2 1 Continued Definitions and Biconditionals Geometry Goals
2. 1 Continued: Definitions and Biconditionals Geometry
Goals n n Recognize and use definitions. Understand Biconditionals 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 2
Definition: Perpendicular Lines Two lines that intersect to form a right angle. n Notation: m 11 December 2021 m n Geometry 2. 1 Cont Definitions and Biconditionals 3
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. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 4
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. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 5
Example 1 Statement D, X, and B are collinear. A D X C B Truth Value TRUE Reason Definition of collinear points. If points are collinear, then they are on the same line. If points are on the same line, then they are collinear. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 6
Example 2 Statement A AC DB Truth Value D X B TRUE Reason C Definition of Perpendicular lines Def lines 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 7
Example 3 Statement A CXB is adjacent to BXA Truth Value D X B TRUE Two angles with a common vertex and side Cbut no common interior points. 11 December 2021 Reason Def. of adj. s Geometry 2. 1 Cont Definitions and Biconditionals 8
Example 4 A D X C 11 December 2021 B Statement CXD and BXA are vertical angles. Truth Value TRUE Reason Def. vert. s Geometry 2. 1 Cont Definitions and Biconditionals 9
Example 5 Statement DXA and CXB are adjacent angles. Truth Value A D X B FALSE Reason C 11 December 2021 There is not a common side. (Or, they are vertical angles. ) Geometry 2. 1 Cont Definitions and Biconditionals 10
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. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 11
Biconditionals If 2 s are complementary, then their sum is 90°. True 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… 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 12
Writing 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°. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 13
Shorthand Two angles are complementary if and only if their sum is 90°. 2 s comp. iff sum = 90°. Iff means “if and only if”. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 14
Biconditional 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. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 15
Try it. n n 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. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 16
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. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 17
Example Biconditional or False? x = 5 iff x 2 = 25. True False! Conditional true or False? If x = 5, then x 2 = 25. True Converse False!or False? If x 2 = 25, then x = 5. True 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 18
Definitions 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. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 19
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. 11 December 2021 The symbol “ ” is often read as “implies”. Geometry 2. 1 Cont Definitions and Biconditionals 20
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” 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 21
Logical Statements Conditional: P Q Converse: Q P Biconditional: P Q Inverse: ~P ~Q Contrapositive: ~Q ~P 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 22
What you can do now: n n Identify statements about drawings as true or false. Recognize and write biconditionals. Write a conditional and its converse from a biconditional. Write a counterexample. 11 December 2021 Geometry 2. 1 Cont Definitions and Biconditionals 23
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