Unit 1: Mathematical Proofs

Mathematical Proofs

Objective

In this lesson, you will construct mathematical proofs and apply proof techniques to simple geometric relationships.

G.MP.3, G.MP.7, G.GSR.4.4

Match

Match the following

• a position in space

• a connected set of points that extends infinitely in both directions

• a part of a line with endpoints on either side

• figure formed from two rays sharing a common endpoint

Point

Line

Line Segment

Angle

• Point

• Line

• Line Segment

• Angle

Two lines intersect to form two angles. Ben says that angles AEB and DEC are congruent. But how does he know?

There are two types of proofs that Ben can use to show his statement is true: direct proofs and indirect proofs.

In this lesson, we’ll explore how to use different types of proofs to show that mathematical statements are true or false.

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Introduction

Two adjacent angles whose sides form a straight line are called a linear pair.

Consider the statement: If two angles form a linear pair, then the angles are supplementary.

From the diagram, it certainly looks like this statement is true. But how can we prove that it is true?

One way is to use a direct proof. A direct proof is an argument that establishes the truth of a given conjecture using a logical sequence of statements. All statements in the proof are supported by evidence.

Before we build the argument, let’s learn about the evidence that we can use in proofs.

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Direct Proofs

In this course, the evidence and arguments we will use derive from an axiomatic system. In an axiomatic system, a set of accepted truths, or axioms, provides the basis for deriving all other conclusions.

The axiomatic system of geometry consists of:

• the undefined terms point, line, and plane

• defined terms created from undefined terms

• axioms and postulates, which are true statements accepted without proof.

• theorems, which are statements that must be proven true

Defined terms, axioms and postulates, and previously proven theorems all serve as evidence for proofs. On the next slide, you will review some of these axioms and postulates.

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Axiomatic System

​X=X

Reflexive

X=Y, then Y=X

Symmetric

X=Y and Y=Z, then X=Z

​​Transitive

If a=b, then a can be substituted for b in any expression and vice versa

​​Substitution

​Properties of Equality

Reflexive

Symmetric

​​Transitive

​​Substitution

​Properties of Congruence

​Angle Addition Property: If B is a point on the interior of ∠AOC, the measure of ∠AOC is equal to the sum of the measures of ∠AOB and ∠BOC.

​Angle and Segment Addition

​Two additional properties that will be useful in developing proofs are the angle addition and the segment addition properties.

​In other words, the measure of an angle or line segment is the sum of the measures of its parts.

On the next screen, you'll use angle addition to explore the relationship between angles in a linear pair. You'll make a conjecture from your findings, then see how to develop a proof for the conjecture.

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Open Ended

∠ADB and ∠BDC represent a linear pair because points A, D, and C lie on a straight line. Calculate the sum of m∠ADB and m∠BDC. Then, move point B around and see how the angles change. What happens to the sum of m∠ADB and m∠BDC as you move point B around?

Open Ended

What can you say about ∠ADB and ∠BDC  based on the sum of their angles? How are the angles related?

Open Ended

Based on your observations, what is the relationship between a linear pair and supplementary angles?

In the activity, you used observations to demonstrate the linear pair theorem: if two angles form a linear pair, then the angles are supplementary. To prove the statement, you can use a direct proof.

Types of Proofs

​A two-column proof uses a table of statements and reasons to logically move from one step to another, starting with the given and ending with what is being proved.

• The statements are either given at the beginning or deductions from previous statements.

• The reasons are the definitions, axioms, and previously proven theorems that provide evidence for the statements.

A paragraph proof writes out the statements and the reasons in a running narrative form. The proof will have the same steps as other types of proof.

A flowchart proof represents each step of the proof in a flowchart. Each statement is written inside a box, and an arrow leads the reader from step to logical step.

Some statements lead directly from one step another. Other statements require the combination of two or more previous steps. The reason for each step is written outside the box.

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Let's look at how to use a flow chart proof to prove the statement: If two angles are supplementary to the same angle, then the two angles are congruent.

• State the given.

• Apply the definition of supplementary angles to write two equations.

• Apply the transitive property of equality to equate the equations.

• Apply the subtraction property of equality to isolate the angles of interest.

• Apply the definition of congruence.

We've now proven the congruent supplements theorem.

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Congruent Supplements Theorem

Using the congruent supplements theorem, we can now prove the congruent and supplementary angles theorem. This theorem states: If two congruent angles are supplementary, then each angle is a right angle.

Let's use a paragraph proof for this theorem.

Let ∠1 and ∠2 be congruent and supplementary angles. Since ∠1 and ∠2 are supplementary, m∠1 + m∠2=180∘ by the definition of supplementary angles. Since ∠1 and ∠2 are congruent, m∠1=m∠2 by the definition of congruent angles. By substitution, m∠1 + m∠2=2⁢(m∠1)=180∘. By the division property of equality, m∠1=90∘. Since m∠1=m∠2, m∠2=90∘ by the transitive property of equality. Therefore, by the definition of a right angle, ∠1 and ∠2 are both right angles.

