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Five-Minute Check (over Lesson 2–5) CCSS Then/Now New Vocabulary Key Concept: Properties of Real Numbers Example 1: Justify Each Step When Solving an Equation Example 2: Real-World Example: Write an Algebraic Proof Example 3: Write a Geometric Proof Lesson Menu

A. A line contains at least two points. In the figure shown, A, C, and lie in plane R, and B is on . Which option states the postulate that can be used to show that A, B, and C are collinear? A. A line contains at least two points. B. A line contains only two points. C. A line contains at least three points. D. A line contains only three points. 5-Minute Check 1

In the figure shown, A, C, and lie in plane R, and B is on In the figure shown, A, C, and lie in plane R, and B is on . Which option states the postulate that can be used to show that lies in plane R? A. Through two points, there is exactly one line in a plane. B. Any plane contains an infinite number of lines. C. Through any two points on the same line, there is exactly one plane. D. If two points lie in a plane, then the entire line containing those points lies in that plane. 5-Minute Check 2

In the figure shown, A, C, and lie in plane R, and B is on In the figure shown, A, C, and lie in plane R, and B is on . Which option states the postulate that can be used to show that A, H, and D are coplanar? A. Through any two points on the same line, there is exactly one plane. B. Through any three points not on the same line, there is exactly one plane. C. If two points lie in a plane, then the entire line containing those points lies in that plane. D. If two lines intersect, then their intersection lies in exactly one plane. 5-Minute Check 3

In the figure shown, A, C, and lie in plane R, and B is on In the figure shown, A, C, and lie in plane R, and B is on . Which option states the postulate that can be used to show that E and F are collinear? A. Through any two points, there is exactly one line. B. A line contains only two points. C. If two points lie in a plane, then the entire line containing those points lies in that plane. D. Through any two points, there are many lines. 5-Minute Check 4

In the figure shown, A, C, and lie in plane R, and B is on In the figure shown, A, C, and lie in plane R, and B is on . Which option states the postulate that can be used to show that intersects at point B? A. The intersection point of two lines lies on a third line, not in the same plane. B. If two lines intersect, then their intersection point lies in the same plane. C. The intersection of two lines does not lie in the same plane. D. If two lines intersect, then their intersection is exactly one point. 5-Minute Check 5

Which of the following numbers is an example of an irrational number? D. 34 5-Minute Check 6

Preparation for G.CO.9 Prove theorems about lines and angles. Content Standards Preparation for G.CO.9 Prove theorems about lines and angles. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. CCSS

Use algebra to write two-column proofs. You used postulates about points, lines, and planes to write paragraph proofs. Use algebra to write two-column proofs. Use properties of equality to write geometric proofs. Then/Now

algebraic proof two-column proof Vocabulary

Algebraic Properties

What are Algebraic Properties of Equality? In mathematics equality is a relationship between two mathematical expressions, asserting that the quantities have the same value. Algebraic Properties of Equality help us to justify how we solve equations and inequalities.

Properties of Equality Definition Examples 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, then a / c = b / c (c≠0) If x = 12, then x + 3 = 12 + 3 then x – 3 = 12 – 3 then x ∙ 3 = 12 ∙ 3 If x = 12, then x / 3 = 12 / 3

Commutative Properties Commute = travel (move) It doesn’t matter how you swap addition or multiplication around…the answer will be the same! Samples: Commutative Property of Addition 1+2 = 2+1 AB + CD = CD + AB Commutative Property of Multiplication (2x3) = (3x2) AB(CD) = CD(AB) Rules: Commutative Property of Addition a+b = b+a Commutative Property of Multiplication ab = ba The next slide will discuss how these do not apply to subtraction and division.

Associative Properties Associate = group It doesn’t matter how you group (associate) addition or multiplication…the answer will be the same! Rules: Associative Property of Addition (a+b)+c = a+(b+c) Associative Property of Multiplication (ab)c = a(bc) Samples: Associative Property of Addition (1+2)+3 = 1+(2+3) (AB+CD)+EF=AB+(CD+EF) Associative Property of Multiplication (2x3)4 = 2(3x4) (ABxCD)EF=AB(CDxEF) Later we will discuss how these do not apply to subtraction and division.

Properties of real numbers are only for Addition and Multiplication STOP and THINK! Does the Associative Property hold true for Subtraction and Division? Does the Commutative Property hold true for Subtraction and Division? Is (5-2)-3 = 5-(2-3)? Is (6/3)-2 the same as 6/(3-2)? Leave time to discuss prior to clicking examples. Is 5-2 = 2-5? Is 6/3 the same as 3/6? Properties of real numbers are only for Addition and Multiplication

Distributive Property If something is sitting just outside a set of parenthesis, you can distribute (multiply) it through the parenthesis and remove the parenthesis. Rule: a(b+c) = ab+ac a(b-c) = ab - ac Samples: 4(3+2)=4(3)+4(2)=12+8=20 2(x+3) = 2x + 6 -(3+x) = -3 - x Discuss/illustrate how arrows can help a student stay on track

Properties of Equality Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality Substitution 1-4 Solving Equations

Any quantity is equal to itself Reflexive Property Any quantity is equal to itself a = a AB = AB or AB = BA

Symmetric Property If one quantity equals a second quantity, then the second quantity equals the first. If a = b, then b = a If 10 = 4 + 6, then 4 + 6 = 10 IF AB = CD then CD = AB

AB = CD & CD = EF then AB = EF Transitive Property If one quantity equals a second quantity and the second quantity equals a third quantity, then the first equals the third. If a = b and b = c, then a = c If 2+5 = 7 and 7 = 3+4, Then 2+5 = 3+4 AB = CD & CD = EF then AB = EF

REMEMBER 123… RST COMPARING 1 THING is REFLEXIVE COMPARING 2 THINGS is SYMMETRIC COMPARING 3 THINGS is TRANSITIVE 1-4 Solving Equations

Substitution Property If a = b, then a can be replaced by b. a=b & a=c then b=c AB = CD & AB = EF then CD = EF

Write an Algebraic Proof Statements Reasons Proof: 1. Given 1. d = 20t + 5 2. d – 5 = 20t 2. 3. = t 4. Example 2

1. A B; mC = 45 2. 2.5 3. 3. mA = 2mC 4. mA = 2(45) 5. mA = 90   Statements Reasons Proof: 1. A B; mC = 45   2. 2.5 3. 3. mA = 2mC 4. mA = 2(45) 5. mA = 90 5. Example 3

2. Multiplication Property of Equality ? Statements Reasons Proof: 3. 4. 1. Given 1. 2. _____________ 2. Multiplication Property of Equality ? Example 2

1. Given 1. 2. 3. AB = RS 3. 4. AB = 12 4. Given 5. RS = 12 5. Proof: Statements Reasons Proof: 1. Given 1. 2. 3. AB = RS 3. 4. AB = 12 4. Given 5. RS = 12 5. Example 3

Concept

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