Proving Theorems

Proving TheoremsLesson 2.3Pre-AP Geometry

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.

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.

Midpoint Theorem

If M is the midpoint of AB, then AM = ½AB and MB = ½AB

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)

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

The Midpoint Formula

Application:

Find the midpoint of the segment defined by the points A(5, 4) and B(-3, 2).

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).

Vocabulary

Angle BisectorA ray that divides an angle into two adjacent angles that are congruent.

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

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

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 .

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.”

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.

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Computer Key-In Project