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Pre-Algebra View details

3. Geometric Proof

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Pre-Algebra /

Chapter 9

Geometric Proof

Student Learning Objectives:
Use inductive and deductive reasoning to determine a pattern Justify a conjecture using a proof

Why

Geometric proofs are essential for demonstrating the validity of mathematical statements using logical reasoning. This lesson covers key methods such as inductive and deductive reasoning, helping students distinguish between observations and formal logic. It explains how to create conjectures and validate them using direct proofs, theorems, and structured formats like paragraph and two-column proofs. By mastering these techniques, students can develop critical thinking and problem-solving skills applicable in geometry and broader contexts. Understanding geometric proofs enables clear communication of ideas and fosters confidence in tackling complex mathematical problems.

Lesson Settings & Tools

	14 Theory slides
	11 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Geometric Proof

Slide of 14

When police detectives want to solve a crime, they scan the crime scene for clues. They develop possible scenarios based on what they find and then focus on the scenario that seems most likely based on these limited observations. This type of analytical thinking is known as inductive reasoning. Conclusions are drawn from an incomplete picture, so they may be incorrect.

Another type of analytical thinking is deductive reasoning. In deductive reasoning, claims are supported by facts and are connected to consequent facts. Coming to a conclusion is like putting together the pieces of a jigsaw puzzle. Since all of the pieces fit neatly together, conclusions drawn by this type of reasoning tend to be stronger than those drawn by inductive reasoning.

For example, if a house was robbed and there is only one forced door, it is a fact that the thief entered through that door. Interested in knowing more about these types of reasoning? Stay and enjoy!

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Which Is More Reliable?

Kevin loves movies and television shows where mysteries are solved. When he grows up he would like to be a detective 🕵🏾‍♂️. He is always helping his friends to clear their doubts. Last week, Dylan and Maya came to him for help.

In what situation is Kevin's advice based on observations? In what situation is Kevin's advice based on facts? Which of Kevin's tips is more reliable?

Discussion

Looking for Patterns

When not all the information about a particular situation is known, the existing information can be analyzed to look for some kind of pattern. If such a pattern exists, it can lead to some conclusions. This way of thinking is known as inductive reasoning.

Concept

Inductive Reasoning

Inductive reasoning is the process of finding patterns in specific observations and writing a conclusion or conjecture. Since the conjecture is based on observations, it might be false. For example, suppose an observer notices that all the birds around them are white. The observer might inductively reason that all birds in the world are white, which is not true.

Man watching four white doves, then makes a conjecture. Look and behold, he sees a red bird!

Inductive reasoning is a practical method used in geometry and algebra for recognizing visual and numerical patterns. For instance, use the first three figures in the following diagram to guess the shape and the number of cubes in the fourth figure.

After observing the first three figures, it is reasonable to conclude that the number of cubes increases by 3 from one figure to the next — one on the top, one on the front, and one on the right. Since the figure starts with 1 cube and each step adds 3 cubes, an expression can be written to model the number of cubes in each figure. 1 + 3n Here, n represents the number of steps after Figure 1. For example, Figure 2 is one step after Figure 1, so n=1. This expression allows the calculation of the number of cubes in any figure of the pattern.

	n	Number of Cubes
Figure 1	0	1+ 3* 0 = 1
Figure 2	1	1+ 3* 1 = 4
Figure 3	2	1+ 3* 2 = 7
Figure 4	3	1 + 3* 3 = 10
Figure 121	120	1+3* 120 = 361

Example

The Thief's Pattern

It is Saturday night and Kevin, his parents, and his sister are gathered in the living room watching an action movie. The movie is about a thief who leaves riddles for the police at every crime scene. The police have collected the following cards from the first four robberies.

Each card has a riddle written on the back that the police must solve to determine where the next robbery will take place.

Determine how many green triangles will be on the card that the thief will leave after his next hit.

The thief wrote the following riddle on the back of the fourth card.

What is the number of the house that the gang plans to rob?

Hint

Use inductive reasoning. Start by counting the number of green triangles on each card. Notice that card 2 has two more green triangles than Card 1 and Card 3 has three more green triangles than Card 2.

Since the seventeenth card is quite far into the pattern, try to find an algebraic expression representing the number of green triangles on each card. Notice that Card 2 has 1+2 green triangles and Card 3 has 1+2+3 green triangles.

Solution

It is asked to find the number of green triangles on the fifth card. This can be found by observing the first four cards in the pattern — in other words, finding the answer requires using inductive reasoning. Start by counting the number of green triangles in each of the cards.

