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Conditionals, Biconditionals, and Deductive Reasoning and algebraic properties Be able to write a converse of a conditional statement (Geometry and Spatial Sense, GR9,1) Be able to write a biconditional (Geometry and Spatial Sense, GR9,1) Use law of detachment and law of syllogism to draw conclusions (Geometry and Spatial Sense, GR9,1) Use algebraic properties to prove a statement (Geometry and spatial sense, GR9, 1) Lets begin with an OGT question…. The graph as shown represents the amount of money Sarah can earn at her part-time job Which of the following equations best represents the relationship between Sarah’s pay and the hours she works? A. y = 4x B. y = 6.5x C. y = 4x + 10 D. y = 6.5x + 10 If-Then Statements Hypothesis – If statement Conclusion – Then statement Identify the Hypothesis and Conclusion: ◦ If two lines are parallel, then they are coplanar Identify the Hypothesis and Conclusion: ◦ If two lines are parallel, then they are coplanar ◦ Hypothesis: Two lines are parallel ◦ Conclusion: They are coplanar Write the statement as a conditional ◦ An acute angle measures less than 90º Identify the Hypothesis and Conclusion: ◦ If two lines are parallel, then they are coplanar ◦ Hypothesis: Two lines are parallel ◦ Conclusion: They are coplanar Write the statement as a conditional ◦ An acute angle measures less than 90º ◦ If an angle is acute, then it measures less than 90 Converse The converse converse of of aaconditional conditional statement if • The statement is formed by exchanging the hypothesis and the formed by in switching the hypothesis and the conclusion the conditional. conclusion Example 3: in the conditional. ◦ Conditional- If a figure is a triangle, then it has three – angles. Conditional- If a figure is a triangle, then it has three angles. ◦ Converse- If a figure has three angles, then it is a triangle. – Converse- If a figure has three angles, then it is a triangle. If x = 5, then x + 15 = 20 ◦ Write the converse: If x = 5, then x + 15 = 20 ◦ Write the converse: ◦ If x + 15 = 20, then x = 5 ◦ Write the Biconditional: If x = 5, then x + 15 = 20 ◦ Write the converse: ◦ If x + 15 = 20, then x = 5 ◦ Write the Biconditional: ◦ X = 5 if and only if x + 15 = 20 Write two statements that form this biconditional: ◦ Lines are skew if and only if they are noncoplanar Write two statements that form this biconditional: ◦ Lines are skew if and only if they are noncoplanar ◦ If lines are skew, then they are noncoplanar ◦ If lines are noncoplanar, then they are skew If a conditional is true and its hypothesis is true – then its conclusion is true If p implies q is true and p is true then q is true Can you make a conclusion?… ◦ Every High School student likes music. ◦ Ling likes music If a conditional is true and its hypothesis is true – then its conclusion is true If p implies q is true and p is true then q is true Can you make a conclusion?… ◦ Every High School student likes music. Ling likes music ◦ If you are a high school student, then you like music If p implies q is true and q implies r is true, then p implies r is true P Q and QR, then PR If a quadrilateral is a square, then it contains 4 right angles. If a quadrilateral contains 4 right angles, then it is a rectangle A gardener knows that if it rains, the garden will be watered. It is raining. If you want to build a skyscraper, start with a good foundation. The foundation is a concrete form that supports the rest of the structure. The foundation of geometry is made of statements called postulates. These are accepted as true. Addition Property Subtraction Property Multiplication Property Division Property Reflexive Property Symmetric Property Transitive Property Substitution Property Distributive Property Fill in… 3x 2x + 1 AB + BC = AC 3x + 2x + 1 = 36 5x + 1 = 36 5x = 35 X= Simplify AC = 36 Images are copyright of, and used with permission from Clipart.com, © [2010], Jupiterimages Corporation, a wholly owned subsidiary of Jupiter media Corporation. All rights reserved.
