# 2.5 Reasoning in Algebra and Geometry

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2.5 Reasoning in Algebra and Geometry
Objective: To connect reasoning in algebra and geometry

Properties of Equality
Let a, b, and c be any real number. Addition If a = b, then a + c = b + c Subtraction If a = b, then a – c = b – c Multiplication If a = b, then a ∙ c = b ∙ c Division If a = b and c ≠ 0, then a/c = b/c Reflexive a = a Symmetric 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 expression Distributive a(b + c) = ab + ac a(b – c) = ab – ac

Justify Steps When Solving
M C A (2x + 30)° What is the value of x? ∠𝐴𝑂𝑀 and ∠𝑀𝑂𝐶 Angles that form a linear are supplementary pair are supplementary 𝑚∠𝐴𝑂𝑀 + 𝑚∠𝑀𝑂𝐶 = Definition of suppl. Angles (2x + 30) + x = Substitution Property 3x + 30 = Distributive Property 3x = Subtraction Prop. of Eq. X = Division Prop. of Eq.

Try again. What is the value of x? Given: 𝐴𝐵 bisects ∠𝑅𝐴𝑁 B R N A x°

Extra Practice D F E C (2x – 15)° What is the value of x?

Equality and Congruence
Reflexive Property 𝐴𝐵 ≅ 𝐴𝐵 ∠𝐴≅∠𝐴 Symmetric Property If 𝐴𝐵 ≅ 𝐶𝐷 , then 𝐶𝐷 ≅ 𝐴𝐵 If ∠𝐴≅∠𝐵, then ∠𝐵≅∠𝐴 Transitive Property If 𝐴𝐵 ≅ 𝐶𝐷 and 𝐶𝐷 ≅ 𝐸𝐹 , then 𝐴𝐵 ≅ 𝐸𝐹 If ∠𝐴≅∠𝐵 and ∠𝐵≅∠𝐶, then ∠𝐴≅∠𝐶 If ∠𝐵≅∠𝐴 and ∠𝐵≅∠𝐶, then ∠𝐴≅∠𝐶

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 = 10 B. ∠𝑂≅∠𝑊 and ∠𝑊≅∠𝐿 ∠𝑂≅∠𝐿 C. 𝑚∠𝐸=𝑚∠𝑇 𝑚∠𝑇=𝑚∠𝐸

Proof Proof – convincing argument that uses deductive reasoning; logically shows why a conjecture is true Two-column proof – lists each statement on the left and the justification/reason on the right

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?

𝑚∠1=𝑚∠3 Given 𝑚∠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∠𝐷𝐸𝐵 𝑚∠𝐴𝐸𝐶=𝑚∠𝐷𝐸𝐵 Substitution Prop

Again… Given: 𝐴𝐵 ≅ 𝐶𝐷 Prove: 𝐴𝐶 ≅ 𝐵𝐷 A C B D

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