Write the results in a table to organize the information. Make a column for the number of the card and a column for the number of green triangles on the card.

Card	Green Triangles
1	1
2	3
3	6
4	10

The next step is trying to find a relation between the numbers in the table. Comparing the rows, the following relations can be written.

Card 2 has two more triangles than Card 1.
Card 3 has three more triangles than Card 2.
Card 4 has four more triangles than Card 3.

If this pattern continues, then Card 5 would have five more triangles than Card 4. Therefore, the fifth card will have 15 green triangles.

Since the 17^\text{th} card is quite far into the pattern, it would take a while to draw the diagram for the 17^\text{th} figure. The police do not have much time to prevent the next robbery. Use the table made in Part A to find an algebraic expression that represents the number of green triangles on each card.

Card	Green Triangles
1	1
2	3
3	6
4	10

Card 2 has 3 green triangles, which can be written as 1+ 2. Similarly, Card 3 has 6 green triangles, which can be written 1+2+ 3. The same happens with Card 4.

Card	Green Triangles
1	1=1
2	1+ 2=3
3	1+2+ 3=6
4	1+2+3+ 4=10

If this pattern continues, the number of green triangles in the {\color{#A800DD}{17}}^\text{th} card is equal to the sum of the first 17 natural numbers. 1+2+⋯ + 17 = 153 Consequently, Card 17 will have 153 green triangles. With this information, the police now have the complete address of the next robbery!

Discussion

Logical Reasoning

Another way of analyzing facts is through deductive reasoning. Unlike inductive reasoning, deductive reasoning does not rely on approximations or guesses. It is a process of logical reasoning through which two or more pieces of information are combined to arrive at a conclusion.

Concept

Deductive Reasoning

Deductive reasoning is the process of reaching logical conclusions from given statements. As long as the given statements are true, the conclusions drawn using deductive reasoning are also true. Consider the following two statements.

Two statements combined to obtain a conclusion

The conclusion is true because the given statements are true. Unlike inductive reasoning, deductive reasoning uses facts, definitions, properties, and the laws of logic to reach a conclusion. The two main laws of logic used to form a conclusion in deductive reasoning are the Law of Detachment and the Law of Syllogism.

Example

An Odd Riddle

The police arrived at house 153 on Boulevard Avenue too late and the gang got away. At the new crime scene, the police found the fifth card, which had the following riddle.

Determine the answer to the thief's question using both inductive and deductive reasoning.

Answer

See solution.

Hint

Inductive reasoning is the process of finding patterns in specific observations and writing a conclusion. Find the sum of random odd numbers and analyze the results. Deductive reasoning is the process of reaching logical conclusions from given statements. Any odd number can be written as 2n + 1, with n an integer number.

Solution

Start by recalling what inductive and deductive reasoning are.

	Definition
Inductive Reasoning	The process of finding patterns in specific observations and writing a conclusion
Deductive Reasoning	The process of reaching logical conclusions from given statements

Keeping the definitions in mind, the answer will be found by applying each reasoning procedure.

Using Inductive Reasoning

The thief's question involves the sum of two odd numbers. Think of two arbitrary odd numbers and calculate their sum.

Odd Numbers		Sum
-21	-13	-21+(-13) = -34
-7	3	-7+3 = 4
5	-5	5+(-5) = 0
1	5	1+5 = 6
11	51	11+51 = 62
33	45	33+45 = 78

Take a look at the numbers in the right-hand side column. All of the numbers are even, so it is natural to say that the sum of two odd numbers is always even.

The sum of two odd numbers is an even number.

Note that this conclusion is based on the pattern observed in the table. In other words, there is no 100 % certainty that the conclusion is true for any two odd numbers. This is not good news to the detective.

Using Deductive Reasoning

Considering random odd numbers does not qualify as deductive reasoning because this type of reasoning is not based on guesses. Instead, consider two odd numbers written in a general form. Recall that any odd number can be written as 2n+1, where n is an integer number. Let A and B be two different odd numbers. A &= 2n +1 B &= 2m+1 Since the purpose is to study the sum of two odd numbers, add A and B and simplify as much as possible.

A + B = 2n+1 + 2m+1

CommutativePropAdd

Commutative Property of Addition

A + B = 2n + 2m + 1 + 1

AddTerms

Add terms

A + B = 2n + 2m + 2

FactorOut

Factor out 2

A + B = 2(n+m+1)

Notice that the sum of A and B is equal to 2 times (n+m+1). Since n and m are integers, the sum inside the parentheses is an integer. Then, the sum of the two odd numbers has the form 2K, where K=n+m+1. A + B = 2K This form matches the general form of even numbers. Therefore, A+B is an even number.

