Most of the geometry concepts and theorems that are learned in high school today were first discovered and proved by mathematicians such as Euclid thousands of years ago. Given that these geometry concepts and theorems have been known to be true for thousands of years, why is it important that you learn how to prove them for yourself?
Theorems and Proofs
In geometry, a postulate is a statement that is assumed to be true based on basic geometric principles. An example of a postulate is the statement “through any two points is exactly one line”. A long time ago, postulates were the ideas that were thought to be so obviously true they did not require a proof. A theorem is a mathematical statement that can and must be proven to be true. You've heard the word theorem before when you learned about the Pythagorean Theorem. Much of your future work in geometry will involve learning different theorems and proving they are true.
What does it mean to “prove” something? In the past you have often been asked to “justify your answer” or “explain your reasoning”. This is because it is important to be able to show your thinking to others so that ideally they can follow it and agree that you must be right. A proof is just a formal way of justifying your answer. In a proof your goal is to use given information and facts that everyone agrees are true to show that a new statement must also be true.
Suppose you are given the picture below and asked to prove that \begin{align*}\overline{AD} \cong \overline{DC}\end{align*}. This means that you need to give a convincing mathematical argument as to why the line segments MUST be congruent.
Here is an example of a paragraph-style proof. This is similar to a detailed explanation you might have given in the past.
\begin{align*}{\color{red}\overline{AB}\cong \overline{BC}}\end{align*} because it is marked in the diagram. Also, \begin{align*}{\color{red}\angle ADB}\end{align*} and \begin{align*}{\color{red}\angle CDB}\end{align*} are both right angles because it is marked in the diagram. This means that \begin{align*}{\color{red}\triangle ADB}\end{align*} and \begin{align*}{\color{red}\triangle CDB}\end{align*} are right triangles because right triangles are triangles with right angles. Both triangles contain segment \begin{align*}{\color{red}\overline{BD}}\end{align*}. \begin{align*}{\color{red}\overline{BD}\cong \overline{BD}}\end{align*} because of the reflexive property that any segment is congruent to itself. \begin{align*}{\color{red}\triangle ADB \cong \triangle CDB}\end{align*} by \begin{align*}{\color{blue}HL \cong}\end{align*} because they are right triangles with a pair of congruent legs and congruent hypotenuses. \begin{align*}{\color{red}\overline{AD}\cong \overline{DC}}\end{align*} because they are corresponding segments and corresponding parts of congruent triangles must be congruent.
There are two key components of any proof -- statements and reasons.
- The statements are the claims that you are making throughout your proof that lead to what you are ultimately trying to prove is true. Statements are written in red throughout the previous proof.
- The reasons are the reasons you give for why the statements must be true. Reasons are written in blue throughout the previous proof. If you don't give reasons, your proof is not convincing and so is not complete.
When writing a proof, your job is to make everything as clear as possible, because you need other people to be able to understand and believe your proof. Skipping steps and using complicated words is not helpful!
There are many different styles for writing proofs. In American high schools, a style of proof called the two-column proof has traditionally been the most common (see Example 3). In college and beyond, paragraph proofs are common. An example of a style of proof that is more visual is a flow diagram proof (see Example 4). No matter what style is used, the key components of statements and reasons must be present. You should be familiar with different styles of proof, but ultimately can use whichever style you prefer.
Learning to write proofs can be difficult. One of the best ways to learn is to study examples to get a sense for what proofs look like.
Rewriting Proofs
1. Rewrite the proof from the guidance in a two-column format.
Using the picture below, prove that \begin{align*}\overline{AD} \cong \overline{DC}\end{align*}.
In a two-column proof, the statements and reasons are organized into two columns. All of the same logic that was used in the paragraph proof will be used here. Look at the proof below and compare it to the paragraph proof from the guidance.
