How is related to logically? We make a truth table to find out. Begin with all the possible truth values of . This is very simple; can be either true , or false .
Next we write the corresponding truth values for . has the opposite truth value as . If is true, then is false, and vice versa. Complete the truth table by filling in the column.
Now we construct truth tables for slightly more complex logic.
Example 2
Draw a truth table for and written .
Begin by filling in all the combinations possible for and .
How can and be true? Common sense tells us that and is false whenever either or is false. We complete the last column accordingly.
Another way to state the meaning of the truth table is that is true only when is true and is true.
Let’s do the same for or . Before we do that, we need to clarify which “or” we mean in mathematics. In ordinary speech, or is sometimes used to mean, “this or that, but not both.” This is called the exclusive or (it excludes or keeps out both). In mathematics, or means “this, that, or both this and that.” This is called the inclusive or. Knowing that or is inclusive makes the truth table an easy job.
Example 2
or is true, because is true.
or is true, because is true.
or is true because is true and is true.
or is false because is false and is false.
Example 3
Draw a truth table for or , which is written .
Begin by filling in all the combinations possible for and . Keeping in mind the definition of or above (inclusive), fill in the third column. or will only be false when both and are false; it is true otherwise.
Lesson Summary
Do we all have our own version of what is logical? Let’s hope not—we wouldn’t be able to agree on what is or isn’t logical! To avoid this, there are agreed-on rules for logic, just like there are rules for games. The two most basic rules of logic that we will be using throughout our studies are the law of detachment and the law of syllogism.
Points to Consider
Rules of logic are universal; they apply to all fields of knowledge. For us, the rules give a powerful method for proving new facts that are suggested by our explorations of points, lines, planes, and so on. We will structure a specific format, the two-column proof, for proving these new facts. In upcoming lessons you will write two-column proofs. The facts or relationships that we prove are called theorems.
Review Questions
Must the third sentence be true if the first two sentences are true? Explain your answer.
People who vote for Jane Wannabe are smart people.
I am a smart person.
I will vote for Jane Wannabe.
If Rae is the driver today then Maria is the driver tomorrow.
Ann is the driver today.
Maria is not the driver tomorrow.
All equiangular triangles are equilateral.
is equiangular.
is equilateral.
What additional statement must be true if the given sentences are true?
If West wins, then East loses. If North wins, then West wins.
If then . If then .
.
Fill in the truth tables.
When is true?
For what values of is the following statement true?
or
For what values of is the following statement true?
or
Review Answers
No (converse error).
No (inverse error).
Yes.
If North wins, then East loses.
. (also )
Note that
is
never
true.
Note that
is
always
true.
Note that
is true only when
and
are
both
false.
is always true except when , and are all false.
none,
Algebraic Properties
Learning Objectives
Identify and apply properties of equality.
Recognize properties of congruence “inherited” from the properties of equality.
Solve equations and cite properties that justify the steps in the solution.
Solve problems using properties of equality and congruence.
Introduction
We have begun to assemble a toolbox of building blocks of geometry (points, lines, planes) and rules of logic that govern deductive reasoning. Now we start to expand our geometric knowledge by applying logic to the geometric building blocks. We’ll make a smooth transition as some fundamental principles of algebra take on new life when expressed in the context of geometry.
Properties of Equality
All things being equal, in mathematics the word “equal” means “the same as.” To be precise, the equal sign means that the expression on the left of the equal sign and the expression on the right represent the same number. So equality is specifically about numbers—numbers that may be expressed differently but are in fact the same.
Some examples:
Basic properties of equality are quite simple and you are probably familiar with them already. They are listed here in formal language and then translated to common sense terms.
Properties of Equality
For all real numbers , and :
Reflexive Property:
That is, any number is equal to itself, or the same as itself.
Example:
Symmetric Property: If then .
You can read an equality left to right, or right to left.
Example: If then
Example: If , then .
Sometimes it is more convenient to write than . The symmetric property allows this.
Transitive Property: If and then .
Translation: If there is a “chain” of linked equations, then the first number is equal to the last number. (You can prove that this applies to more than two equalities in the review questions.)
Example: If and , then .
