Introduction to Geometry Proofs This PowerPoint is meant to used in class at a math station. In my classroom, I have a mini-lab that has one student computer. I also have students work on there laptops and this PowerPoint could be posted to my website for multiple student access. An alternative use could be to have students study for a quiz or test on proofs and to study in groups by following through and checking the different types of proofs presented in the slides. My student population consists of 9th and 10th graders enrolled in Honors Geometry, but this could be used for any high school Geometry class. Some direct instruction would have occurred prior to this presentation as the students would need knowledge of some of the postulates and theorems related to lines an angles to be able to complete and justify the proofs.

Proof Vocabulary Axiom Postulate Theorem Click here to look up these words on Merriam Webster’s website. Write them in your Geometry notebook in your own words. Look in the appendices in your textbook and find an example of each.

Logical Argument in Algebra Click here to watch an example. Given x + y = 60 Given x = 5 Prove y = 55 Use your algebra knowledge to write a proof. Justify each step you write.

Algebra Proof Solution Follow the steps. x + y = 60 x = 5 5 + y = 60 y = 55 Justify the steps. Given Substitution Property of Equality Subtraction Property of Equality The last step in the left hand column is lowered to line it up with the reason in the right hand column.

Types of Geometry Proof Vocabulary cards are at Classzone.com 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. 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 Proof Given BA ┴ BC Prove angle 3 and 4 are complementary A D 3 4 B C Because BA ┴ BC, angle ABC is a ________ and the measure = _______. According to the ________ Postulate, the measure of angle 3 + the measure of angle 4 = the measure of angle ABC. So, by the substitution property of equality, ________ + ________ = ________. By definition, angle 3 and angle 4 are complementary.

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 angles 5 and 6 are a linear pair. measure of 5 = measure of 6 angle 5 is a right angle

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.

Two Column Proofs Ask Dr. Math is a great place to start. 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 Proof Fill in the blanks to complete the proof of the Reflexive Property of the Congruence of Angles. Statements A is an angle. Measure of A = Measure of A Angle A is congruent to Angle A Reasons ______________________

Two Column Proof Check your solution for the proof of the Reflexive Property of the Congruence of Angles. 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

One last 2 Column Proof Statements Reasons n m 2 3 1 2 3 1 Complete the following proof by filling in the blanks. Given: Angle 1 and Angle 2 are supplementary Prove: 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) ______________________

One last 2 Column Proof Statements Reasons n m 2 3 1 2 3 1 Check your work to see how well you are doing. Given: Angle 1 and Angle 2 are supplementary Prove: 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