Statements | Reasons |
\begin{align*}{\color{red}\overline{AB} \cong \overline{BC}}\end{align*} | Given |
\begin{align*}{\color{red}\angle ADB}\end{align*} and \begin{align*}{\color{red}\angle CDB}\end{align*} are right angles | Given |
\begin{align*}{\color{red}\Delta ADB}\end{align*} and \begin{align*}{\color{red}\Delta CDB}\end{align*} are right triangles | definition of right triangles |
\begin{align*}{\color{red}\overline{BD} \cong \overline{BD}}\end{align*} | reflexive property |
\begin{align*}{\color{red}\Delta ADB \cong \Delta CDB}\end{align*} | \begin{align*}{\color{blue}HL \cong}\end{align*} |
\begin{align*}{\color{red}\overline{AD} \cong \overline{DC}}\end{align*} | CPCTC (corresponding parts of congurent triangles must be congurent) |
There are a couple of points to note about two-column proofs.
- For a two-column proof, instead of saying “it is marked in the diagram” as a reason, you just write “given”. You can use the reason “given” for anything that was stated up front or marked in a diagram. Typically, the first few rows of your proof will always be the “givens”.
- In a two-column proof you will use less words than in a paragraph proof, because you are not writing in complete sentences.
- \begin{align*}HL \cong\end{align*} and the other criteria for triangle congruence are always acceptable reasons if you have shown in earlier rows that each part of the criteria has been met. You do not need to write a sentence explaining why you can use \begin{align*}HL \cong\end{align*}.
- Instead of stating right triangles are triangles with right angles as a reason, you can just say “definition of right triangles”. Definitions are always acceptable reasons.
- CPCTC is an abbreviation for the statement “corresponding parts of congruent triangles are congruent”. The abbreviation was developed because this reason is used often, and it can be cumbersome to write it over and over.
2. Rewrite the proof from the guidance in a flow diagram format.
Using the picture below, prove that \begin{align*}\overline{AD} \cong \overline{DC}\end{align*}.
In the proof below, statements are written in red and reasons are written in blue. In a flow diagram, the statements and reasons will be organized into boxes that are connected with arrows to show the flow of logic. Look at the proof below and compare it to the two-column and paragraph versions of the same proof.
There are a couple of points to note about flow diagram proofs.
- Statements are written inside the boxes and the reasons the statements must be true are written below the boxes.
- The arrows show the flow of logic. If two boxes are connected by arrows it means that the statement in the lower box can be made because the statement in the upper box is true. Notice that three boxes point towards the statement that \begin{align*}\Delta ADB \cong \Delta CDB\end{align*}. This is because all three of those statements were necessary for making the conclusion that the two triangles are congruent.
- Just like in the two-column format, “given” is the reason used for anything that was stated up front or marked in the diagram. The “given” reasons will be towards the top of the flow diagram.
- Just like in the two-column format, you use abbreviations where possible. \begin{align*}HL \cong\end{align*}, other triangle congruence criteria, CPCTC, and definitions are all acceptable reasons.
Identifying Mistakes
Each proof below has a mistake, can you figure out where the mistake is and why it is a mistake?
Using the picture below, prove that \begin{align*}\overline{AD} \cong \overline{DC} \end{align*}.
PROOF A:
Statements | Reasons |
\begin{align*}{\color{red}\overline{AB} \cong \overline{BC}}\end{align*} | Given |
\begin{align*}{\color{red}\Delta ADB \cong \Delta CDB}\end{align*} | \begin{align*}{\color{blue}HL \cong}\end{align*} |
\begin{align*}{\color{red}\overline{AD} \cong \overline{DC}}\end{align*} | CPCTC (corresponding parts of congruent triangles must be congruent) |
PROOF B:
\begin{align*}{\color{blue}\triangle ABD}\end{align*} looks to be the same size and shape as \begin{align*}{\color{blue}\triangle CBD}\end{align*}, so the two triangles are congruent. \begin{align*}{\color{red}\overline{AD} \cong \overline{DC}}\end{align*} because they are corresponding segments and corresponding parts of congruent triangles must be congruent.