The sum of two odd numbers is an even number.

Conclusion

As shown, both conclusions are the same but were found using different types of reasoning. ccc Deductive & & Inductive Reasoning & & Reasoning ↘ & & ↙ c The sum of two odd numbers is an even number. However, deductive reasoning is the most formal way to solve the question because it shows that the sum is an even number, no matter what odd numbers are chosen. Understanding this and fed up with being played for a fool by the gang, the detective put all his efforts into catching them and discovered the next target very quickly.

Discussion

Unproven Statements

Mathematicians often make claims after detecting patterns. However, these claims cannot be accepted as fact until they are rigorously verified. The term used to describe this type of claim is conjecture.

Concept

Conjecture

A conjecture is an unproven statement based on observations of a pattern. It is an educated guess that holds true for many supporting cases. A conjecture about the sum of any three consecutive numbers, for example, is shown in the table.

Sum of Any Three Consecutive Integers
Observation I	Observation II	Observation III
2+ 3+4& =9 & =3* 3	7+ 8+9& =24 & =3* 8	13+ 14+15& =42 & =3* 14
Conjecture: The sum of any three consecutive integers is three times the second number.

However, it is unknown whether any given conjecture holds true for all cases. It could be false under some circumstances and, therefore, cannot be used to support other claims. A counterexample is enough to prove that a conjecture is false.

Extra

Goldbach's Conjecture

One of the best-known conjectures is Goldbach's Conjecture, named after the German mathematician Christian Goldbach.

Goldbach's Conjecture
Every even whole number greater than 2 is the sum of two prime numbers.

For example, the even numbers below follow the rule. 14 & = 3+ 11 24 & = 11+13 40 & = 17+23 As of 2013, the conjecture has been verified by a computer for all integers less than 4 * 10^(18). In March 2000, it was announced that anyone who could prove Goldbach's Conjecture and whose proof was accepted by other mathematicians would be awarded a one million dollar prize. Although the prize was kept open for two years, nobody claimed it.

Discussion

Justifying a Conjecture Logically

The process of verifying that a conjecture is true is called a proof.

Concept

Proof

In mathematics, a proof is a series of logical steps of reasoning that lead to a conclusion. The reasoning should be strict enough so that the conclusion must be true if the given circumstances it uses are true.

There are several different methods that can be used to construct and visually present a mathematical proof.

Proof Methods
Visual Styles	Proof Types
Paragraph Proof Two-Column Proof	Direct Proof

The preferred method and visual style in any given case are highly dependent on the nature of the problem. The most common way of proving a certain statement is through direct proof.

Method

Direct Proof

A direct proof is a proof that uses the given information and other known facts until the statement is shown to be true. Consider the following statement.

If n is an odd number, then n^2 is also an odd number.

A direct proof is dependent on what information is available, and what is the statement to be proven. The following steps summarize, in general, how to do a direct proof.

Understand the Given Statement

The aim is to prove that given any odd number n, its square n^2 is also odd. To better understand the statement some example cases can be worked.

n	n^2	Is n^2 odd?
1	1	Yes
3	9	Yes
5	25	Yes
7	49	Yes

In the above table, the statement was proven to be true for just a few odd numbers, but the goal is to prove that it is true for any odd number.

Reach the Conclusion Using Given Information, Known Facts, and Logical Reasoning

The given information and other known facts should be intertwined using logical reasoning several times. These can be other theorems, definitions, axioms, and so on. This step is repeated several times until a conclusion is reached.

Statement	Reason
n is odd	Given.
n= 2k+ 1	Every odd number is equal to twice an integer plus 1.
n^2 = (2k+1)^2	Raise the equation to the power of 2.
n^2 = 4k^2+4k+1	Expand the square.
n^2 = 2(2k^2+2k)+ 1	Factor out 2.
n^2 is odd	It is written as twice an integer plus 1.

It is important to keep in mind that whether other theorems, definitions, or axioms are to be used or not depends on which type of statement is to be proven.

Discussion

Proven Statements

Once a conjecture is proven, it is no longer called a conjecture but a theorem.

Concept

Theorem

A theorem is a statement which is not self-evident but has been proven to be true using deductive reasoning. Many theorems come in the form of conditional statements — if-then statements that combine a hypothesis and a conclusion.

