Reasoning in Algebra & Geometry Skill 12

Objective HSG-CO.9/10/11: Students are responsible for using and connecting reasoning in algebra and geometry.

Definitions A proof is a convincing argument that uses deductive reasoning and logically shows why a conjecture is true.

Algebra Properties If 𝒂=𝒃, then 𝒂+𝒄=𝒃+𝒄 If 𝒂=𝒃, then 𝒂−𝒄=𝒃−𝒄 Addition Property of Equality If 𝒂=𝒃, then 𝒂+𝒄=𝒃+𝒄 Subtraction Property of Equality If 𝒂=𝒃, then 𝒂−𝒄=𝒃−𝒄 Multiplication Property of Equality If 𝒂=𝒃, then 𝒂∙𝒄=𝒃∙𝒄 Division Property of Equality If 𝒂=𝒃 and 𝒄≠𝟎, then 𝒂 𝒄 = 𝒃 𝒄

Algebra Properties 𝒂=𝒂 If 𝒂=𝒃, then 𝒃=𝒂 If 𝒂=𝒃 and 𝒃=𝒄, then 𝒂=𝒄. Reflexive Property of Equality 𝒂=𝒂 Symmetric Property of Equality If 𝒂=𝒃, then 𝒃=𝒂 Transitive Property of Equality If 𝒂=𝒃 and 𝒃=𝒄, then 𝒂=𝒄. Substitution Property of Equality If 𝒂=𝒃, then 𝒃 can replace 𝒂 in any expression.

𝒂 𝒃+𝒄 =𝒂𝒃+𝒃𝒄 𝒂 𝒃−𝒄 =𝒂𝒃−𝒃𝒄 Algebra Properties Distributive Property of Equality 𝒂 𝒃+𝒄 =𝒂𝒃+𝒃𝒄 𝒂 𝒃−𝒄 =𝒂𝒃−𝒃𝒄

Geometry Properties 𝑨𝑩 ≅ 𝑨𝑩 ∠𝑨≅∠𝑨 If 𝑨𝑩 ≅ 𝑪𝑫 , then 𝑪𝑫 ≅ 𝑨𝑩 . Reflexive Property of Equality 𝑨𝑩 ≅ 𝑨𝑩 ∠𝑨≅∠𝑨 Symmetric Property of Equality If 𝑨𝑩 ≅ 𝑪𝑫 , then 𝑪𝑫 ≅ 𝑨𝑩 . If ∠𝑨≅∠𝑩, then ∠𝑩≅∠𝑨. Transitive Property of Equality If 𝑨𝑩 ≅ 𝑪𝑫 and 𝑪𝑫 ≅ 𝑬𝑭 , then 𝑨𝑩 ≅ 𝑬𝑭 . If ∠𝑨≅∠𝑩 and ∠𝑩≅∠𝑪, then ∠𝑨≅∠𝑪.

Example 1; Justifying when Solving Equations (a) What is the value of x in the picture? Justify each step. 1) ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 are supplementary. 1) Linear Pair Postulate 2) 𝑚∠𝐴𝑂𝑀+𝑚∠𝑀𝑂𝐶=180 2) Def. of Supplementary 3) 2𝑥+30+𝑥=180 3) Substitution Property 4) 3𝑥+30=180 4) Combine Terms 5) 3𝑥=150 𝟐𝒙+𝟑𝟎 ° 𝒙 ° 𝑨 𝑶 𝑪 𝑴 5) Subtraction Property 6) 𝑥=50 6) Division Property Q.E.D.

Example 1; Justifying when Solving Equations (b) What is the value of x in the picture? Justify each step. 1) ∠𝑅𝐴𝐵≅∠𝐵𝐴𝑁 1) Def. of ∠ Bisector 2) 𝑚∠𝑅𝐴𝐵=𝑚∠𝐵𝐴𝑁 2) Def. of ≅ ∠’s 3) 𝑥=2𝑥−75 3) Substitution Property 4) 𝑥+75=2𝑥 4) Addition Property 5) 𝑥=75 5) Subtraction Property 𝑹 𝑨 𝑵 𝑩 𝒙° 𝟐𝒙−𝟕𝟓 ° Q.E.D.

Example 2; Name the Property What is the property that justifies going from the first step to the second step? (a) 2𝑥+9=19 (b) ∠𝑂≅∠𝑊 and ∠𝑂≅∠𝐿 2𝑥=10 ∠𝑂≅∠𝐿 Subtraction Property Transitive Property (c) 𝑚∠𝐸=𝑚∠𝑇 𝑚∠𝑇=𝑚∠𝐸 Symmetric Property

Example 3; Writing a Proof Write a two column proof. 1) 𝑚∠1=𝑚∠3 1) Given 2) 𝑚∠2=𝑚∠2 2) Reflexive Property 3) 𝑚∠1+𝑚∠2=𝑚∠3+𝑚∠2 3) Addition Property 4) 𝑚∠1+𝑚∠2=𝑚∠𝐴𝐸𝐶 and 𝑚∠2+𝑚∠3=𝑚∠𝐷𝐸𝐵 4) ∠ Addition Post. A B C D E 1 2 3 5) 𝑚∠𝐴𝐸𝐶=𝑚∠𝐷𝐸𝐵 5) Substitution Property Q.E.D.

#12: Reasoning in Algebra and Geometry Questions? Summarize Notes Homework Worksheet Quiz