1. 2.1 Continued: Definitions and Biconditionals
Geometry

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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