Inverse The statement formed by negating the hypothesis and conclusion of a conditional statement Symbolic Notation: Contrapositive The equivalent statement formed by exchanging AND negating the hypothesis and conclusion of a conditional statement Symbolic Notation: Equivalent Statements Two statements that are both true or both false Ex. Conditional and Contrapositive; Converse and Inverse Biconditional Statement A statement that contains the phrase “if and only if.” Combines a conditional and its converse when both are true. Ex. Definitions are biconditionals Perpendicular Lines Two lines that intersect to form right angles Notation: Show: Ex 1: Rewrite the conditional statement in if-then form. All whales are mammals. If an animal is a whale, then it is a mammal. Three points are collinear when there is a line containing them. If there is a line containing three points, then the points are collinear. Ex 2: Write the if-then form, the converse, the inverse, and the contrapositive of the statement, then determine the validity of each statement. Statement: Soccer players are athletes Conditional: If you are a soccer player, then you are an athlete. True. Converse: If you are an athlete, then you are a soccer player. False. Inverse: If you are not a soccer player, then you are not an athlete. False. Contrapositive: If you are not an athlete, then you are not a soccer player. True.
Warm Up: Key Concepts: Algebraic Properties of Equality let a, b, and c are real numbers Addition Property If a = b, then a + c = b + c. Subtraction Property If a = b, then a – c = b – c. Multiplication Property If a = b, then ac = bc. Division Property If a = b and c 0, then . Substitution Property If a = b, then a can be substituted for b in any equation or expression. Distributive Property Reflexive Properties of Equality Real Numbers For any real number a, a = a. Segment Lengths For any segment AB, AB = AB. Angle Measures For any angle A, . Symmetric Properties of Equality Real Numbers For any real numbers a and b, if a = b , then b = a.
Division Prop. of Eq. Ex 3: The cost C of using a certain cell phone can be modeled by the plan rate formula when m represents the number of minutes over 300. Solve the formula for m. Write a reason for each step. Steps Reasons Given Distributive Prop. Simplify Add. Prop. Of Eq Div Prop of Eq Try it: Ex 4: The formula for the area of a trapezoid is . Solve the formula for . (Depending on your solution method, you may not fill out all the blanks below.) Steps Reasons Given Distributive Prop. Subtraction Prop, of Eq. Multiplication Prop. of Eq. Division Prop. of Eq. Ex 5: intersect at so that . Show that Steps Reasons Given
Subtraction Property of Eq. Section: 2– 6 Prove Statements about Segments and Angles Essential Question How do you write a geometric proof? Warm Up: Key Vocab: Proof A logical argument that shows a statement is true. Two-Column Proof A type of proof written as numbered statements and corresponding reasons that show an argument in a logical order. Theorem A true statement that follows as a result of other true statements. Must be proven to be true Theorems: Congruence of Segments Segment congruence is reflexive, symmetric and transitive Reflexive Symmetric If Transitive If Congruence of Angles Angle congruence is reflexive, symmetric, and transitive.
Prove: Statements Reasons Given ; Definition of Angle Bisector Substitution Ex 4: Given: Prove: Statements Reasons Given Definition of Congruence Reflexive Prop. of Eq. Addition Prop. of Eq. Segment Add. Post. Substitution Definition of Congruence Section: 2 – 7 Prove Angle Pair Relationships Essential Question What is the relationship between pairs of vertical angles, between pairs of angles that are supplementary/complementary to the same angle? Warm Up:
Theorems: Right Angles Congruence Theorem All right angles are congruent Vertical Angles Congruence Theorem Vertical angles are congruent. Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. are supplementary & are supplementary Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent. are complementary & are complementary Linear Pair Postulate If two angles form a linear pair, then they are supplementary.
Ex 4: Prove the Vertical Angles Congruence Theorem Given: are vertical angles Prove: Statements Reasons Angle Addition Post. Substitution (Transitive) Reflexive Prop. of Eq. = Subtraction Prop. of Eq. Definition of Congruence Ex 5: Prove the Congruent Supplements Theorem Given: are supplements are supplements Prove: Statements Reasons are supplements are supplements Given Definition of Supplementary Angles Substitution Prop. of Eq. (Transitive) Reflexive Prop. of Eq. = Subtraction Prop. of Eq. Definition of Congruence Ex 6: Given: