles are \(\angle AED\) and \(\angle BEC\text{.}\)

Checkpoint 2.2.17.

What must be true about the measures of any pair of vertical angles? Give a thorough explanation of why this must be true using Principle 2.2.13.

Exercises 2.2.6 Exercises

Skills and Recall

1.
Use Figure 2.2.18 to identify the following:
  1. All labeled points that lie on segment \(\overline{EF}\text{.}\)
  2. All labeled points that lie on line \(\overleftrightarrow{EF}\text{.}\)
  3. All labeled points that lie on ray \(\overrightarrow{FE}\text{.}\)
  4. Another name for ray \(\overrightarrow{FE}\)
  5. Two other names for angle \(\angle{HEB}\text{.}\)
  6. The sum \(m(\angle{HEB})+m(\angle{AEH})\text{.}\)
  7. A point in the interior of \(\angle{CEJ}\text{.}\)
  8. Four pairs of congruent angles.
  9. All labeled points that lie on both \(\overleftrightarrow{AB}\) and \(\overleftrightarrow{CE}\text{.}\)
  10. All labeled points that lie on both \(\overrightarrow{GB}\) and \(\overrightarrow{AB}\text{.}\)
Three lines intersect at a point E. The order of points on each line are as follows: A, E, B, and then G on line 1; C, H, E, F, and D on line 2; and I, E, and J on line 3. Going clockwise around point E, we have A, C, I, B, F, and J.
Figure 2.2.18. Sketch for Exercise 2.2.6.1 and Exercise 2.2.6.2.
2.
Use Figure 2.2.18 to name the following. Answers may be lines, line segments, rays, angles, single points, or the empty set.
  1. The set of all points that lie on \(\overrightarrow{IE}\) or \(\overrightarrow{JE}\) or both.
  2. The set of all points that lie on \(\overrightarrow{EI}\) or \(\overrightarrow{EF}\) or both.
  3. The set of points that lie on both \(\overrightarrow{GE}\) and \(\overrightarrow{BG}\text{.}\)
  4. The set of points that lie on both \(\overrightarrow{CH}\) and \(\overrightarrow{EF}\text{.}\)
  5. The set of points that lie on both \(\overrightarrow{CH}\) and \(\overrightarrow{BG}\text{.}\)
  6. The set of points that lie on both \(\overrightarrow{CH}\) and \(\overrightarrow{JI}\)
3.
In Figure 2.2.19,
\begin{align*} m(\angle{BED})\amp = 89^\circ\\ m(\angle{CAD})\amp = 37^\circ\\ m(\angle{DCE})\amp = 67^\circ\\ m(\angle{EFC})\amp = 52^\circ\\ m(\angle{ECF})\amp = 89^\circ \end{align*}
Determine the measures of other angles as directed.
  1. \(\displaystyle \angle{ACD}\)
  2. \(\displaystyle \angle{ADC}\)
  3. \(\displaystyle \angle{BCE}\)
  4. \(\displaystyle \angle{BCF}\)
  5. \(\displaystyle \angle{BFC}\)
  6. \(\displaystyle \angle{CBF}\)
  7. \(\displaystyle \angle{CEF}\)
  8. \(\displaystyle \angle{ACF}\)
  9. \(\displaystyle \angle{CDE}\)
  10. What do we know about \(\angle{ABC}\) and \(\angle{BAC}\text{?}\) Explain.
Quadrilateral ABED. Diagonals AE and BD meet at point C. Point F lies on side BE.
Figure 2.2.19. Sketch for Exercise 2.2.6.3.

Extending the Concepts

4.
Must the right angle be the largest angle of a right triangle? Explain how this follows from results of this section.
5.
Sketch any convex quadrilateral on paper or using the GeoGebra drawing tool.
Figure 2.2.20. An interactive GeoGebra applet for Quadrilateral Angle Sum Exercise.
Connect opposite vertices \(B\) and \(D\) with a line segment, called a diagonal, to divide the quadrilateral into two shapes.
  1. What two shapes did this create?
  2. 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.
  3. 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?
Hint.
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.
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