Pythagorean Theorem
If a triangle has a right angle, then the hypotenuse squared equals the sum of the squares of the legs.

Proofs of theorems are based on definitions, axioms, corollaries, or other theorems. Therefore, theorems can be used as reasons to justify statements in the proofs of other theorems.

Discussion

Writing a Proof in a Paragraph

There are various ways of visually presenting the proof of a certain statement. One way is to write all of the statements and reasons in a single paragraph.

Method

Paragraph Proof

A paragraph proof, or informal proof, is a way of presenting a mathematical proof that consists of statements and reasons written as complete sentences in a paragraph. The reasons can be postulates, theorems, or other mathematical reasoning that the reader is assumed to be able to follow without difficulty. For example, consider the following prompt.

Let C be a point on AE, B be a point on AC, and D be a point on CE, such that AB ≅ DE and BC ≅ CD. Prove that AC ≅ CE.

The following steps can be used to prove this particular statement.

Draw a Diagram

According to the given information, C is a point on AE.

Also, it is given that B is a point on AC and D is a point on CE.

From the last piece of given information, AB is congruent to DE and BC is congruent to CD.

Apply the Definition of Congruence

The given congruence statements imply that the congruent segments have equal lengths. AB ≅ DE and BC ≅ CD ⇓ AB= DE and BC= CD

Use the Segment Addition Postulate

The Segment Addition Postulate says that the length of a segment is the sum of the lengths of its parts. AC&= AB+ BC CE&= CD+ DE These relationships can be visualized on the diagram.

Use the Commutative Property of Addition

The Commutative Property of Addition guarantees that the order of the terms in a sum can be changed. CE = CD + DE ⇓ CE = DE + CD

Use the Substitution Property of Equality

List all the four equations written before. { & AB= DE & (I)& & BC= CD & (II)& & AC = AB + BC & (III)& & CE = DE + CD & (IV) & . According to the Substitution Property of Equality, equal values can replace each other in equations. Substitute Equations (I) and (II) into Equation (III). AC = DE + CD The right-hand side sum is the same as in Equation (IV). Then, substitute the left-hand side of Equation (IV) into this last equation. AC = CE

Apply the Definition of Congruence

If two segments have equal lengths, then the segments are congruent. AC = CE ⇓ AC≅CE It is concluded that AC is congruent to CE.

Once the proof is done, it can be compactly summarized in a paragraph. Given: & Cis a point onAE, Bis a point onAC, & Dis a point onCE, AB ≅ DE, & BC ≅ CD Prove: & AC ≅ CE

Paragraph Proof: According to the definition of congruence, AB=DE and BC=CD. The Segment Addition Postulate states that AC=AB+BC and CE=CD+DE. By the Substitution Property of Equality and the Commutative Property of Addition, it follows that AC=CE. Since segments with equal lengths are congruent, this completes the proof that AC ≅ CE.

Example

The Detective's Hit

When the gang reached their next target, the police were already waiting for them. Finding themselves cornered, the gang split up to escape out to sea. One group flew out to sea in a helicopter. However, due to bad weather, their escape was foiled and the police captured them.

Helicopter trajectory during the storm in the sea.

According to the helicopter's radar, the angles α and β of the trajectory are supplementary. Write a paragraph proof that justifies this statement.

Answer

See solution.

Hint

Start by writing the given information and label some points on the trajectory to make it easier to refer to them. The triangular hatch marks indicate that the lines are parallel. When a transversal cuts two parallel lines, the alternate interior angles are congruent. If two angles form a linear pair, their measures add up to 180^(∘).

Solution

The first step to proving a statement is identifying what information is given and what the final goal is. Labeling some points in the given diagram will make it easier to refer to them throughout the proof. For simplicity, consider only the helicopter's trajectory.

The triangular hatch marks indicate that AB and CE are parallel. This fact is not derived, so it is considered as given information. No more information is given, so continue by identifying what is desired to prove.

Given	Prove
AB ∥ CE	Angles α and β are supplementary ⇕ α + β = 180^(∘)

Develop logical conclusions that lead to the desired statement using the given information. For example, the alternate interior angles ABC and BCE are congruent because AB ∥ CE. ∠ ABC &≅ ∠ BCE &⇓ α &= m∠ BCE Additionally, the angles BCE and DCE form a linear pair. Therefore, these two angles are supplementary.

