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Introduction to Two-Column Proofs Lesson 1.7 Geometry as a Language Which is more important to a problem, the statement of the problem or the given diagram? Explain. Things to Assume Straight lines and angles Collinearity of points Betweenness of points Things to definitely NOT Assume Right angles Congruency Relative s size of segments or angles Postulate: If a statement is a postulate, then it is an unproved assumption. Theorem: If a statement is a theorem then it can be proved. What are examples of postulates we have covered so far? Why do we need them? What do you notice about the structure of each of the statements? If___________, then__________. Conditional Statement: If a statement is a conditional statement , then it is written in the form “if…then…” Parts of a conditional statement • Hypothesis: the “if” part of the statement • Conclusion: the “then” part of the statement Hypothesis: You don’t study Conclusion: You won’t pass your exam Write the following statements as conditional statements: 1. Expensive cars are reliable 2. Right angles are congruent. If it’s a theorem, then… What does a proof look like Reason Statement 1. 2. 3. 4. 5. 6. This is where the equations belong. Where we want to see what we want to prove GEOMETRIC LANGUAGE 1. 2. 3. 4. 5. 6. The reasoning and the “why” for each of the Statements What allows you to write each of these statements ENGLISH LANGUAGE Theorem: If two angles are right angles, then they are congruent Statement 1. Two angles are right angles 2. 𝑚⊾ = 90 3. all right angles are congruent Reason 1. Given 2. Definition of Right Angles 3. Definition of Congruency Things to know about proofs 1. Once a theorem is proven, it can be used to prove others (it does not have to be proven again) 2. It’s simple for something to be proven and called a theorem, but only things that are going to be consistently used will classify as theorems 3. Most theorems connect different definitions like congruency and measure (definitions are used to prove theorems) 1. Given: A is a right angle C is a right angle Prove: A C Statement Reason Given: Diagram as shown Prove: GKJ IKH Statement Reason 1.Given: Diagram as shown Prove: KHI JHG 2. Given: mIGJ 50 mJGK 40 L is a right angle Prove: IGK L 3. Given: GI 2.5cm JK 2.5cm Prove: GI JK