2.1 Conditional Statements
EQ: What are conditional statements and how are they used?
Conditional statement: a two part statement with a hypothesis (if) and a conclusion (then).
If it is 9:00 am in Kernersville, then it is 8:00 am in Dallas, Texas.
hypothesis conclusion
Write the conditional statements in “if, then” form.
A number divisible by two is an even number.
You may go to the game Friday night if you clean up your room.
All mammals are warm blooded.
If it is 7:00 am, then I am walking my dog Forrest.
Converse: a conditional statement formed by switching the hypothesis (if) and the conclusion (then).
Negation: the negative of a statement
I am 48 years old. I am not 48 years old.
2 + 3 = 5 2 + 3 ≠ 5
Give me a few other statements that we can easily negate:
If it is 7:00 am, then I am walking my dog Forrest.
Inverse: a conditional statement formed by negating the hypothesis (if) and the conclusion (then) of a conditional statement.
Contrapositive: a conditional statement formed by negating the hypothesis (if) and the conclusion (then) of the converse of a statement.
Writing conditionals, converses, inverses, and contrapositives.
If I studied, then I passed the test.
Equivalent statements: when two statements are both true or both false. Conditionals and contrapositives are always equivalent.
75: 1, 4-6, 9-13, 18-24 even, 46-49
2.1 Conditional Statements:
EQ: What are the postulates that deal with points, lines, and planes?
Postulates:
1. Through any two points there exists exactly one line.
2. A line contains at least two points.
3. If two lines intersect, then their intersection is exactly one point.
4. Through any three noncollinear points there exists exactly one plane.
5. A plane contains at least three noncollinear points.
6. If two points lie in a plane, then the line containing them lies in the plane.
7. If two planes intersect, then their intersection is a line.
Classwork: Page 75: 2-3, 7-8, 14-17, 22-24
Homework: Page 76: 29-43, 65-70
2.2 Definitions and Biconditionals
Perpendicular lines: two lines that intersect to form right angles.
A line perpendicular to a plane is a line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it.
True or False: D
Points A, B, and C are collinear.
AC is perpendicular to DE. C B A
<ABD is adjacent to <DBC.
<ABD is vertical to <EBA.
E
Additional Forms of Conditional Statements
“only if”
It is Saturday only if I am mowing my lawn.
means the same as
If it is Saturday, then I am mowing my lawn.
“if and only if”
A biconditional statement contains the statement “if and only if”. A biconditional is equivalent to writing a conditional and its converse.
Three lines are coplanar if and only if they lie in the same plane.
means the same as
If three lines are coplanar, then they lie in the same plane.
and
If three lines lie in the same plane, then they are coplanar.
Analyzing a Biconditional Statement
x = 3 if and only if x² = 9
Is the statement a biconditional?
Is the statement true?
Writing a biconditional statement?
The following statement is true. Write the converse and decide whether or not it is true.
If x² = 4, then x = 2 or x = -2.
Classwork: page 82: 1-12.
Homework: page 82: 13-20, 22, 24-31, 38-43
JL
2.3 Deductive Reasoning
You know a conditional has a hypothesis and a conclusion.
Conditionals may be written using symbolic notation where:
p represents the hypothesis
q represents the conclusion
- represents implies
conditional
p q: If the sun is out, then the weather is good
converse
q p: If the weather is good, then the sun is out.
inverse
~p ~q: If the sun is not out, then the weather is not good.
contrapositive
~q ~p: If the weather is not good, then the sun is not out.
Biconditional
p q: The sun is out if and only if the weather is good.
Let p be “<3 measures 90 degrees” and let q be “<3 is not acute.”
Write the converse, inverse, and contrapositive. Are they T or F.
Law of Detachment: if p q is a true conditional and p is true, then q is true.
Law of Syllogism: if p q and q r are true conditional statements, then p r is true.
Use the Laws of Syllogism to write the statement that follows from the given statements:
1. If it is sunny, then I am at the pool.
2. If I am at the pool, then I am in the water.
1. If you do all your homework, then you will pass the class.
2. If you pass the class, then you have a happy life.
Use the Law of Detachment to determine if the argument is valid.
Tyler knows if he misses practice, then he will not start the game. Tyler missed practice on Wednesday so he concludes that he will not start the game.
If two angles form a linear pair, then they are supplementary.
<A and <B are supplementary, so <A and <B form a linear pair.
Classwork: page 92: 8-13 all, 14-20 even, 30-35 all
Homework: page 91: 36-42, 45-50
Reasoning with Properties from Algebra
EQ: What are proofs and how do they work?
