Homework 10: p.147: 17-41, 45

2-4B: Writing Proofs Homework 10: p.147: 17-41, 45 Learning Objectives: Analyze figures to identify and use postulates about points, lines and planes Analyze and construct viable arguments in several proof formats Entry Task: Determine whether the stated conclusion is valid based on the given information. If not, choose invalid. Given: A and B are supplementary. Conclusion: m A + m B = 180 valid Given: Polygon RSTU has 4 sides. Conclusion: Polygon RSTU is a square. invalid Given: A and B are congruent. Given: m Y in ΔWXY = 90. Conclusion: A and B are vertical angles invalid Conclusion: ΔWXY is a right triangle. valid Given: A and B are congruent. Conclusion: ΔABC exists. invalid

Concept: Properties of Equality

Example 1: Turning Algebra into a Proof Solve the equation 4m 8 = 12. Write a justification for each step. 4m 8 = 12 +8 +8 4m = 4 4 4 Given Addition Property of Equality Simplify Division Property of Equality m = 1 Simplify

Student Led Example 1: Turning Algebra into a Proof Solve the equation 1 t = 7. Write a 2 justification for each step. 1 2 t = 7 Given 2 1 2 t = 2( 7) Multiplication Property of Equality t = 14 Simplify

Concept: Using Properties of Congruence

Example 2: Identifying Properties Identify the property that justifies each statement. A. QRS QRS Reflex. Prop. of. B. m 1 = m 2 so m 2 = m 1 Symm. Prop. of = C. AB AB CD CD and CDCD EF, EF, so so ABAB EF EF. Trans. Prop of D. 32 = 32 Reflex. Prop. of =

Student Led Example 2: Identifying Properties Identify the property that justifies each statement. A] DE = GH, so GH = DE. Sym. Prop. of = B] 94 = 94 Reflex. Prop. of = C] 0 = a, and a = x. So 0 = x. Trans. Prop. of = D] A Y, so Y A Sym. Prop. of

Example 3: Writing Justifications Find m B Write a justification for each step, given that A and B are supplementary and m A = 45. 1. A and B are supplementary. m A = 45 Given information 2. m A + m B = 180 Def. of supp s 3. 45 + m B = 180 Subst. Prop of = 4. m B = 135 Subtr. Prop of =

Student Led Example 3: Writing Justifications Show that BC EF. Write a justification for each step, given that B is the midpoint of AC and AB EF. B is the midpoint of AC. AB BC AB EF BC EF Given information Def. of mdpt. Given information Trans. Prop. of

Concept: Postulates about Points, Lines and Planes Textbook: p.141

Concept: Postulates about Points, Lines and Planes Textbook: p.141

Example 4A: Sometimes, Always, Never Determine whether the following statement is always, sometimes, or never true. Explain. If plane T contains EF and EF contains point G, then plane T contains point G. Answer: Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane.

Example 4B: Sometimes, Always, Never Determine whether the following statement is always, sometimes, or never true. Explain. GH contains three noncollinear points. Answer: Never; noncollinear points do not lie on the same line by definition.

Student Led Example 4A: Sometimes, Always, Never Determine whether the statement is always, sometimes, or never true. Plane A and plane B intersect in exactly one point. A. always B. sometimes C. never

Student Led Example 4B: Sometimes, Always, Never Determine whether the statement is always, sometimes, or never true. Point N lies in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R. A. always B. sometimes C. never

Concept: The Proof Process

Example 5: Baby Steps XY = XY Statements 1. 1. XY Given Reasons 2. 2.. XY XY Reflexive Prop or 3.. 3. XY = XY Def. of

Student Led Example 5: Baby Steps a. 1 and 2 are supp., and 2 and 3 are supp. b. m 1 + m 2 = m 2 + m 3 c. Subtr. Prop. of = d. 1 3

Concept: A Couple of Theorems

Example 6: A Little Bigger Given: 1 and 2 are supplementary, and 1 3 Prove: 3 and 2 are supplementary. Statements Reasons 1. 1 and 2 are supplementary. 1. Given 1 3 2. 2.. m 1 + m 2 = 180 Def. of supp. s 3. m 1. = m 3 3. 4. m 3 + m 2 = 180 4. 5. 5. 3 and 2 are supplementary Def. of s Subst. Def. of supp. s

Concept: Theorems About Complements

Student Led Example 6: A Little Bigger

End of Lesson Write a justification for each step, given that m ABC = 90 and m 1 = 4m 2. 1. m ABC = 90 and m 1 = 4m 2 Given 2. m 1 + m 2 = m ABC Add. Post. 3. 4m 2 + m 2 = 90 Subst. 4. 5m 2 = 90 5. m 2 = 18 Simplify Div. Prop. of =.