StatementsReasons
AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯Given
∠ADB and ∠CDB are right anglesGiven
ΔADB and ΔCDB are right trianglesdefinition of right triangles
BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯reflexive property
ΔADB≅ΔCDBHL≅
AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯CPCTC (corresponding parts of congurent triangles must be congurent)
Step-by-step explanation:
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 AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯. 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.
AB¯¯¯¯¯¯¯¯≅BC¯¯¯¯¯¯¯¯ because it is marked in the diagram. Also, ∠ADB and ∠CDB are both right angles because it is marked in the diagram. This means that △ADB and △CDB are right triangles because right triangles are triangles with right angles. Both triangles contain segment BD¯¯¯¯¯¯¯¯. BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯ because of the reflexive property that any segment is congruent to itself. △ADB≅△CDB by HL≅ because they are right triangles with a pair of congruent legs and congruent hypotenuses. AD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ 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.