PROOF A is incorrect because it is missing steps. You can't say that the two triangles are congruent by \begin{align*}HL \cong\end{align*} without having shown that all the parts of the \begin{align*}HL\end{align*} criteria have been met (congruent leg pair, congruent hypotenuse pair, right triangles). Be careful when writing proofs that you don't skip over steps, even if the steps seem obvious.
PROOF B is incorrect because it did not convincingly explain why the two triangles have to be congruent. Looking congruent is not a good enough reason. For proving triangles are congruent, there are five triangle congruence criteria to use. If you don't have enough information to use one of those five criteria, you can't prove that the triangles are congruent.
Remember, your goal when writing a proof is to convince everyone else that what you are trying to show is true actually is true. If you skip steps or use reasons that aren't convincing, other people won't believe your proof.
Examples
Example 1
Earlier, you were asked why is it important to learn how to prove geometry concepts and theorems yourself.
Most of the geometry concepts and theorems that are taught at the high school level today were first discovered and proved by mathematicians such as Euclid thousands of years ago. Given that these geometry concepts and theorems have been known to be true for thousands of years, why is it important that you learn how to prove them for yourself?
There are many reasons why it is valuable to learn to write proofs for yourself. Even though all of the theorems you will learn in geometry have already been proven, mathematicians today are working on trying to prove new ideas that will hopefully help to advance science/technology/medicine. Writing proofs in geometry class allows you to see what proofs are all about and practice writing them. That way, when you someday want to prove something new, you can feel confident in your proof writing abilities.
Writing proofs is all about logic. If you get good at writing proofs, this logical thinking can transfer to other subjects. Writing a persuasive essay about any topic is very similar to writing a paragraph proof. Knowing how to persuade others to believe your way of thinking can be very helpful in many careers and life in general.
Given: \begin{align*}C\end{align*} is the midpoint of \begin{align*}\overline{BE}\end{align*} and of \begin{align*}\overline{AD}\end{align*}. \begin{align*}\angle ACB \cong \angle DCE\end{align*}.
Prove: \begin{align*}\overline{AB} \cong \overline{DE}\end{align*}
No matter which style of proof you use, before starting to write you should brainstorm what you will say in your proof. Start by looking at the given information and thinking about what you know based on each given fact.
- The fact that \begin{align*}C\end{align*} is a midpoint means it is right in the middle of the two line segments. This means there are two pairs of segments that must be congruent. Mark these congruent segments on the diagram as you brainstorm. This will help you to keep track of what you know!
- You also are given that \begin{align*}\angle ACB \cong \angle DCE\end{align*}. This should be marked on the diagram as well.
Next think about what other conclusions you can make based on what you have now marked on the diagram. You have SAS \begin{align*}\cong\end{align*} criteria marked, so you can say that the two triangles are congruent. This will allow you to be able to say that \begin{align*}\overline{AB} \cong \overline{DE}\end{align*}, because they are corresponding parts of the triangles.
Once you have thought through the proof and your approach, start writing. In all proofs, the statements have been written in red and the reasons have been written in blue.
Example 2
Write a paragraph proof that shows that \begin{align*}\overline{AB} \cong \overline{DE}\end{align*}.
\begin{align*}{\color{red}C}\end{align*} is the midpoint of \begin{align*}{\color{red}\overline{BE}}\end{align*} and \begin{align*}{\color{red}\overline{AD}}\end{align*} because it is given information. This means that \begin{align*}{\color{red}\overline{AC} \cong \overline{CD}}\end{align*} and \begin{align*}{\color{red}\overline{EC} \cong \overline{CB}}\end{align*}, because midpoints divide segments into two congruent segments. Also, \begin{align*}{\color{red}\angle ACB \cong \angle DCE}\end{align*} because it is given information. \begin{align*}{\color{red}\Delta ACB \cong \Delta DCE}\end{align*} by \begin{align*}{\color{blue}SAS \cong}\end{align*} because they are triangles with two pairs of corresponding sides congruent and included angles congruent. \begin{align*}{\color{red}\overline{AB} \cong \overline{DE}}\end{align*} because they are corresponding segments and corresponding parts of congruent triangles must be congruent.
