Unit 1: Introduction to Proof Prove geometric theorems both formally and informally using a variety of methods. G.CO.9 Prove and apply theorems about lines and angles. Theorems include but are not restricted to the following: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. Learning Goals: Lesson 1: Justifying with Properties from Algebra Solve algebraic equations and verify steps with properties from algebra Understand if-then conditional statements and interpret geometry postulates Identify Counterexamples to geometric statements Lesson 2: Proving Statements about Segments Justify statements about congruent segments using properties of geometric congruence Organize arguments into a 2-column or paragraph proof Lesson 3: Proving Statements about Angles Justify statements about angle congruence and angle relationships using definitions from chapter 1 Organize arguments into a 2-column or paragraph proof
I. Counterexamples Lesson 1: Introduction to Proof using Algebra A counterexample is a specific example (words or diagram) that proves a statement wrong. Only one counterexample is necessary to prove a statement wrong. Example 1: Provide a counterexample to disprove the statement. a. All animals have four legs. b. Only the cookie monster likes to eat cookies. c. All lines intersect in a point. d. If two lines do not intersect, then they are parallel. e. A point can lie on at most 2 lines. f. A triangle and a square can not have the same area. II. Justifying with properties from algebra Example 2: Solve the equation and write the property used at each step. STEP: 6x + 20 + x = 4x 13 Property Used: Given Equation Example 3: Identify the algebraic property of equality shown. If m K = 20, then 10 + m K = 30. If m K = 20, then 4(m K) = 80. If AB = 3, and AC = AB + 5, then AC = 8.
III. Using geometric properties to justify statements. A conditional statement is an if-then statement that is common in algebraic equality properties and geometry postulates and theorems. We learned several geometric postulates and theorems in our first unit. See if you can identify the correct geometric property shown below. Example 4: Identify the postulate, theorem, or definition used to make the conclusion. a. If the triangle is given, then 3! + 4! = 5!. b. If the segment is given, then AB + BC = AC. c. If B is a midpoint, then AB = BC. d. If angles a and b are complementary, then a + b = 90. e. If AB MN, then m MNB is a right angle.
Lesson 2: Proving Statements about Segments
Example 1: Name the property that the statement illustrates. a. If GH JK, then JK GH b. DE = DE c. If P Q and Q R, then P R Example 2: In the diagram, M is the midpoint of AB. Show that AB = 2 AM. Example 3: Fill in the missing reasons in this logical argument. Statements: Reasons: MN = NP NP = PQ MN = PQ Example 4: Complete the following 2-column proof Given: EF = GH Prove: EG FH Statements Reasons
Lesson 3: Proving Statements about Angles I. Recall from Introduction Unit: A theorem is a proven true statement that follows from other true statements. A postulate (or axiom) is a statement that is accepted as truth without proof.
Example 1: Identify the angle pairs as adjacent or nonadjacent. Example 2: Find the following angles measures. Explain your reasoning. Example 3: Find the following angles measures. Explain your reasoning. 3 is a right angle, and m 5 = 57. Find m 1, m 2, m 3, and m 4.
II. Writing a proof using angle properties Example 4: Fill in the missing reasons in this two-column proof. Example 5: Explain how to set up the proof.
Example 6: Identifying Reasons from a Partial Proof Example 7: Writing your own proof