A formal proof uses general statements along with algebra rules to prove a conjecture is true or false. Variables like x or a instead of specific number values, are used to represent quantities. Watch the following video to learn how to construct a formal proof.
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Using a series of general statements to define a specific end-result is known as deductive reasoning. Whereas using specific examples, or observations, that lead to a general statement, or conjecture, is an example of inductive reasoning. It is important to verify whether a conjecture is true using logic and reasoning. In this video, we will focus our attention on deductive reasoning. Deductive reasoning in Geometry often requires logical arguments called proofs, to be written in two columns. In one column we make our statements, or arguments, and in the second column we provide reasons, or explanations for each statement in the argument. Some Geometry students spend much of the year learning about Proofs, and we will spend a fair amount of time learning them ourselves. Here are two examples of proofs that follow clear and logical steps to show that a conjecture is true.
Our first example of a proof is really just a simple algebra problem. In Algebra, we were taught to solve for the unknown variable. In this case, we would like to prove that x equals three, if the quantity of three times x plus five divided by two is seven. All two-column proofs require us to write a series of logical statements supported by reasons for those statements. Allow me to demonstrate. First, I will begin my argument with a piece of known information, sometimes call the condition. Three x plus five over two equals seven is the given information. Next you might find that three x plus five equals fourteen because you multiplied both sides of the equation by two. Then you might find that three x is equal to 9 because you subtracted both sides by five. Finally you see that x is equal to three after dividing both sides of the equation by 3. Our proof is complete since our final statement matches our conjecture that x was equal to three.
Our next example is commonly seen in a Geometry class. Pay close attention to the structure of the proof – the mathematics that supports the proof will come to you in time. We must prove that two triangles are congruent, that is the exact same shape and size, with only two pieces of given information. In order to begin this two-column proof, we will write the given information on the first two lines. D is the midpoint of line segment A-C and triangle A-B-C is isosceles; given information. Next, I will say that segment A-D is congruent to segment C-D because of the definition of Midpoint. The first piece of given information was very helpful. Now, let us point out that segment B-D is congruent to itself by the reflexive property. And finally, I would like to make use of the other piece of given information; segment A-B must be congruent to segment C-B. The definition of isosceles triangles indicates two sides must be congruent. Now that I have shown three pairs of sides in the given triangles are congruent, then I know both triangles are completely congruent – angles too – by the side-side-side congruence postulate. Our proof is complete since our final statement matches our conjecture that triangle A-B-D is congruent to triangle C-B-D.
This concludes our video on proof examples that describe the process for proving a conjecture. Feel free to refer the examples in this video to help you construct your own proofs.
Good luck!
Question
How is a formal proof written? Describe the process you should use to construct a formal proof.
