______ Reasoning Accepted As Logical from Agreed-Upon Assumptions and Proven Facts

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 Original equation

b) Step Reason

5x2 + 19x – 45 = 5x( x + 2 ) Original equation

c) Step Reason

4x + 3(2 – x) = 8 – 2x Original equation

Example 2: In each diagram, AC bisects obtuse Ð BAD. Classify ÐBAD, ÐDAC, and ÐCAB as acute, right, or obtuse. Then complete the conjecture.

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

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

Given: AB bisects CE at point D

c) True statement: Two angles are supplementary if the sum of their measures is 180°

Given: ÐA and ÐB are supplementary

Investigation: Overlapping Segments

In each segment, AB≅ CD .

Step 1 From the markings on each diagram, determine the length of AC and BD. What do you

discover about these segments?

Step 2 Draw a new segment. Label it AD. Place your own points B and C on AD so that AB≅ CD.

Step 3 Measure AC and BD. How do these lengths compare?

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 ReasoningPage 1