Algebraic Proof LESSON 2–6

Lesson Menu 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

Over Lesson 2–5 5-Minute Check 1 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. 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?

Over Lesson 2–5 5-Minute Check 2 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. 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?

Over Lesson 2–5 5-Minute Check 3 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. 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?

Over Lesson 2–5 5-Minute Check 4 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. 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?

Over Lesson 2–5 5-Minute Check 5 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. 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?

Over Lesson 2–5 5-Minute Check 6 Which of the following numbers is an example of an irrational number? A.–7 B. C. D.34

TEKS 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)

Then/Now 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.

Vocabulary algebraic proof two-column proof formal proof

Concept

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

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

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

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

Example 2 Write an Algebraic Proof 2. d – 5 = 20t2. Addition Property of Equality StatementsReasons Proof: 1. Given 1. d = 20t Symmetric Property of Equality 3.3. Division Property of Equality = t

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 StatementsReasons 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 = (b 1 + b 2 )h B. C. D.

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

Example 3

StatementsReasons Proof: 1. Given _______________ ? 3. AB = RS3. Definition of congruent segments 4. AB = 124. Given 5. RS = 125. Substitution

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

Algebraic Proof LESSON 2–6