Download Geometry Lesson 2.6

State the hypothesis and conclusion of the conditional,
then write the statement’s converse.
If it snows, then school will be cancelled.
hypothesis
conclusion
Converse :
If school is cancelled, then it snowed.
2.6 Deductive Reasoning
Goal 1: How to identify special angle relationships
Goal 2: How to use deductive reasoning to verify angle relationships
Power Standard #11
Apply skills of conjecture, analysis, and counter-example
to formulate a hypothesis and test it. (3.2.1)
[2.4-2.6, 3.3-3.6, 4.3, 4.4, 6.4, 8.5]
Angle Relationships
•
•
•
•
•
•
•
•
Vertical Angles
Linear Pair (of angles)
Complementary Angles
Supplementary Angles
Linear Pair Postulate
Congruent Supplements Theorem
Congruent Complements Theorem
Vertical Angles Theorem
Vertical Angles
Vertical Angles are the non adjacent angles
formed by two intersecting lines.
3
2
1
1 & 2 are a pair of
vertical angles.
4
3 & 4 are also a pair of
vertical angles.
6
5
5 & 6 are not a pair of
vertical angles.
Linear Pair
If the noncommon sides of adjacent angles
are opposite rays then the angles are a
linear pair.
B
1
A
2
O
statement
reason
1. OA & OC are
1. Given
opposite rays
2. AOB & COB 2. Definition
are a linear pair
C
Complementary Angles
If the sum of the measures of two angles is 90, then the
angles are complementary.
Each angle is the complement of
the other
1
2
If m1 + m2 = 90 degrees, then the angles are
complementary
statement
reason
Reversible
1. m1 + m2 = 90
1. Given
2. 1 & 2 are
complementary
2. Definition of
Complementary angles.
Supplementary Angles
If the sum of the measures of two angles is 180, then the
angles are supplementary.
1
2
Each angle is the supplement of the other
If m1 + m2 = 180, then the angles are supplementary
statement
Reversible
reason
1. m1 + m2 = 180
1. Given
2. 1 & 2 are
supplementary
2. Definition of
Supplementary angles.
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
(m1 + m2 = 180)
1
statement
•
•
1. 1 & 2 are a linear
pair.
2. m1 + m2 = 180
2
reason
• 1. Given
• 2. Linear Pair Postulate
1. Solve:
x  4 3x  2

12
28
2. If the product of the slopes of two lines is -1, then
the lines are perpendicular.
a) write the converse
If the lines are perpendicular, then the product of
their slopes is -1.
b) write the statement represented by p↔q
The product of the slopes of two lines is -1, if and
only if, the lines are perpendicular.
3. Write an example of the transitive property.
Warm-up
1
State the relationship between the angles
3
4
2
1. 1 and 2 are Vertical 's.
2. 1 and 3 are Supplementary 's and a
3. 3 and 4 are Vertical 's.
4. 2 and 4 are Supplementary 's.
Linear Pair .
Warm-up
State the relationship between the angles
3
1
2
1. 1 and 2 are Complementary's.
2. 1 and 3 are Supplementary 's and a
Linear Pair .
Congruent Supplements Theorem
Two 's are supplementary to the same or  's
 the 's are  to each other.
A
B
C
A and B are supplements to C
 A  B
Congruent Supplements Theorem
Two 's are supplementary to the same or  's
 the 's are  to each other.
A
B
C
D
A and C are supplements and
B and D are supplements
C  D  A  B
Congruent Complements Theorem
Two 's are complementary to the same or  's
 the 's are  to each other.
A
B
C
A and B are complements to C
 A  B
Vertical Angles Theorem
If two angles are vertical,
then they are congruent.
3
2
1
4
1  2
3  4
10x + 40
20x - 50
Find the measure of all four 's.
10x  40  20x  50
x 9
10x
10x
130
40  10x  50
50
50
50
50
130
90  10x
10 10
10x + 40
20x - 50
Solve for x
10x  40  20x  50  180
30x 10  180
10 10
30x  190
30 30
x  19 / 3
x
2x 1
Solve for x
x  2 x  1  90
3x  1  90
3x  91
x  91/ 3
2.6 Deductive Reasoning
2.6/13-18, 21, 23, 25, 26, 27-35 odd
2.6 Deductive Reasoning
2.6 Even Answers
14) True
16) True
18) True
26) 1. Given, 2. Given, 3. Given,
4. Transitive Prop., 5. Substitution,
6.  prop. of =
2.6 Deductive Reasoning
Quiz
Assume the 1 and 2 are complementary. Copy
and complete the table.
m1
m2
1
10
20
30
40
50
60
70
80
89
Warm-up
1. Write the correct notation for each figure
AB B
A
E EF F
R
CD

ROZ
O
C
D
Z
2. Draw opposite rays AB and AC.
B A C
3. Find the length of AB if CD bisects AB at T
and AT  12. Draw a sketch.
T D
A
B
AB  24
C
4. If c  d , then d  c.
Write the converse of this statement
and prove the converse using a two-column
proof with statements and reasons.
If d  c, then c  d
5. Solve for x
3x + 2
4x - 4
statement
reason
1. d  c
2. c  d
1. given
2. Symmetric Prop.
3x  2  4 x  4  180
7 x  2  180
7 x  182
x  26