Example 3
Write a two-column proof that shows that \begin{align*}\overline{AB} \cong \overline{DE}\end{align*}.
Statements |
Reasons |
\begin{align*}{\color{red}C}\end{align*} is the midpoint of \begin{align*}{\color{red}\overline{BE}}\end{align*} and \begin{align*}{\color{red}\overline{AD}}\end{align*} |
Given |
\begin{align*}{\color{red}\overline{AC} \cong \overline{CD}}\end{align*} and \begin{align*}{\color{red}\overline{EC} \cong \overline{CB}}\end{align*} |
definition of midpoint |
\begin{align*}{\color{red}\angle ACB \cong \angle DCE}\end{align*} |
Given |
\begin{align*}{\color{red}\Delta ACB \cong \Delta DCE}\end{align*} | \begin{align*}{\color{blue}SAS \cong}\end{align*} |
\begin{align*}{\color{red}\overline{AB} \cong \overline{DE}}\end{align*} |
CPCTC |
Example 4
Write a flow diagram proof that shows that \begin{align*}\overline{AB} \cong \overline{DE}\end{align*}.
Review
1. What’s the difference between a postulate and a theorem?
2. What are the two main components of any proof?
3. What does it mean when a reason in a proof is “given”?
4. What should the last line/sentence/box for any proof be?
5. What are three styles of proof?
For 6-8, consider the proof below.
Given triangles \begin{align*}\Delta ACB\end{align*} and \begin{align*}\Delta ACD\end{align*} as marked, prove that \begin{align*}\overline{AB} \cong \overline{AD}\end{align*}.
Statements |
Reasons |
\begin{align*}???\end{align*} |
Given |
\begin{align*}\angle BAC \cong \angle DAC\end{align*} |
Given |
\begin{align*}\overline{AC} \cong \overline{AC}\end{align*} | ??? |
\begin{align*}\Delta ACB \cong \Delta ACD\end{align*} | ??? |
\begin{align*}???\end{align*} |
CPCTC (corresponding parts of congruent triangles must be congruent) |
6. Fill in the missing statements and reasons.
7. Rewrite this proof as a paragraph proof.
8. Rewrite this proof as a flow diagram proof.
For 9-11, consider the proof below.
Given: Circle \begin{align*}G\end{align*} with center \begin{align*}G\end{align*}. \begin{align*}\angle HGI \cong \angle JGK\end{align*}.
Prove: \begin{align*}\Delta HGI \cong \Delta JGK\end{align*}
_______________________ because it is given information. Point \begin{align*}G\end{align*} is the center of the circle because _______________________. \begin{align*}\overline{HG}\end{align*}, \begin{align*}\overline{GI}\end{align*}, \begin{align*}\overline{GJ}\end{align*}, \begin{align*}\overline{GK}\end{align*} are all radii of the circle, because they are segments that connect the center of the circle with the circle. \begin{align*}\overline{HG} \cong \overline{GK}\end{align*} and \begin{align*}\overline{GI} \cong \overline{GJ}\end{align*} because all ______ are congruent. \begin{align*} \Delta HGI \cong \Delta JGK\end{align*} _______ because they are triangles with two pairs of corresponding sides congruent and included angles congruent.
9. Fill in the blanks.
10. Rewrite this proof as a two-column proof.
11. Rewrite this proof as a flow diagram proof.
For 12-14, consider the proof below.
Given: Square \begin{align*}ABCD\end{align*}
Prove: \begin{align*}\Delta ABD \cong \Delta CBD\end{align*}
12. Fill in the missing boxes/reasons.
13. Rewrite this proof as a paragraph proof.
14. Rewrite this proof as a two-column proof.
15. Give an example of a real life situation where being able to persuade someone else that something is true would be helpful.
Review (Answers)
To see the Review answers, open this PDF file and look for section 4.1.