From the definition of supplementary angles, their measures add up to 180^(∘). Write an equation representing this fact. AnglesBCE andDCE are supplementary ⇓ m∠ BCE + β = 180^(∘) Next, substitute α for m∠ BCE. α + β = 180^(∘) The last equation confirms the fact that the angles α and β are supplementary, just as the helicopter's radar said. Finally, condense all the information written above into one paragraph. Given: & AB ∥ CE Prove: & α + β = 180^(∘) Proof: Since AB ∥ CE, the alternate interior angles ABC and BCE are congruent. Therefore, α = m∠ BCE. Also, ∠ BCE and ∠ DCE form a linear pair, which means they are supplementary. As such, m∠ BCE + β = 180^(∘). By the Substitution Property of Equality, α+β = 180^(∘). Consequently, the angles α and β are supplementary.

Discussion

Writing a Proof in a Table

Sometimes, compacting an entire proof into one paragraph results in a very long block of text that might be difficult to follow. A different way of presenting the proof is through a two-column table.

Method

Two-Column Proof

A two-column proof, or formal proof, is a compact way of showing the reasoning behind a mathematical proof. It consists of two columns, one for the statements and one for the reasons. The reasons can be postulates, theorems, or other mathematical reasoning the reader is assumed to be able to follow without difficulty. For example, consider proving the following statement.

If M is the midpoint of AB, then AB=2AM.

Three main steps can be followed when writing a two-column proof.

Write the Given Information

In the first row, write the given statement in the left-hand side column. This statement is given, not derived, so write given in the right-hand column.

Statements	Reasons
1. 1. M is the midpoint of AB	1. 1. Given

If possible, draw a diagram that helps to derive the information that will be written in the table. This diagram will not be included in the table, though.

Develop Logical Conclusions

Starting from what is given, develop logical statements that help to prove the desired statement. Since point M is the midpoint of AB, it divides the segment into two parts of equal length. MB = AM Write this equation in the next row in the statements column. Since this equation came from the definition of a midpoint, write that in the reasons column.

Statements	Reasons
1. 1. M is the midpoint of AB	1. 1. Given
2. 2. MB=AM	2. 2. Definition of Midpoint

Repeat the Step 2 as Many Times as Necessary

The statements written so far are not enough to reach to the desired conclusion, so continue deriving information and combining it until it points to the desired statement. Remember that postulates, theorems, or other mathematical reasoning can be used.

Statements	Reasons
1. 1. M is the midpoint of AB	1. 1. Given
2. 2. MB=AM	2. 2. Definition of Midpoint

The Segment Addition Postulate says that the length of a segment equals the sum of the lengths of its parts. Then, the following equation can be derived. AB = AM + MB As before, write the equation in the left-hand side and the reason in the right-hand side.

Statements	Reasons
1. 1. M is the midpoint of AB	1. 1. Given
2. 2. MB=AM	2. 2. Definition of Midpoint
3. 3. AB=AM+MB	3. 3. Segment Addition Postulate

Next, use the Substitution Property of Equality to substitute the equation written in the second row into the equation written in the third row.

Statements	Reasons
1. 1. M is the midpoint of AB	1. 1. Given
2. 2. MB=AM	2. 2. Definition of Midpoint
3. 3. AB=AM+MB	3. 3. Segment Addition Postulate
4. 4. AB=AM+AM	4. 4. Substitution Property of Equality

The right-hand side of the last equation can be simplified by adding the two terms.

Statements	Reasons
1. 1. M is the midpoint of AB	1. 1. Given
2. 2. MB=AM	2. 2. Definition of Midpoint
3. 3. AB=AM+MB	3. 3. Segment Addition Postulate
4. 4. AB=AM+AM	4. 4. Substitution Property of Equality
5. 5. AB=2AM	5. 5. Simplify

Notice that the last statement is the desired one. Therefore, the proof is done!

Example

Police Chase

It is the climax of the movie. The police are chasing the rest of the gang who had escaped the last crime scene. Suddenly, the Internet connection is lost and the movie stops — right in the middle of the police chase! While waiting for the connection to be restored, Kevin tells his parents that the beams of the police car's light bar make vertical angles.

Kevin's sister also mentions that these angles are congruent, something that impressed Kevin. Write a two-column proof to support Kevin's sister's claim.

Answer

See solution.

Hint

Start by identifying the given information and the desired statement. Label two vertical angles and consider an angle that is between them. This angle forms a linear pair with each of the labeled angles. The measures of supplementary angles add up to 180^(∘).

Solution

Consider a pair of vertical angles, as Kevin talks about this type of angles. A diagram will be very helpful here.

