Presentation on theme: "2.5 Reasoning in Algebra and Geometry"— Presentation transcript:
1 2.5 Reasoning in Algebra and Geometry Objective:To connect reasoning in algebra and geometry
2 Properties of Equality Let a, b, and c be any real number.Addition If a = b, then a + c = b + cSubtraction If a = b, then a – c = b – cMultiplication If a = b, then a ∙ c = b ∙ cDivision If a = b and c ≠ 0, then a/c = b/cReflexive a = aSymmetric If a = b, then b = a.Transitive If a = b and b = c, then a = c.Substitution If a = b, then b can replace a in any expressionDistributive a(b + c) = ab + ac a(b – c) = ab – ac
3 Justify Steps When Solving MCAx°(2x + 30)°What is the value of x?∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 Angles that form a linearare supplementary pair are supplementary𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = Definition of suppl. Angles(2x + 30) + x = Substitution Property3x + 30 = Distributive Property3x = Subtraction Prop. of Eq.X = Division Prop. of Eq.
4 Try again. What is the value of x? Given: 𝐴𝐵 bisects ∠𝑅𝐴𝑁 B R N A x°
5 Extra PracticeDFECx°(2x – 15)°What is the value of x?
6 Equality and Congruence Reflexive Property𝐴𝐵 ≅ 𝐴𝐵 ∠𝐴≅∠𝐴Symmetric PropertyIf 𝐴𝐵 ≅ 𝐶𝐷 , then 𝐶𝐷 ≅ 𝐴𝐵If ∠𝐴≅∠𝐵, then ∠𝐵≅∠𝐴Transitive PropertyIf 𝐴𝐵 ≅ 𝐶𝐷 and 𝐶𝐷 ≅ 𝐸𝐹 , then 𝐴𝐵 ≅ 𝐸𝐹If ∠𝐴≅∠𝐵 and ∠𝐵≅∠𝐶, then ∠𝐴≅∠𝐶If ∠𝐵≅∠𝐴 and ∠𝐵≅∠𝐶, then ∠𝐴≅∠𝐶
7 Using Equality and Congruence What property of equality or congruence is used to justify going from the first statement to the second statement?2x + 9 = 19 2x = 10B. ∠𝑂≅∠𝑊 and ∠𝑊≅∠𝐿 ∠𝑂≅∠𝐿C. 𝑚∠𝐸=𝑚∠𝑇 𝑚∠𝑇=𝑚∠𝐸
8 ProofProof – convincing argument that uses deductive reasoning; logically shows why a conjecture is trueTwo-column proof – lists each statement on the left and the justification/reason on the right
9 Here we go… Given: 𝑚∠1=𝑚∠3 Prove: 𝑚∠𝐴𝐸𝐶=𝑚∠𝐷𝐸𝐵 What do we know? What do we need to do?What is our plan?
10 𝑚∠1=𝑚∠3 Given𝑚∠2=𝑚∠2 Reflexive Prop of =𝑚∠1+𝑚∠2=𝑚∠3+𝑚∠2 Addition Prop of =𝑚∠1+𝑚∠2=𝑚∠𝐴𝐸𝐶 Angle Add. Post. 𝑚∠3+𝑚∠2=𝑚∠𝐷𝐸𝐵𝑚∠𝐴𝐸𝐶=𝑚∠𝐷𝐸𝐵 Substitution Prop