As a reminder, here are some properties of equality that you used heavily when you learned to solve equations in algebra.
Substitution Property: If then can be put in place of anywhere or everywhere.
Example: Given that and that . Then .
Addition Property of Equality: If , then .
Translation: You can add the same number to both sides of an equation and retain the equivalency.
Example: If , then .
Multiplication Property of Equality: If , then
Translation: You can multiply the same number on both sides of an equation and retain the equivalency.
Example: If , then .
Keep in mind that these are properties about numbers. As you go further into geometry, you can apply the properties of equality to anything that is a number: lengths of segments and angle measures, for example.
Properties of Congruence
Let’s review the definitions of congruent segments and angles.
Congruent Segments: if and only if .
Remember that, although and are segments, and are lengths of those segments, meaning that and are numbers. The properties of equality apply to and .
Congruent Angles: if and only if =
The comment above about segment lengths also applies to angle measures. The properties of equality apply to and .
Any statement about congruent segments or congruent angles can be translated directly into a statement about numbers. This means that each property of equality has a corresponding property of congruent segments and a corresponding property of congruent angles.
Here are some of the basic properties of equality and the corresponding congruence properties.
Given that and are real numbers.
Reflexive Property of Equality:
Reflexive Property of Congruence of Segments:
Reflexiv
e Property of Congruence of Angles:
Symmetric Property of Equality: If , then .
Symmetric Property of Congruence of Segments: If , then
Symmetric Property of Congruence of Angles: If , then .
Transitive Property of Equality: If and , then
Transitive Property of Congruence of Segments
If and then
Transitive Property of Congruence of Angles
If , and then .
Using Congruence Properties in Equations
When you solve equations in algebra you use properties of equality. You might not write out the logical justification for each step in your solution, but you know that there is an equality property that justifies that step.
Let’s see how we can use the properties of congruence to justify statements in deductive reasoning. Abbreviated names of the properties can be used.
Example 1
Given points , and , with , and .
Are , , and collinear?
Example 2
Given that and .
Prove that is an acute angle.
The deductive reasoning scheme in example 2 is called a proof. The final statement must be true if the given information is true.
Lesson Summary
We built on our previous knowledge of properties of equality to derive corresponding properties of congruence. This enabled us to test statements about congruence, and to create new properties and relationships about congruence. We had our first introduction, in informal terms, to logical proof.
Points to Consider
In the examples and review questions, terms like given, prove, and reason were used. In upcoming lessons we’ll see how to identify the given facts, how to draw a diagram to represent a statement that we need to prove, and how to organize proofs more formally. As we move ahead we’ll prove many important geometric relationships called theorems. We have now established the framework of logic that we’ll use repeatedly in future work.
Review Questions
Given: , and are real numbers.
Use the given property or properties of equality to fill in the blank in each of the following questions.
Symmetric: If , then ___________________.
Reflexive: If , then ___________________.
Transitive: If and then __________________.
Symmetric: If , then ____________________.
Reflexive: If , then _____________________.
Substitution: If and , then _________________________.
Use the transitive property of equality to write a convincing logical argument (a proof) that the statement below is true.
If and and and , then .
Note that this chain could be extended with additional links.
Let be the relation “is the mother of.” Let be the relation “is the brother of.”
Is symmetric? Explain your answer.
Is symmetric? Explain your answer.
Is transitive? Explain your answer.
Is transitive? Explain your answer.
Let and be real numbers. Prove: If and , then
The following statement is not true. “Let and be points. If and , then ” Draw a diagram with these points shown to provide a counterexample.
Review Answers
( or ).
If and then (transitive property). If and then (transitive property). If and then (transitive property).
No. If Maria is the mother of Juan, it does NOT follow that Juan is the mother of Maria!
Yes. For example, if Bill is Frank’s brother, then Frank is Bill’s brother.
No. If were transitive, then “Maria is Fern’s mother and Fern is Gina’s mother” would lead to “Maria is Gina’s mother.” However, Maria would actually have to be Gina’s grandmother!
Yes. If Bill is Frank’s brother and Frank is Greg’s brother, then Bill is Greg’s brother. You might say the brother of my brother is (also) my brother.
and (given); (reflexive); (substitute for and for ).