Then, Kevin's sister basically states that if two angles are vertical, then they are congruent. This statement will be proven using a two-column proof. Remember, the first step when writing a proof is to identify the given and desired statements. In this case, they can be written as follows. Given:& ∠ 1 and∠ 2 are vertical angles Prove:& ∠ 1 ≅ ∠ 2 It seems that just because ∠ 1 and ∠ 2 are vertical is not enough information to confirm that they are congruent. Some additional information is needed. Let ∠ 3 be one of the angles between ∠ 1 and ∠ 2.

From the diagram, ∠ 1 and ∠ 3 form a linear pair, as do ∠ 2 and ∠ 3. This means that the sum of the measures of each pair of angles is 180^(∘). Write two equations representing this information. m∠ 1 + m∠ 3 = 180^(∘) m∠ 2 + m∠ 3 = 180^(∘) Since both equations are equal to 180^(∘), by the Substitution Property of Equality, they can be equated. m∠ 1 + m∠ 3 = m∠ 2 + m∠ 3 Finally, subtract m∠ 3 from both sides. m∠ 1 = m∠ 2 The last equation is the desired statement, so the proof is done! To finish, summarize all the steps in a two-column table.

Statements	Reasons
1. 1. ∠ 1 and ∠ 2 are vertical angles	1. 1. Given
2. 2. ∠ 1 and ∠ 3 form a linear pair ∠ 2 and ∠ 3 form a linear pair	2. 2. From the diagram
3. 3. m∠ 1+m∠ 3 = 180^(∘) m∠ 2 + m∠ 3 = 180^(∘)	3. 3. Definition of linear pair
4. 4. m∠ 1 +m∠ 3 = m∠ 2 + m∠ 3	4. 4. Substitution Property of Equality
5. 5. m∠ 1 = m∠ 2	5. 5. Subtraction Property of Equality

Once the connection was restored, the movie continued and the police caught the rest of the gang, recovering all the stolen articles. What a movie, Kevin said!

Closure

Other Types of Proofs

How to prove a statement largely depends on the statement itself. Writing direct proofs may not be always easy. For example, consider the following statement.

sqrt(2) is an irrational number.

Proving this statement using direct proof is not as simple as it may sound. In cases like these, alternative ways of proving statements come into action. The most classic way of proving the above statement is via indirect proof.

	Description
Indirect Proof or Proof by Contradiction	A claim is proven by showing how the opposite conclusion of the claim creates a contradiction.

The first step in writing an indirect proof for the given statement is to temporarily assume that sqrt(2) is not irrational. Based on this new hypothesis, logical conclusions are developed until a contradiction is obtained. Other types of proofs are listed in the table below.

	Description
Proof by Contrapositive	A claim is proven by stating and proving the contrapositive of the claim.
Proof by Mathematical Induction	One special case is shown to be true. Next, it is shown that if the statement is true for any special case, then it is also true for some other special case.
Coordinate Proof	It is used to prove geometric statements by placing geometric figures in a coordinate plane and assigning variables to the coordinates of points.

When it comes to visually displaying the proof of a statement, there are three main styles: paragraph proofs, two-column proofs, and flowchart proofs. This last style consists of a flowchart with statements in boxes, reasons below each box, and arrows indicating the flow of the reasoning.

This flowchart proves that when a transversal cuts two parallel lines, the alternate interior angles are congruent. As a final remark, mathematicians traditionally end proofs with an abbreviation that tells that the proof is complete, often by adding a small right-aligned square □ or writing Q.E.D., which comes from a Latin phrase that means what was to be shown.

Level 1

Level 2

Level 3

Geometric Proof

Exercise 2.1

Complete the following paragraph proof by pairing each missing part with the appropriate expression or phrase. Given:& AB = 2BC & M is the midpoint ofAB Prove:& AM ≅ BC Proof: Since M is the midpoint of AB, it divides AB into two equal parts. That is, AM=MB. Applying the Segment Addition Postulate, (I)=2AM. Next, by the (II), (III)=2BC. Simplifying the equation gives AM=BC. Consequently, AM≅BC by the (IV).

Let's write the proof ourselves so we can determine the missing parts of the given proof. Let's start by making a diagram that illustrates the given information. We have to consider a segment AB, its midpoint M, and a point C such that 2BC is equal to AB.

Since M is the midpoint, it divides AB into two equal parts — that is, AM=MB. Then, the length of AB is equal to the sum of AM and MB by the Segment Addition Postulate.

