1. Section 1.7 The Formal Proof of a Theorem
Statement: States the theorem to be proved.
Drawing: Represents hypothesis of the theorem.
Given: Describes the drawing according to the information found in the hypothesis.
Prove: Describes the drawing according to the claim made in the conclusion of the theorem.
Proof: Orders a list of claims (Statements) and justifications (Reasons), beginning with the Given and ending with the Prove; there must be a logical flow in this Proof.
H: Hypothesis ← statement of proof
P: Principle ← reason of proof
∴C: Conclusion← next statement in proof
Ex 2. p 55
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2. Converse of a Statement
Ex. Consider the statement:
If I study, then I will do well on my test.
A new statement can be formed:
If I did well on my test, then I studied
The new statement formed is called the converse of the original statement.
The converse of a true statement may not be assumed to be true.
Another ex.
If I am 12 years old, then I am not eligible to vote.
Converse:
If I am not eligible to vote, then I am 12 years old.
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3. Proving a Theorem
Theorem 1.7.1: If two lines meet to form a right angle, then these lines are perpendicular
Fig 1.66
Strategy:
Look at the diagram
What are we given?
What does the given tell us to use?
How will using these postulates, theorems, definitions help us to prove our theorem
Example 3 p. 56
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4. Additional Theorems on Angles
Theorem 1.7.2: If two angles are complementary to the same angle (or to congruent angles), then these angles are congruent. (Ex. 21)
Theorem 1.7.3: If two angles are supplementary to the same angle (or to congruent angles), then these angles are congruent .(Ex. 22)
Theorem 1.7.4: Any two right angles are congruent.
Theorem 1.7.5: If the exterior sides of two adjacent acute angles form perpendicular rays, then these angles are complementary. Picture proof p. 57
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5. Other Theorems to use in Proofs
Theorem 1.7.6: If the exterior sides of two adjacent angles form a straight line, then these angles are supplementary. Proof: Ex 4 p. 57.
Theorem 1.7.7: If two line segments are congruent, then their midpoints separate these areas into four congruent segments.
Theorem 1.7.8: If two angles are congruent, then their bisectors separate these angles into four congruent angles.
Example in class #32: Prove: The bisectors of two adjacent supplementary angles form a right angle .
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