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 5/3/2019

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. 5/3/2019

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 5/3/2019

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 5/3/2019

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 . 5/3/2019