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 + 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

3. Justify Steps When SolvingM C A x°
(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.

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

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

6. 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

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

8. 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

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 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

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

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