CP - Building Proofs Lesson

Building Proofs

A proof is an argument that something will always be true given the same information. A proof is not about the specific figures you have in front of you, but it is about all possible figures. The idea is, if I know the same information, no matter what the size of the segments or angles, I know this will always be true.

Now that you have your toolbox full, it is time to build some proofs. You already have seen and built simple proofs in the previous lessons. Now we are going to formalize what you have been doing. Each proof will have the same basic format.

Given: You start your proof by stating the information you have been given. The given is the list of facts about the situation that you have been told to start from. You do not have to prove these things. They are GIVEN.

Building Blocks: This is the main part of the proof, and it is the part where you will have to do the most thinking. In this part of the proof, you are taking the given information and using your tools to build your argument. The thought process here is, "Since I already know _____________________, I also know _________________________ because of _______________________.

Conclusion: Once you have put all of your pieces together and have built your argument, you are ready to conclude your proof. In this part you are basically saying, "Since I know all these things are true, I can say ______________________". This part of the proof should be the statement you were asked to prove.

Here is an example:

example of a proof (triangle with a mid-segment dividing into two right triangles)

The picture tells me that $LaTeX: \overline{AB}\cong\overline{BD}$ $\overline{AB}\cong\overline{BD}$ and $LaTeX: \overline{CB}\perp \overline{AD}$ $\overline{CB}\perp \overline{AD}$ . Since the segments are perpendicular, I know that $LaTeX: \angle ABC\cong \angle DBC$ $\angle ABC\cong \angle DBC$ because they are both right angles. I also know that $LaTeX: \overline{CB}\cong \overline{CB}$ $\overline{CB}\cong \overline{CB}$ because of the reflexive property. With these corresponding parts being congruent, I can now say that the triangles are congruent by the SAS Congruence Postulate.

This means that any time I have this same situation (congruent sides in the same place and same segments perpendicular) I know the triangles will be congruent by the same argument no matter what their shape or size.

The proof can also be written more quickly and easily using what is called a Two-Column Proof. Notice how the table below says the exact same thing as the paragraph above.

	Statement	Reason
Given	$LaTeX: \overline{AB}\cong \overline{BD}$ $\overline{AB}\cong \overline{BD}$	Given
Given	$LaTeX: \overline{CB}\perp \overline{AD}$ $\overline{CB}\perp \overline{AD}$	Given
Building the Argument	$LaTeX: \angle ABC\cong \angle DBC$ $\angle ABC\cong \angle DBC$	Right Angles are Congruent
Building the Argument	$LaTeX: \overline{CB}\cong \overline{CB}$ $\overline{CB}\cong \overline{CB}$	Reflexive Property
Conclusion	$LaTeX: \bigtriangleup ABC\cong \bigtriangleup DBC$ $\bigtriangleup ABC\cong \bigtriangleup DBC$	SAS Congruence Postulate

If you wanted to prove that any of the corresponding parts of the triangles are congruent, you could now be able to say that because Corresponding Parts of Congruent Triangles are Congruent. For example, if we wanted to prove that $LaTeX: \overline{CA}\cong \overline{CD}$ $\overline{CA}\cong \overline{CD}$ , the proof would look like this.

	Statement	Reason
Given	$LaTeX: \overline{AB}\cong \overline{BD}$ $\overline{AB}\cong \overline{BD}$	Given
Given	$LaTeX: \overline{CB}\perp \overline{AD}$ $\overline{CB}\perp \overline{AD}$	Given
Building the Argument	$LaTeX: \angle ABC\cong \angle DBC$ $\angle ABC\cong \angle DBC$	Right Angles are Congruent
	$LaTeX: \overline{CB}\cong \overline{CB}$ $\overline{CB}\cong \overline{CB}$	Reflexive Property
	$LaTeX: \bigtriangleup ABC\cong \bigtriangleup DBC$ $\bigtriangleup ABC\cong \bigtriangleup DBC$	SAS Congruence Postulate
Conclusion	$LaTeX: \overline{CA}\cong \overline{CD}$ $\overline{CA}\cong \overline{CD}$	CPCTC

image stating: "you must have already proven triangles congruent before using CPCTC!"

Look here for some more examples of building proofs.

Practice your proof building skills with the activity below.

Although many of the proofs you will do in this module deal with congruent triangles, there are some proofs that use similar triangles. The following proofs will show you what these proofs can look like.

Click here to download a handout that will help you walk through the following interactive. Links to an external site.

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