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Proving TheoremsLesson 2.3Pre-AP Geometry
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Proofs
Geometric proof is deductive reasoning at work.
Throughout a deductive proof, the “statements” that are made are specific examples of more general situations, as is explained in the "reasons" column.
Recall, a theorem is a statement that can be proved.
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VocabularyMidpoint
The point that divides, or bisects, a segment into two congruent segments.
BisectTo divide into two congruent parts.
Segment BisectorA segment, line, or plane that intersects a segment at its midpoint.
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Midpoint Theorem
If M is the midpoint of AB, then AM = ½AB and MB = ½AB
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Proof: Midpoint Formula
Given: M is the midpoint of Segment AB
Prove: AM = ½AB; MB = ½AB Statement
1. M is the midpoints of segment AB2. Segment AM= Segment MB, or AM = MB 3. AM + MB = AB4. AM + AM = AB, or 2AM = AB 5. AM = ½AB 6. MB = ½AB
Reason
1. Given2. Definition of midpoint 3. Segment Addition Postulate4. Substitution Property (Steps 2 and 3) 5. Division Prop. of Equality6. Substitution Property. (Steps 2 and 5)
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The Midpoint FormulaThe Midpoint Formula
If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the midpoint of segment AB has coordinates:
2
,2
2121 yyxxM
221 xx
M x
221 yy
M y
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The Midpoint Formula
Application:
Find the midpoint of the segment defined by the points A(5, 4) and B(-3, 2).
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Midpoint Formula
Application:
Find the coordinates of the other endpoint B(x, y) of a segment with endpoint C(3, 0) and midpoint M(3, 4).
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Vocabulary
Angle BisectorA ray that divides an angle into two adjacent angles that are congruent.
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Angle Bisector Theorem
If BX is the bisector of ∠ABC, then the measure of ∠ABX is one half the measure of ∠ABC and the measure of ∠XBC one half of the ∠ABC.
A
X
CB
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Proof: Angle Bisector TheoremGiven: BX is the bisector of ∠ABC.Prove: m ∠ABX = ½ m ∠ABC; m ∠XBC = ½m ∠ABC
Statement Reason
1. BX is the bisector of ∠ABC 1. Given
2. m ∠ABX + m ∠XBC = m ∠ABC
2. Angle addition postulate
3. m∠ ABX = m ∠XBC 3. Definition of bisector of an angle
4. m∠ ABX + m ∠ABX = 2 m ∠ABC; m ∠XBC = m ∠XBC =2 m ∠ABC
4. Addition property
5. m ∠ABX = ½ m ∠ABC; m ∠XBC = ½ m ∠ABC
5. Division property
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Deductive Reasoning
• If we take a set of facts that are known or assumed to be true, deductive reasoning is a powerful way of extending that set of facts.
• In deductive reasoning, we say (argue) that if certain premises are known or assumed, a conclusion necessarily follows from these.
• Of course, deductive reasoning is not infallible: the premises may not be true, or the line of reasoning itself may be wrong .
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Deductive Reasoning
For example, if we are given the following premises:
A) All men are mortal,
B) and Socrates is a man,
then the conclusion Socrates is mortal follows from
deductive reasoning.
In this case, the deductive step is based on the logical principle that "if A implies B, and A is true, then B is true.”
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Written Exercises
Problem Set 2.3A, p. 46: # 1 – 12
Problem Set 2.3B, P. 47: # 13 – 22
Challenge: p.48, Computer Key-In Project (optional)Submit a print out of your results from running the program along with your answers to Exercises 1 – 3.
Download BASIC at: http://www.justbasic.com
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Computer Key-In Project