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Day 7 Reasoning in Geometry (2) Conditional and Related Statements • Write a conditional statement and its converse, its inverse and its contrapositive. Vocabulary: Conditional Statement: a statement that can be written as an if-then statement. (After “If”Hypothesis; After “Then”- Conclusion) ! (Ex) If it’s rain, then Bonding Day is postponed. ! (Ex) Two angles are supplementary if they are a linear pair. ! ! ! Negation of a statement: the opposite of the original statement. ! (Ex) Original: It is snow. Negation: It is NOT snow. ! (Ex) Original: A right triangle has a right angle. Negation: A right triangle does not have a right angle. Verifying Statement: ! Conditional Statements can be true or false. To show that a conditional statement is true, you must prove that the conclusion is true every time the hypothesis is true. To show that a conditional statement is false, you need to give only one counterexample. ! (ex) Write a conditional statement and decide if it is true or false. ! (1) Congruent Segments have equal measures. ! ! ! (2) Supplementary angles form a linear pair. ! Related Conditional Statements ! Converse of a conditional statement: exchange the hypothesis and the conclusion. ! Inverse of a conditional statement: negate both hypothesis and the conclusion. ! Contrapositive or a conditional statement: Firstly write the converse and then negate both the hypothesis and the conclusion. ! (Ex) Write the converse, inverse and contrapositive of given conditional statements. Decide if each statement is true or false. ! (a) If m∠A = 99° , then ∠A is obtuse. ! ! (b) If a polygon is equilateral, then the polygon is regular. ! ! (c) If a figure is a square, then it has four right angles. ! Biconditional Statements: When a conditional statement and its converse are both true, you can write them as a single biconditional statement. ! (ex) Definition of Perpendicular Lines Definition: if two lines intersect to form a right angle, then they are perpendicular. ! Converse: If two lines are perpendicular, then they intersect to form a right angle. ! Biconditional Statement: Two lines are perpendicular if and only if they intersect to form a right angle. ! Practice: Write a converse, inverse and contrapositive statement of each conditional statement. Then decide if each is true or false. (a) Linear Pair Conjecture: If two angles form a linear pair, then the measures of the angles add up to 180 degree. ! ! (b) Vertical Angles Conjecture: If two angles are vertical angles, then they have equal measures. ! ! With Deductive Reasoning, you use facts, definitions, and properties to draw conclusions and prove that conjectures are true. ! Given: Two lines are intersecting and form 4 angles. Conjecture: ∠1 and ∠3 are vertical angles. ! ! ! ! Explain why. !