1.2 Segments and Angles
Lots of different kinds of thinking go into geometry. You could start by just playing around with figures looking for interesting relationships. You could draw diagrams, you could make paper models, you could ask what if this, what if that. These activities could lead to more organized research in which you develop some interesting conjectures. There's great satisfaction in suddenly seeing or discovering a relationship, but there's a need to convince yourself and others that what you believe is, in fact, true, and this calls for proof. There's a particular satisfaction in nailing a proof. If it happens that you prove more than one idea, then there's a need to organize those ideas. Over time quite a number of geometric relationships get discovered, proved, and organized into a coherent whole. Any geometry textbook represents such a deductively organized collection of ideas.
Geometry starts with undefined terms. Then we develop definitions of some important ideas. This is no easy task because it has to be done rigorously and precisely. Great care must be taken to get the words exactly right; otherwise undesirable results, if not contradictions, will appear down the road. It is no wonder that mathematicians love puns since they spend so much time massaging ambiguity out of words. Along with definitions come basic assumptions called postulates or axioms. Axioms combined with definitions and undefined terms lead to conjectures that are proved. After a conjecture is proved, it is called a theorem. Theorems then lead to proofs of subsequent theorems and a lengthy deductive chain is created. We are ready to begin.
This geometry course is especially concerned with numerical relationships. We will start with some basic definitions and then postulate measurements for segments and angles. One must strike a balance between rigor and coverage. There are many subtle issues in these ideas, but in an introductory course, we can't spend a month on a concept such as betweenness, so we choose to be less precise and rigorous, thereby passing over a number of issues. But we hope to cover more thoroughly such topics as congruence, similarity, area, circle relationships, and right triangles by doing so.
- Definitions: Segment AB, denoted $\overline{AB}$, consists of points $A$ and $B$ and all points between them.
The length of $\overline{AB}$ is denoted $AB$.
When we have a segment $\overline{AB}$, the points $A$ and $B$ are the endpoints of the segment. Similarly, $A$ is the endpoint of $\overrightarrow{AB}$ and the ray proceeds in the direction of $B$. Note also that we are distinguishing between a segment and its length. $\overline{AB}$ is a set of points, whereas $AB$ is a number. We write $AB=5$, not $\overline{AB}=5$.
The following postulate allows us to draw a line between any two points. Furthermore, even if a diagram doesn't show a line connecting two points, it is there, and you can draw it.
- Postulate (The Line Postulate):
Through any two points there is exactly one line.
This postulate is often expressed as "Two points determine a line."
Now let us state that distances can be measured.
- Postulate (The Ruler Postulate): The set of real numbers and the points on a line can be put into a one-to-one correspondence. A scale can be obtained by assigning 0 to one point and 1 to another point.
We need to be able to combine segments and to do that we need another postulate.
- Postulate (The Segment Addition
Postulate): If $B$ is between $A$ and $C$, then $AB+BC=AC$.
Here are several important definitions involving segments.
- Definition:
Congruent segments are
segments that have the same length.
We express the fact that segment $\overline{AB}$ is congruent to segment $\overline{CD}$ by writing $\overline{AB}\cong\overline{CD}$; however, we will allow "$AB=CD$" to mean that the segments are congruent. We do not write $\overline{AB}=\overline{CD}$ except in the extraordinary case that we have used different letters for the same points so that the segments actually are the same collections of points.
- Definition: The midpoint of a segment is the point that divides the segment into two congruent parts.
- Definition: A bisector of a segment is a line, ray, or segment that intersects the segment at its midpoint.
- Definitions: An angle is a plane figure formed by two different rays that have the same endpoint. The rays are called the sides of the angle and the common endpoint is called the vertex.
- Postulate (The Protractor Postulate): To every angle there corresponds a real number $x$ between 0 and 180, not including 0, but including 180; i.e., $0<x<180$.
The measure of angle ABC is denoted $m\angle ABC$. Note we are distinguishing between the angle as a set of points and its measure. We will write $m\angle1=90$, not $\angle1=90$.
In the figure below $m\angle AOB=40$, $m\angle COA=60$, $m\angle DOE=10$, and $m\angle EOB=180$. There is no $0^{\circ}$ angle $\overline{OB}$. Since the measure of an angle is a number, we write $m\angle AOB=40$ and not $m\angle AOB=40^{\circ}$.
The Protractor Postulate does several important things. It defines the measure of an angle in terms of the arc the angle cuts off in a circle. Essentially, we are dividing the circumference of a circle into 360 equal arcs and then we define the measure of an angle in terms of the number of arcs the angle cuts off. We also allow for parts of arcs. Thus, you shouldn't be surprised to see a whole chapter on the relationships between the measure of angles and the measures of the arcs that they cut off. Secondly, we will use the Protractor Postulate in proofs to justify the construction of any angle that we wish.
We need to be able to add angles and so we have the following two-part postulate:
- Postulate (The Angle Addition Postulate): If $C$ lies in the interior of $\angle AOB$, then $m\angle AOC + m\angle COB = m\angle AOB$
- Postulate: If $C$ does not lie on $\overleftrightarrow{AOB}$, then $m\angle AOC + m\angle COB=180$.
We need some additional definitions of angles. The crucial one is the definition of a right angle. We will need to discuss this later when we take up perpendicular lines.
- Definitions:
An acute
angle is an angle with a measure satisfying $0<m\angle
A<90$.
An obtuse angle is an angle with measure satisfying $90<m\angle A<180$.
A straight angle is an angle with a measure satisfying $m\angle A=180$.
A right angle is an angle with a measure satisfying $m\angle A=90$.
Congruent angles are angles that have equal measures.
Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points.
An obtuse angle is an angle with measure satisfying $90<m\angle A<180$.
A straight angle is an angle with a measure satisfying $m\angle A=180$.
A right angle is an angle with a measure satisfying $m\angle A=90$.
Congruent angles are angles that have equal measures.
Adjacent angles are two angles in a plane that have a common vertex and a common side but no common interior points.
As with segments we can write either $\angle A\cong \angle B$ or $m\angle A= m\angle B$ but not $\angle A=\angle B$.
Adjacent angles are frequently misunderstood. In the diagram below, angles 1 and 2 are adjacent while angles 3 and 4 as well as angles 5 and 6 are not.
- Definition: The bisector of an angle is a ray or segment in the interior of the angle that divides the angle into two congruent adjacent angles.
The angle bisector definition means that if we are told that $\overline{OB}$ bisects $\angle AOC$, then we can conclude that $m\angle AOB=m\angle BOC$ or that $\angle AOB\cong\angle BOC$ depending on which is convenient. Also, if we know that $m\angle AOB=m\angle BOC$ or $\angle AOB\cong\angle BOC$, then we can conclude that $\overline{OB}$ bisects $\angle AOC$. Our reason in all cases is the definition of an angle bisector.