From this last equation, we can say that the first missing part is AB. (I) → AB Remember that it was given that AB=2BC. Therefore, we can substitute 2AM for AB into the given equation. 2AM = 2BC This equation is true thanks to the Substitution Property of Equality. We have just determined the second and third missing parts! (II) & → Substitution Property of Equality (III) & → 2AM Let's continue the proof even though we know what the fourth missing part is. Simplifying the equation 2AM=2BC gives that AM=BC. That is, AM has the same length as BC. Therefore, by the definition of congruent segments, AM≅BC. This gives us the fourth missing part. (IV) → definition of congruent segments

Proof: Since M is the midpoint of AB, it divides AB into two equal parts. That is, AM=MB. Applying the Segment Addition Postulate, AB=2AM. Next, by the Substitution Property of Equality, 2AM=2BC. Simplifying the equation gives AM=BC. Consequently, AM≅BC by the definition of congruent segments.

In the following diagram, AE ∥ FG, BC∥ DH, and m∠ GFH = 60^(∘).

The following two-column table summarizes the proof that the measure of ∠ ABC is 120^(∘).

Statements	Reasons
AE ∥ FG, BC∥ DH, m∠ GFH = 60^(∘)	a.
∠ EDF ≅ ∠ GFH	Two parallel lines and a transversal generate congruent corresponding angles
b.	Definition of congruent angles
m∠ EDF = 60^(∘)	c.
∠ ABC and ∠ EDF are supplementary angles	Two parallel lines and a transversal generate supplementary same-side exterior angles
d.	Definition of supplementary angles
e.	Substitution Property of Equality
m∠ ABC = 120^(∘)	f.

Complete the proof by pairing each missing part with the appropriate expression or phrase.

Let's write the proof ourselves so we can determine the missing parts. Consider the given diagram again.

When we want to write a two-column proof, the first step is writing the given information. We are told that AE ∥ FG, BC∥ DH, and m∠ GFH = 60^(∘), so let's fill in the first row of the table.

Statements	Reasons
AE ∥ FG, BC∥ DH, m∠ GFH = 60^(∘)	a. Given

Since DH is a transversal that cuts a pair of parallel lines, the corresponding angles are congruent. This means that ∠ EDF and ∠ GFH are congruent.

Let's write this in the second row.

Statements	Reasons
AE ∥ FG, BC∥ DH, m∠ GFH = 60^(∘)	a. Given
∠ EDF ≅ ∠ GFH	Two parallel lines and a transversal generate congruent corresponding angles

By definition, congruent angles have the same measure. This means that m∠ EDF=m∠ GFH. Since we know that m∠ GFH=60^(∘), we can substitute 60^(∘) into the equation. From this, we can write two more rows of the proof.

Statements	Reasons
AE ∥ FG, BC∥ DH, m∠ GFH = 60^(∘)	a. Given
∠ EDF ≅ ∠ GFH	Two parallel lines and a transversal generate congruent corresponding angles
b. m∠ EDF= m∠ GFH	Definition of congruent angles
m∠ EDF = 60^(∘)	c. Substitution Property of Equality

We also have that AE is a transversal that cuts another pair of parallel lines.

Notice that ∠ ABC and ∠ EDF are same-side exterior angles. Since the same-side exterior angles are supplementary, ∠ ABC and ∠ EDF are also supplementary angles. As such, the sum of their measures is 180^(∘). We can add another couple of rows to the proof!

Statements	Reasons
AE ∥ FG, BC∥ DH, m∠ GFH = 60^(∘)	a. Given
∠ EDF ≅ ∠ GFH	Two parallel lines and a transversal generate congruent corresponding angles
b. m∠ EDF= m∠ GFH	Definition of congruent angles
m∠ EDF = 60^(∘)	c. Substitution Property of Equality
∠ ABC and ∠ EDF are supplementary angles	Two parallel lines and a transversal generate supplementary same-side exterior angles
d. m∠ ABC + m∠ EDF = 180^(∘)	Definition of supplementary angles

Next, we can substitute the equation written in the fourth row into this last equation. After that, we can subtract 60^(∘) from both sides to finally get the measure of ∠ ABC.

