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Unit 2.2 Proofs NAME_______________________ PERIOD___ Standard: BellWork CO.9 Find the perimeter and area of the figures below. 1. 2. Objective: To complete two column proofs. 3.2 NOTES 3. Proofs In mathematics, a _PROOF__ is a logical argument that uses a _HYPOTHESIS_ to prove a conjecture. Once the conjecture is proved, it is called a theorem. Each STATEMENT in a proof must follow _FACTS from what has come before and must have a _REASON_ to support it. The table states some properties of equality that you seen in Algebra. PROPERTY EXAMPLE ADDITION PROPERTY Properties SUBTRACTION PRPOERTY of Equality MULTIPLICATION PROPERTY DIVISION PROPERTY INVERSE PROPERTY TRANSITIVE PROPERTY SYMMETRIC PROPERTY REFLEXIVE PROPERTY The same algebraic properties of equality can apply to geometry with segments and angles. Congruence Segments and Angles Congruent line segments: _____________ Congruent angles: _____________ Properties of Congruence Congruence of Segments Congruence of Angles Example 1 Identify the property that justifies the property. a. Y Y (STATEMENT) REASON?___________________ b. If ST UV and UV WX, then ST WX (STATEMENT) c. If FG JK, then JK FG (STATEMENT) REASON?________ REASON?___________________ Checkpoint Identify the property that justifies the property. a. IF E I and I O, then E O (STATEMENT) b. IF ABC DEF, then DEF ABC (STATEMENT) c. BC BC (STATEMENT) REASON?_________ REASON?___________ REASON?___________________ Example 2 Given: Statements 1. ____ + ____ = ____ Reasons 1. Checkpoint Given: Statements 1. ____ and ____ are supplementary angles 2. ____ + ____ = ____° Reasons 1. 2. Example 3 W, X, Y, and Z are collinear as shown in the figure. 2 COLUMN PROOFS Given: X is the midpoint of WY, Y is the midpoint of XZ Prove: WX = YZ Statements 1. X is the midpoint of WY, Y is the midpoint of XZ 2. WX = XY and XY = YZ 3. XY = XY 4. WX = YZ Checkpoint Reasons 1. 2. 3. 4. Given: Prove: AB = 17 Statements 1. ____ + ____ = ____ 2. AB + 18 = 35 3. AB = 17 Reasons 1. 2. 3. Example 4 Given: Prove: mVXW = mZXY Statements 1. VXW and ZXY are formed by intersecting lines 2. VXW and ZXY are vertical angles 3. VXW and WXZ are a linear pair; WXZ and ZXY are a linear pair 4. VXW and WXZ are supplementary 5. VXW + WXZ = 180° 6. WXZ and ZXY are supplementary 7. WXZ + ZXY = 180° 8. mVXW + mWXZ = mWXZ + mZXY 9. mVXW = mZXY Reasons 1. 2. 3. 4. 5. 6. 7. 8. 9.