# Proofs

Warm-Up: 2. What is the difference between the Definition of a Linear Pair and the Linear Pair Theorem?

Today we will prove/justify:

Congruent Supplements Theorem

Vertical Angles Theorem

Corresponding Angles Postulate

Alternate Exterior Angles Theorem

Key Ideas for Proofs

You can only use substitution for equations

The linear pair theorem does not involve an equation!

The definition of supplementary involves an actual equation

You have to explicitly state the switch from congruence to equality, or equality to congruence

The transitive property is stated as: Two things that are equal or congruent to the same thing, are equal or congruent to each other.

Strategies for Writing Proofs

Restate the given information, and draw a conclusion using valid reasoning (a correct theorem, definition, postulate, or property).

State information given from the diagram, draw a conclusion using valid reasoning.

Use newly generated information (ideas you previously got from the diagram or given information) combined with a theorem/postulate/property/definition to get more new information.

Combine multiple pieces of information together.

When using equations, use substitution and other properties of equality to rearrange information to get new ideas.

Work Backwards: Keep the goal in mind: What are you trying to prove?  Is it about congruence, an equation, or just a new statement?  Can you imagine what the second to last step might look like?  What about before that?

Always ask yourself: What can I do with the information I have?  Don’t worry about if its right or not!  You need to give your mind some options of paths to take in a proof!  You will make mistakes, but that’s the only way you’ll get better!

Formulating a Proof

Recall the linear pair theorem:

Linear Pair Theorem- If two angles form a linear pair, then they are supplementary.

Note that the language was very general. It was talking about two angles, but the angles didn’t have names.  If you look back on our proof, we had to get slightly more specific, so we labeled the angles, and said “Suppose angle QRS and SRT are a linear pair.”

Congruent Supplements Theorem

• Def. of Supp.- Two angles are supplementary if and only if their measures add up to 180
• Subtraction Prop. Of Equality- If two things are equal to each other, then you can subtract the same thing from both sides of an equation.
• Substitution- If two things are equal to each other, you can substitute one for the other in any equation
• Definition of Congruent Angles- Two angles are congruent if and only if they have equal measures.

Vertical Angles Theorem

• Linear Pair Theorem- If two angles form a linear pair, then the angles are supplementary.
• Congruent Supplements Theorem- If two angles are supplements to the same angle, then those two angles are congruent.

Corresponding Angles Postulate

If a transversal crosses two parallel lines, then the corresponding angles are congruent.

Alternate Exterior Angles Theorem

• Vertical Angles Theorem- If two angles are vertical angles, then they are congruent.
• Corresponding Angles Postulate
• Transitive Property- If two things are congruent to the same thing, then they are congruent to each other. (This can be used for equality as well).
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