Statements	Reasons
AE ∥ FG, BC∥ DH, m∠ GFH = 60^(∘)	a. Given
∠ EDF ≅ ∠ GFH	Two parallel lines and a transversal generate congruent corresponding angles
b. m∠ EDF= m∠ GFH	Definition of congruent angles
m∠ EDF = 60^(∘)	c. Substitution Property of Equality
∠ ABC and ∠ EDF are supplementary angles	Two parallel lines and a transversal generate supplementary same-side exterior angles
d. m∠ ABC + m∠ EDF = 180^(∘)	Definition of supplementary angles
e. m∠ ABC + 60^(∘) = 180^(∘)	Substitution Property of Equality
m∠ ABC = 120^(∘)	f. Subtraction Property of Equality

Now that we have completed the proof, we can identify the missing parts. a.& → Given b.& → m∠ EDF = m∠ GFH c.& → Substitution Property of Equality d.& → m∠ ABC + m∠ EDF = 180^(∘) e.& → m∠ ABC + 60^(∘) = 180^(∘) f.& → Subtraction Property of Equality

In the following diagram, AB and DE are parallel, m∠ A=50^(∘), and m∠ B=40^(∘).

The following flowchart shows the proof that ∠ ACB is a right angle.

Consider the following four statements. I. & m∠ DCA = m∠ A & m∠ ECB = m∠ B II. & Parallel lines and a transversal generate & alternate interior angles that are congruent III. & 50^(∘) + m∠ ACB + 40^(∘) = 180^(∘) IV. & Given Complete the proof by pairing each missing part with the appropriate statement.

Let's follow the flow of the proof and find the missing parts. We will start with the top-left box, which contains the given information. Now, let's take a look at the box that the first box is pointing to. That box contains two pairs of congruent angles. ∠ DCA ≅ ∠ A ∠ ECB ≅ ∠ B a. Let's mark these four angles in the diagram. This may help us to determine the reason that justifies the congruence statements.

If we take AC as a transversal, we can see that ∠ DCA and ∠ A are alternate interior angles. Similarly, if we take BC as a transversal, ∠ ECB and ∠ B are alternate interior angles. In both cases, the alternate interior angles are congruent because AB∥DE. This is the reason that goes in the second box, which corresponds to statement II.

Blank	Statement
a.	II. Parallel lines and a transversal generate alternate interior angles that are congruent

Let's continue from where we left off. This time a statement is missing. b. Def. of Congruent Angles At this point, we know that the previous statement and the definition of congruent angles are used to derive this missing statement. The previous box contained statements about congruent angles, so let's apply the definition of congruence to those congruent angles. ∠ DCA ≅ ∠ A ∠ ECB ≅ ∠ B ⇓ m∠ DCA = m∠ A m∠ ECB = m∠ B The equations inside this last box fill in the second blank. These equations correspond to statement I.

Blank	Statement
a.	II. Parallel lines and a transversal generate alternate interior angles that are congruent
b.	I. m∠ DCA = m∠ A m∠ ECB = m∠ B

Let's move on to the third blank. m∠ A = 50^(∘) m∠ B = 40^(∘) c. This box has two main characteristics. It has no incoming arrow and the angle measures in it can be found in the diagram. This indicates that this box contains given information. Consequently, the third blank corresponds to statement IV.

Blank	Statement
a.	II. Parallel lines and a transversal generate alternate interior angles that are congruent
b.	I. m∠ DCA = m∠ A m∠ ECB = m∠ B
c.	IV. Given

Finally, the fourth blank corresponds to the equation obtained by substituting m∠ DCA = 50^(∘) and m∠ ECB = 40^(∘) into the equation m∠ DCA + m∠ ACB + m∠ ECB = 180^(∘). We know this from the arrows pointing to the box.

The resulting equation fills in the fourth blank. This equation corresponds to statement III.

Blank	Statement
a.	II. Parallel lines and a transversal generate alternate interior angles that are congruent
b.	I. m∠ DCA = m∠ A m∠ ECB = m∠ B
c.	IV. Given
d.	III. 50^(∘) + m∠ ACB + 40^(∘) = 180^(∘)

We have completed the flowchart proof!

Consider the following four figures.

Which of the following options corresponds to the fifth figure of the pattern?

Let's study the given pattern of figures and try to state a conjecture about the fifth figure. In other words, let's use inductive reasoning. We can see that each figure has a different number of slices, so let's start by counting the number of slices in each one.

The first figure has 2 slices, the second figure has 3, the third has 4, and so on. The number of slices increases by one from one figure to the next. From this, we can say that the fifth figure will have 6 slices. Since option A has 7 slices and option D has 8 slices, we can discard them.

Options B and D both have 6 slices. We need to identify another characteristic in the given pattern so we know which one is the fifth figure. Taking another look at the given pattern, we can see that the slices in each figure are the same size. In light of this, we can conclude that option C is the fifth figure in the pattern.

Geometric Proof

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