Download Lesson 2.2 Deductive Reasoning notes

2.2 Deductive Reasoning
________________________ – Reasoning accepted as logical from agreed-upon assumptions and
proven facts.
Example 1: Solve each equation for x. Give a reason for each step in the process.
a)
Step
Reason
3(2x + 1) + 2(2x + 1) + 7 = 42 – 5x
b)
c)
Original equation
Step
Reason
5x2 + 19x – 45 = 5x( x + 2 )
Original equation
Step
Reason
4x + 3(2 – x) = 8 – 2x
Original equation
Geometry Lesson 2.2 Deductive Reasoning
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⃗⃗⃗⃗⃗ bisects obtuse  BAD. Classify BAD, DAC, and CAB as
Example 2: In each diagram, AC
acute, right, or obtuse. Then complete the conjecture.
C
B
C
B
A
D
m BAD = 112
m BAD = 148
A
B
D
C
A
D
m BAD = 130
Conjecture: If an obtuse angle is bisected, then the two newly formed congruent angles are
_____________.
Example 3: Use deductive reasoning to write a conclusion for each pair of statements.
a) All whole numbers are real numbers
2 is a whole number
b) All integers are rational numbers
9 is an integer
c) All whole numbers are integers
6 is a whole number
Geometry Lesson 2.2 Deductive Reasoning
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Example 4: Use each true statement and the given information to draw a conclusion.
a) True statement: An equilateral triangle has three congruent sides
Given: ∆ABC is equilateral
b) True statement: A bisector of a line segment intersects the segment at its midpoint
̅̅̅̅ bisects 𝐶𝐸
̅̅̅̅ at point D
Given: 𝐴𝐵
c) True statement: Two angles are supplementary if the sum of their measures is 180
Given: A and B are supplementary
Investigation: Overlapping Segments
̅̅̅̅ ≅ CD
̅̅̅̅ .
In each segment, AB
25 cm
A
B
36 cm
25 cm
75 cm
C
D
A
80 cm
B
36 cm
C
D
Step 1 From the markings on each diagram, determine the length of ̅̅̅̅
AC and ̅̅̅̅
BD. What do you
discover about these segments?
Geometry Lesson 2.2 Deductive Reasoning
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Step 2 Draw a new segment. Label it ̅̅̅̅
AD. Place your own points B and C on ̅̅̅̅
AD so that ̅̅̅̅
AB ≅ ̅̅̅̅
CD.
̅̅̅̅ and BD
̅̅̅̅. How do these lengths compare?
Step 3 Measure AC
Step 4 Complete the conclusion of this conjecture:
If ̅̅̅̅
AD has points A, B, C, and D in that order with ̅̅̅̅
AB ≅ ̅̅̅̅
CD, then …________________________
________________________________________________________________________________
Now use deductive reasoning and algebra to explain why the conjecture in Step 4 is true.
Step 5 Use deductive reasoning to convince your group that AC will always equal BD. Take turns
explaining to each other. Write your argument algebraically.
pp. 103 – 105 => 1 – 9; 11 - 29
Geometry Lesson 2.2 Deductive Reasoning
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