Below is an example:
A correct response is a diagram showing:
.
.
.
If , , and are collinear and , , and are not collinear then the conditions are satisfied.
Diagrams
Learning Objectives
Provide the diagram that goes with a problem or proof.
Interpret a given diagram.
Recognize what can be assumed from a diagram and what can not be.
Use standard marks for segments and angles in diagrams.
Introduction
Geometry is about objects such as points, lines, segments, rays, planes, and angles. If we are to solve problems about these objects, our work is made much easier if we can represent these objects in diagrams. In fact, for most of us, diagrams are absolutely essential for problem solving in geometry.
Basic Postulates—Another Look
Just as undefined terms are building blocks that other definitions are built on, postulates are the building blocks of logic. We’re now ready to restate some of the basic postulates in slightly more formal terms, and to use diagrams.
Postulate 1 Through any two distinct points, there is exactly one line.
Comment: Any two points are collinear.
Postulate 2 There is exactly one plane that contains any three noncollinear points.
Comment: Sometimes this is expressed as: “Three noncollinear points determine a plane.”
Postulate 3 If two points are in a plane, then the whole line through those two points is in the plane.
Postulate 4 If two distinct lines intersect, then the intersection is exactly one point.
Comments: Some lines intersect, some do not. If lines do intersect, it is in only one point, otherwise one or both “lines” would have to curve, which lines do not do.
Postulate 5 If two distinct planes intersect, then the intersection is exactly one line.
Comments: Some planes intersect, some do not. Think of a floor and a ceiling as models for planes that do not intersect. If planes do intersect, it is in a line. Think of the edge of a box (a line) formed where two sides of the box (planes) meet.
Postulate 6 The Ruler Postulate: The points on a line can be assigned real numbers, so that for any two points, one corresponds to and the other corresponds to a nonzero real number.
Comments: This is how a number line and a ruler work. This also means we can measure any segment.
Postulate 7 The Segment Addition Postulate: Points and are collinear if and only if .
Comment: If and are not collinear, then . We saw examples of this fact in earlier sections of this chapter.
Postulate 8 The Protractor Postulate: If rays in a plane have a common endpoint, can be assigned to one ray and a number between and can be assigned to each of the other rays.
Comment: This means that any angle has a (degree) measure.
Postulate 9 The Angle Addition Postulate: Let and be points in a plane. is in the interior of if and only if + = .
Comment: If an angle is made up of other angles, the measures of the component angles can be added to get the measure of the “big” angle.
Postulate 10 The Midpoint Postulate: Every line segment has exactly one midpoint.
Comments: If is a point on and there is not another point on , let’s say point ,with . The midpoint of a segment is unique.
Postulate 11 The Angle Bisector Postulate: Every angle has exactly one bisector.
Comments: The bisector of an angle is a ray. If bisects , there is not another ray that bisects the angle. The bisector (ray) of an angle is unique.
Using Diagrams
Now we apply our definitions and postulates to a geometric figure. When measures are given on a figure, we can assume that the measurements on the figure are correct. We can also assume that:
Points that appear to be collinear are collinear.
Lines, rays, or segments that appear to intersect do intersect.
A ray that appears to be i
n the interior of an angle is in the interior of the angle.
We cannot assume the following from a diagram:
That lines, segments, rays, or planes are parallel or perpendicular.
That segments or angles are congruent.
These must be stated or indicated in the diagram.
The diagram below shows some segment and angle measures.
Example 1
A. Is the midpoint of Explain your answer.
No. is on , but .
B. Is the midpoint of Explain your answer.
Yes. is on , and = .
C. Name an angle bisector and the angle that it bisects.
bisects .
D. Fill in the blank: _______.
E. Is the bisector of Explain your answer.
No. If bisected , then would be That would make but
Sometimes we use special marks in diagrams. Tick marks show congruent segments. Arc marks show congruent angles. Right angle marks show right angles and perpendicular lines and segments.
When these signs are used, the relationships they represent become part of the given information for a problem.
Example 2
Blue blobs are dots for points and single and double arc marks to show equal angles.
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