2.1 Continued: Definitions and Biconditionals Geometry

Geometry 2.1 Cont Definitions and Biconditionals Goals Recognize and use definitions. Understand Biconditionals June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Definition: Perpendicular Lines Two lines that intersect to form a right angle. m n Notation: m  n June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Definition: Perpendicular Lines Two lines that intersect to form a right angle. All definitions can be read two ways. Both forward and backward as conditionals: If two lines are perpendicular, then they intersect to form a right angle. If two lines intersect to form a right angle, then they are perpendicular. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Justifying Statements In math, deciding if a statement is true or false demands that you can justify your answers. “Just because”, or, “It looks like it” are not sufficient. Justification must come in the form of Postulates, Definitions, or Theorems. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Example 1 Statement A D, X, and B are collinear. Truth Value D X B TRUE Reason Definition of collinear points. C If points are collinear, then they are on the same line. If points are on the same line, then they are collinear. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Example 2 Statement A AC  DB Truth Value D X B TRUE Reason Definition of Perpendicular lines Def  lines C June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Example 3 Statement CXB is adjacent to BXA A Truth Value TRUE D X B Two angles with a common vertex and side but no common interior points. Reason C Def. of adj. s June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Example 4 Statement CXD and BXA are vertical angles. A Truth Value D X B TRUE Reason Def. vert. s C June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Example 5 Statement DXA and CXB are adjacent angles. A Truth Value D X B FALSE Reason There is not a common side. (Or, they are vertical angles.) C June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals In doing proofs, you must be able to justify every statement with a valid reason. To be able to do this you must know every definition, postulate and theorem. Being able to look them up is no substitute for memorization. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals True If 2 s are complementary, then their sum is 90°. Converse If the sum of 2 s is 90°, then they are complementary. True The statement and its converse are both TRUE. This is a BICONDITIONAL… June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Writing Biconditionals Biconditionals are written using the phrase “if and only if” If 2 s are complementary, then their sum is 90°. and If the sum of 2 s is 90°, then they are complementary. can be written: Two angles are complementary if and only if their sum is 90°. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Shorthand Iff means “if and only if”. Two angles are complementary if and only if their sum is 90°. 2 s comp. iff sum = 90°. Iff means “if and only if”. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals If a statement is a biconditional, it means we can write it two ways: as a conditional and as its converse. Biconditional A line is horizontal if and only if its slope is zero. conditional If a line is horizontal, then its slope is zero. converse If the slope of a line is zero, then the line is horizontal. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Try it. An angle is obtuse iff it measures between 90 and 180. Write the biconditional as a conditional and its converse. If an angle is obtuse, then it measures between 90 and 180. If an angle measures between 90 and 180, then it is obtuse. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Truth Values of Biconditionals A biconditional is TRUE if both the conditional and the converse are true. A biconditional is FALSE if either the conditional or the converse is false, or both are false. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Example Biconditional x = 5 iff x2 = 25. Conditional If x = 5, then x2 = 25. Converse If x2 = 25, then x = 5. False! True or False? True or False? true False! True or False? June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals ALL definitions are biconditionals. Example: Definition of Congruent Angles Two angles are congruent iff they have the same measure. Conditional: If two angles are congruent, then they have the same measure. Converse: If two angles have the same measure, then they are congruent. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Mind your Ps and Qs. Conditional: If HYPOTHESIS, then CONCLUSION. Let P represent the HYPOTHESIS. Let Q represent the CONCLUSION. Then the conditional is: If P, then Q. The notation is: P  Q. The symbol “” is often read as “implies”. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Negation Use the symbol ~. Read it as “not”. P is the statement “I like ice cream” ~P is read “Not P” ~P is the statement “I don’t like ice cream” June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals Logical Statements Conditional: P  Q Converse: Q  P Biconditional: P  Q Inverse: ~P  ~Q Contrapositive: ~Q  ~P June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals

Geometry 2.1 Cont Definitions and Biconditionals What you can do now: Identify statements about drawings as true or false. Recognize and write biconditionals. Write a conditional and its converse from a biconditional. Write a counterexample. June 5, 2019 Geometry 2.1 Cont Definitions and Biconditionals