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Reasoning in Algebra & Geometry
Skill 12
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Objective HSG-CO.9/10/11: Students are responsible for using and connecting reasoning in algebra and geometry.
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Definitions A proof is a convincing argument that uses deductive reasoning and logically shows why a conjecture is true.
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Algebra Properties If π=π, then π+π=π+π If π=π, then πβπ=πβπ
Addition Property of Equality If π=π, then π+π=π+π Subtraction Property of Equality If π=π, then πβπ=πβπ Multiplication Property of Equality If π=π, then πβπ=πβπ Division Property of Equality If π=π and πβ π, then π π = π π
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Algebra Properties π=π If π=π, then π=π If π=π and π=π, then π=π.
Reflexive Property of Equality π=π Symmetric Property of Equality If π=π, then π=π Transitive Property of Equality If π=π and π=π, then π=π. Substitution Property of Equality If π=π, then π can replace π in any expression.
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π π+π =ππ+ππ π πβπ =ππβππ Algebra Properties
Distributive Property of Equality π π+π =ππ+ππ π πβπ =ππβππ
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Geometry Properties π¨π© β
π¨π© β π¨β
β π¨ If π¨π© β
πͺπ« , then πͺπ« β
π¨π© .
Reflexive Property of Equality π¨π© β
π¨π© β π¨β
β π¨ Symmetric Property of Equality If π¨π© β
πͺπ« , then πͺπ« β
π¨π© . If β π¨β
β π©, then β π©β
β π¨. Transitive Property of Equality If π¨π© β
πͺπ« and πͺπ« β
π¬π , then π¨π© β
π¬π . If β π¨β
β π© and β π©β
β πͺ, then β π¨β
β πͺ.
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Example 1; Justifying when Solving Equations
(a) What is the value of x in the picture? Justify each step. 1) β π΄ππ and β πππΆ are supplementary. 1) Linear Pair Postulate 2) πβ π΄ππ+πβ πππΆ=180 2) Def. of Supplementary 3) 2π₯+30+π₯=180 3) Substitution Property 4) 3π₯+30=180 4) Combine Terms 5) 3π₯=150 ππ+ππ Β° π Β° π¨ πΆ πͺ π΄ 5) Subtraction Property 6) π₯=50 6) Division Property Q.E.D.
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Example 1; Justifying when Solving Equations
(b) What is the value of x in the picture? Justify each step. 1) β π
π΄π΅β
β π΅π΄π 1) Def. of β Bisector 2) πβ π
π΄π΅=πβ π΅π΄π 2) Def. of β
β βs 3) π₯=2π₯β75 3) Substitution Property 4) π₯+75=2π₯ 4) Addition Property 5) π₯=75 5) Subtraction Property πΉ π¨ π΅ π© πΒ° ππβππ Β° Q.E.D.
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Example 2; Name the Property
What is the property that justifies going from the first step to the second step? (a) 2π₯+9=19 (b) β πβ
β π and β πβ
β πΏ 2π₯=10 β πβ
β πΏ Subtraction Property Transitive Property (c) πβ πΈ=πβ π πβ π=πβ πΈ Symmetric Property
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Example 3; Writing a Proof
Write a two column proof. 1) πβ 1=πβ 3 1) Given 2) πβ 2=πβ 2 2) Reflexive Property 3) πβ 1+πβ 2=πβ 3+πβ 2 3) Addition Property 4) πβ 1+πβ 2=πβ π΄πΈπΆ and πβ 2+πβ 3=πβ π·πΈπ΅ 4) β Addition Post. A B C D E 1 2 3 5) πβ π΄πΈπΆ=πβ π·πΈπ΅ 5) Substitution Property Q.E.D.
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#12: Reasoning in Algebra and Geometry
Questions? Summarize Notes Homework Worksheet Quiz
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