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1.5: Angle Addition Post., Angle Bisectors
(#1, 3, 6, 8, 10, 11, 15, 19-26) _____________________/40pts
1.6: Complementary, Supplementary, Vertical, and Linear Pair Angles
(#2, 4, 5, 7, 9, 12, 13, 14, 16-17)__________/21pts
Overall Test Percentage____________%
Unit 2 Reasoning
&
Proofs
2.1 Conditional Statements
Identify when a conditional statement true or false.
Conditional statements
If ____(Hypothesis )______ , then _____(Conclusion)____.
“p implies q”
p→ q
Examples:
If ∠1 and ∠2 are complementary, then the sum of their measures is 180°
If points A, B, and C are collinear, then they lie on the same line.
Conditional statements can be true or false.
Negation ~
Opposite of the original statement
~p
“Not p”
Examples: Negate the following statements.
a. The ball is red. b. The dog is not barking.
The ball is not red The dog is barking
Determine whether each conditional statement is true or false. Justify your answer.
a. If yesterday was Wednesday, then today is Thursday.
b. If an angle is acute, then it has a measure of 30°.
A hypothesis can either be true or false. The same is true of a conclusion. For a conditional statement to be true, the hypothesis and conclusion do not necessarily both have to be true.
Conditional: | If hypothesis, then conclusion | “If p, then q.” p→ q |
Converse: | If conclusion, then hypothesis | “If q, then p.” q→ p |
Inverse: | negate the conditional | “If not p, then not q.” ~p→ ~q |
Contrapositive: | negate the converse | “If not q, then not p.” ~q→ ~p |
Equivalent statements:
A conditional statement and its contrapositive are either both true or both false.
Similarly, the converse and inverse of a conditional statement are either both true or both false.
In general, when two statements are both true or both false, they are called equivalent statements.
Write the four conditional statements & decide whether each is true or false:
a. conditional statement: p → q
b. converse: q → p
c. inverse: ∼p → ∼q
d. contrapositive: ∼q → ∼p
You Try: Write the four conditional statements & decide whether each is true or false:
Let p be “you are a guitar player” and let q be “you are a musician.”
a. the conditional statement: p → q
b. the converse: q → p
c. the inverse: ∼p→ ∼q
d. the contrapositive: ∼q → ∼q
Solution:
a. Conditional: If you are a guitar player, then you are a musician.
b. Converse: If you are a musician, then you are a guitar player.
c. Inverse: If you are not a guitar player, then you are not a musician.
d. Contrapositive: If you are not a musician, then you are not a guitar player.
true; Guitar players are musicians.
false; Not all musicians play the guitar.
false; Even if you do not play a guitar, you can still be a musician.
true; A person who is not a musician cannot be a guitar player.
Definitions can be written as both a conditional statement or as a converse statement :
Example:
Conditional: If two lines intersect to form a right angle, then they are perpendicular lines.
Converse: If two lines are perpendicular, then they intersect to form a right angle.
Biconditional Statements:
When a conditional statement and its converse are both true.
“if and only if”
p q
Example: “Two lines intersect to form a right angle if and only if they are perpendicular lines.”
Definition of Congruent Angles:
Conditional: If two angles are congruent, then their angle measures are the same.
Converse: If two angles have the same measure, then they are congruent.
Biconditional: Two angles are congruent if and only if their angle measures are the same.