LESSON 2–6 Algebraic Proof

Five-Minute Check (over Lesson 2–5) TEKS 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

Mathematical Processes G.1(A), G.1(G) Targeted TEKS Preparation for G.6(A) Verify theorems about angles formed by the intersection of lines and line segments, including vertical angles, and angles formed by parallel lines cut by a transversal and prove equidistance between the endpoints of a segment and points on its perpendicular bisector and apply these relationships to solve problems. Mathematical Processes G.1(A), G.1(G) TEKS

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 formal proof Vocabulary

Concept

Solve 2(5 – 3a) – 4(a + 7) = 92. Write a justification for each step. Justify Each Step When Solving an Equation Solve 2(5 – 3a) – 4(a + 7) = 92. Write a justification for each step. Algebraic Steps Properties 2(5 – 3a) – 4(a + 7) = 92 Original equation 10 – 6a – 4a – 28 = 92 Distributive Property –18 – 10a = 92 Substitution Property –18 – 10a + 18 = 92 + 18 Addition Property Example 1

–10a = 110 Substitution Property Justify Each Step When Solving an Equation –10a = 110 Substitution Property Division Property a = –11 Substitution Property Answer: a = –11 Example 1

Solve –3(a + 3) + 5(3 – a) = –50. A. a = 12 B. a = –37 C. a = –7 D. a = 7 Example 1

Begin by stating what is given and what you are to prove. Write an Algebraic Proof Begin by stating what is given and what you are to prove. Example 2

2. Addition Property of Equality Write an Algebraic Proof Statements Reasons Proof: 1. Given 1. d = 20t + 5 2. d – 5 = 20t 2. Addition Property of Equality 3. 3. Division Property of Equality = t 4. 4. Symmetric Property of Equality Example 2

Which of the following statements would complete the proof of this conjecture? If the formula for the area of a trapezoid is , then the height h of the trapezoid is given by . Example 2

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

A. 2A = (b1 + b2)h B. C. D. Example 2

Write a Geometric Proof If A B, mB = 2mC, and mC = 45, then mA = 90. Write a two-column proof to verify this conjecture. Example 3

3. Transitive Property of Equality 3. mA = 2mC Write a Geometric Proof Statements Reasons Proof: 1. Given 1. A B; mB = 2mC; mC = 45 2. mA = mB 2. Definition of angles 3. Transitive Property of Equality 3. mA = 2mC 4. Substitution 4. mA = 2(45) 5. mA = 90 5. Substitution Example 3

Example 3

3. Definition of congruent segments Statements Reasons Proof: 1. Given 1. 2. 2. _______________ ? 3. AB = RS 3. Definition of congruent segments 4. AB = 12 4. Given 5. RS = 12 5. Substitution Example 3

A. Reflexive Property of Equality B. Symmetric Property of Equality C. Transitive Property of Equality D. Substitution Property of Equality Example 3

LESSON 2–6 Algebraic Proof