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​Congruent and Supplementary Angles Theorem

Multiple Choice

What is the missing reason in this proof?

• congruent supplements theorem

• addition property of equality

• linear pair theorem

• transitive property of equality

When two lines intersect, the two lines form 4 angles. The angles opposite one another are called vertical angles. In the image, ∠AEB and ∠DEC are vertical angles.

Recall Ben's statement from the introduction: that angles AEB and DEC are congruent. This statement is an application of the vertical angles theorem, which states that vertical angles are congruent.

On the next slide, we'll see how Ben could use a direct proof to prove this theorem.

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Vertical Angles Theorem

Dropdown

It is given that ∠ABE and ∠DBC are vertical angles. By the ​

Select answer

, ∠ABE is congruent to ∠DBC. By the definition of congruence, the measure of ∠ABE must equal the measure of ∠DBC. Then, by the substitution property of equality, ​

Select answer

. Applying the subtraction property of equality gives x=4..

If Jorge takes the highway to school, it always takes less than 15 minutes to get there. One day, it takes him 45 minutes to get to school. Did he take the highway?

It seems unlikely that Jorge took the highway. If Jorge took the highway, it would have taken 15 minutes to get to school. Since it did not take Jorge 15 minutes to get to school, he could not have taken the highway.

This type of argument is called a contradiction. In proofs, it’s possible to use contradiction to prove a statement true. These types of proofs are called indirect proofs.

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Indirect Proofs

​In a direct proof, each statement is derived from the statements before it. Since the given statements are the first steps, each statement following are true statements derived from the givens.

Direct Proofs
Start with given Each statement is true based on the given Leads to conclusion based on the given

​​Direct vs. Indirect Proofs

 In an indirect proof, we first assume that the statement to be proved is false. Each following statement takes the false assumption and leads it to a logical conclusion. If this logical conclusion contradicts the given information, the assumption made at the beginning must have been false. Therefore, the alternative (the statement to be proved) must be true.

Indirect Proofs
Start with assumption Each statement is true based on the assumption Leads to contradiction with the given

Since an indirect proof leads to a contradiction, it is also called a proof by contradiction.

​We'll use an indirect proof to prove the congruent complements theorem: if two angles are complementary to the same angle, then the two angles are congruent.

• Assume that the statement to be proved is false.

• Apply properties to arrive at a contradiction of the given.

Once we contradict the given information, we know our assumption must have been false. Therefore, the alternative (the statement to be proved) must be true.

Compare this method to a direct proof of the congruent complements theorem.

The proofs we’ve been using are examples of deductive reasoning. Deductive reasoning starts with general statements and uses logic to narrow these statements to reach a specific conclusion.

Inductive reasoning begins with specific examples, then makes a broad generalization from the examples. In this case, something that is true for one or a few cases is applied to other cases.

In deductive reasoning, as long as the assumption is valid, the conclusion is valid. For inductive reasoning, it is not necessarily true that the pattern observed in the examples will continue. For this reason, deductive reasoning, but not inductive reasoning, is used for math.

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Inductive Reasoning

Multiple Select

Which of these are examples of deductive reasoning?

• • Milk costs $2.99 per gallon at the first three grocery stores that Gema visited. She concludes that milk costs costs $2.99 per gallon everywhere.

• • Most tulips grow to between 6 and 24 inches tall. Janelia concludes that the flowers in her garden will be shorter than 24 inches.

• • Felix knows that everything that fits in his bookbag will fit in his suitcase. His laptop fits in his bookbag, so he concludes that the laptop will fit in his suitcase.

• • Everytime Matthew has left for school at 7 am, he has been on time. Matthew concludes that if he leaves the house at 7 am today, he will be on time.

• • The school’s schedule states that lunch period is from 12 p.m. to 1 p.m. Eric concludes that it is after 12:00 p.m. because it is currently the lunch period.

In some instances, we might want to show that a statement is false. We can do this with a counterexample: a single example that shows a statement is not true.

• A single example, or even several examples, cannot be used to prove a statement.

• A single example is enough to disprove a statement: if a statement does not hold for one example, it cannot be universally true.

Consider the statement that if two angles are adjacent, they must be supplementary. A counterexample is any pair of adjacent angles whose measure does not sum to 180∘.

Using a counterexample is different from writing a proof: a proof shows the truth of a statement, while a counterexample shows a statement is false.

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Counterexamples

Multiple Choice

Consider the statement:

If a pair of angles is supplementary, both angles are right angles.

Which figure is the best counterexample to disprove the statement?

In this lesson, we saw that Ben could use a mathematical proof to prove that ∠AEB  and ∠DEC  are congruent. Mathematical proofs are a form of a deductive reasoning.

• In a direct proof, a sequence of statements, each supported by evidence, leads to a conclusion.

• In an indirect proof, the statement to be proved is assumed false, and a series of statements leads to a contradiction.

Each type of proof can be presented in a two-column, a paragraph, or a flow-chart format.

Each step in a proof must be supported by evidence, which can include definitions, axioms, and previously proven geometric theorems.

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Summary