Algebraic Properties of Equality
Addition: if a = b, then a + c = b + c
Subtraction: if a = b, then a – c = b – c
Multiplication: if a = b, then ac = bc
Division: if a = b, then a/c = b/c
Reflexive: for any real number a, a = a
Symmetric: if a = b, then b = a
Transitive: if a = b and b = c, then a = c
Substitution: if a = b, then a can be substituted for b in any equation or expression
Give a reason for each step.
5x – 18 = 3x +2 Given
2x – 18 = 2
2x = 20
x = 10
55x – 3(9x + 12) = -64 Given
55x – 27x - 36 = -64
28x – 36 = -64
28x = -28
x = -1
Properties of Equality
Segment Length
Reflexive: for any segment AB, AB = AB
Symmetric: if AB = CD, then CD = AB
Transitive: if AB = CD and CD = EF, then AB = EF
Angle Measure
Reflexive: for any angle A, m<A = m<A
Symmetric: if m<A = m<B, then m<B = m<A
Transitive: if m<A = m<B and m<B = m<C, then m<A = m<C
In the diagram AB = CD. Show that AC = BD
AB = CD 1. Given
AB + BC = BC + CD 2.
AC = AB + BC 3.
BD = BC + CD 4.
AC = BD 5.
Given: m<1 + m<2 = 66
m<1 + m<2 + m<3 = 99
m<3 = m<1
m<1 = m<4
m<1 + m<2 = 66 1. Given
m<1 + m<2 + m<3 =99 2.
66 + m<3 = 99 3.
m<3 = 33 4.
m<3 = m<1, m<1 = m<4 5.
m<3 = m<4 6.
m<4 = 33 7.
Honors Geometry: Page 99: 4 - 15 all, 16 – 28 even, 29 – 31
Regular Geometry: Page 99: 4 - 15 all, 16 – 28 even
2.5 Proving Statements about Segments
EQ: What are two-column proofs?
Theorem: a true statement that follows as a result of other statements.
Two-column proof: proof with numbered statements and reasons that show the logical order of an argument.
Properties of Segment Congruence
Reflexive: For any segment AB, AB = AB.
Symmetric: If AB = CD, then CD = AB.
Transitive: If AB = CD and CD = EF, then AB = EF.
Given: LK = 5, JK = 5, JK = JL
Prove: LK = JL
Statements Reasons
1. LK = 5 1. Given
2. JK = 5 2.
3. LK = JK 3.
4. LK = JK 4.
5. JK = JL 5.
6. LK = JL 6.
Given: X is the midpoint of MN.
MX = RX
Prove: XN = RX
Statements Reasons
1. X is the midpoint of MN 1. Given
2. XN = MX 2.
3. MX = RX 3.
4. XN = RX 4.
Page 104: 1-11, 16-17
2.6 Proving Statements about Angles
Properties of Angle Congruence
Reflexive, Symmetric, and Transitive all hold true.
Right Angle Congruence Theorem: all right angles are congruent.
Congruent Supplements Theorem: if two angles are supplementary to the same (or congruent) angles, then they are congruent.
Congruent Complements Theorem: if two angles are complementary to the same (or congruent) angles, then they are congruent.
Linear Pair Postulate: if two angles form a linear pair, then they are supplementary
Vertical Angles Theorem: vertical angles are congruent
112: 4 – 18, 27 - 30
Review for Chapter Two TEST
We are dismissed early if there is a teacher’s meeting.
1. Write the statement in if-then form:
2. Write the converse:
3. Write the inverse:
4. Write the contrapositive:
5. Write the conditional, converse, inverse, and contrapositive in symbolic notation.
Make a sketch of each of the following:
6. Vertical angles are congruent.
7. A line contains at least 2 points.
8. If two lines intersect, then their intersection is exactly one point.
9. Through any three noncollinear points there exists exactly one plane.
10. If two planes intersect, then their intersection is a line.
11. If two angles form a linear pair, then they are supplementary.
Match each example with the correct Property:
12. If AB = CD, then CD = AB. A. Reflexive
13. For any segment AB, AB = AB. B. Symmetric
14. If AB = CD and CD = EF, then AB = EF C. Transitive
Write each statement as a biconditional. Is it true or false.
15. If x = 5, then x^2 = 25.
16. A rectangle is a square if it has four congruent sides.
17. Write an example of the Law of Syllogism.
18. Solve the equation and give a reason for each step:
23 + 11d – 2c = 12 – 2c given
19. If the conditional statement is true, which other statement must also be true?
20. Which other two statements are always true or always false together?
Homework: page 122: 1 – 11, 14 - 16 all