Introduction to Geometry Proofs

1. Introduction to Geometry Proofs

2. Proof Vocabulary • Postulate • Theorem • Postulate: Rules that are accepted without proof • Theorem: A true statement that follows as a result of other true statements.

3. Logical Argument in Algebra • Given x + y = 60 • Given x = 5 • Prove y = 55 • Use your algebra knowledge to write a proof. Justify each step you write.

4. Follow the steps. x + y = 60 x = 5 5 + y = 60 y = 55 Justify the steps. Given Given Substitution Property of Equality Subtraction Property of Equality Algebra Proof Solution

5. Types of Geometry Proof • Two Column Proofs • This third example is the most commonly used type of proof. We will focus on this type of proof in class. • Paragraph Proofs • Find an example in your textbook and read it to your table partner. • Flow Chart Proofs • Find an example in your textbook and copy the steps into your Geometry notebook.

6. Statements In this column we write the logical steps that lead us to the end result. Reasons For each statement, we must use a postulate or theorem that supports the statement. Two Column Proofs

7. Statements A is an angle. Measure of A = Measure of A Angle A is congruent to Angle A Reasons ______________________ ______________________ ______________________ Two Column ProofFill in the blanks to complete the proof of the Reflexive Property of the Congruence of Angles.

8. Statements A is an angle. Measure of A = Measure of A Angle A is congruent to Angle A Reasons Given Reflexive Property of Equality Definition of Congruent Angles Two Column Proof Check your solution for the proof of the Reflexive Property of the Congruence of Angles.

9. Another 2 Column Proof n m 2 3 1 Complete the following proof by filling in the blanks. Given: Angle 1 and Angle 2 are supplementaryProve: n is parallel to m Statements Reasons 1) Angle 1 and Angle 2 are supplementary. 1)______________________ 2) Angle 1 and Angle 3 are a linear pair. 2)______________________ 3)_____________________________ 3) Linear Pair Postulate 4)_____________________________ 4) Congruent Supplements Theorem 5) n is parallel to m. 5) ______________________

10. One last 2 Column Proof n m 2 3 1 Check your work to see how well you are doing. Given: Angle 1 and Angle 2 are supplementaryProve: n is parallel to m Statements Reasons 1) Angle 1 and Angle 2 are supplementary. 1) Given 2) Angle 1 and Angle 3 are a linear pair. 2) Definition of Linear Pair 3) Angle 1 and Angle 3 are supplementary. 3) Linear Pair Postulate 4) Angle 2 is congruent to Angle 3 4) Congruent Supplements Theorem 5) n is parallel to m. 5) Corresponding Angles Converse

11. Paragraph Proof • See page 102 (bottom of page) “Paragraph Proof” • A proof that can be written in paragraph form is called a paragraph proof. • See example on bottom of page 102

12. Flow Chart Proofs j 5 6 k Given: angle 5 is congruent to angle 6, angle 5 and 6 are a linear pair. Prove: j is perpendicular to k. Put the following statements in the proper order to complete the proof. When you have finished, compare your solution to your partners. j is perpendicular to k 2(measure of 5) = 180° measure of 5 = 90° angle 5 is congruent to angle 6 measure of 5 + measure of 6 = 180° measure of 5 + measure of 5 = 180° angle 5 and angle 6 are supplementary measure of 5 = measure of 6 angles 5 and 6 are a linear pair. angle 5 is a right angle

13. Flow Chart Proofs j 5 6 k Given: angle 5 is congruent to angle 6, angle 5 and 6 are a linear pair. Prove: j is perpendicular to k. • Now that you have the statements in a logical order, add a reason to each statement. Reasons are based on properties, postulates and theorems. • When you have finished, bring your paper to the teacher. You will be asked to explain your reasoning.