1. Flowchart and Paragraph Proofs

2. Flowchart Proof - A style of proof that uses boxes and arrows to show the structure of the proof. A flowchart proof should be read from left to right or from top to bottom. Each part of the proof appears in a box, while the justification for each step is written under the box. The arrows show the progression of the proof’s steps.

3. Use the given flowchart proof to write a two- column proof. Given: ∠1 and ∠3 are congruent. ∠1 and ∠2 are supplementary. Prove: ∠2 and ∠3 are supplementary.

4. Use the given flowchart proof to write a two-column proof. Given: ∠1 and ∠3 are congruent. ∠1 and ∠2 are supplementary. Prove: ∠2 and ∠3 are supplementary. SOLUTION Write the steps and justifications of the proof as a 2-column proof. Statements Reasons 1. ∠1 and ∠3 are congruent. 1. Given 2. ∠1 and ∠2 are supplementary. 2. Given 3. m∠1 = m∠3 3. Def of congruent angles 4. m∠1 + m∠2 = 180° 4. Definition of supp angles 5. m∠3 + m∠2 = 180° 5. Substitution Prop 6. ∠3 and ∠2 are supplementary. 6. Definition of supp angles

5. Flowchart proofs are useful when a proof has two different threads that could be performed at the same time, rather than in sequence with one another. Whenever a proof does not proceed linearly from one step to another, a flowchart proof should be considered.

6. Parallel Postulate Alternate Interior Angles Theorem Alternate Interior Angles Theorem Definition of congruent angles Definition of congruent angles Angle addition postulate and definition of straight angle definition Substitution Property of Equality

7. Paragraph Proof - A style of proof in which statements and reasons are presented in paragraph form. In a paragraph proof, every step of the proof must be explained by a sentence in the paragraph. Each sentence contains a statement and a justification.

8. Use the given paragraph proof to write a two-column proof. Given: ∠1 and ∠2 are complementary. Prove: ∠3 and ∠4 are complementary. ∠1 and ∠2 are complementary, so m∠1 + m∠2 = 90° by the definition of complementary angles. Angle 1 is congruent to ∠4, and ∠2 is congruent to ∠3, by the Vertical Angles Theorem. So m∠1 = m∠4, and m∠2 = m∠3. By substitution, m∠4 + m∠3 = 90°. Therefore, ∠3 and ∠4 are complementary by the definition of complementary angles. SOLUTION Statements Reasons 1. ∠1 and ∠2 are complementary. 1. Given 2. m∠1 + m∠2 = 90° 2. Def of complementary angles 3. ∠1 and ∠4 are congruent. ∠2 and ∠3 are congruent.3. Vertical Angles Theorem 4. m∠1 = m∠4 and m∠2 = m∠3 4. Definition of congruent angles 5. m∠3 + m∠4 = 90° 5. Substitution Property of Equality 6. ∠3 and ∠4 are complementary. 6. Def of complementary angles

9. A paragraph proof is good for short proofs where each step follows logically from the one before. Paragraph proofs are usually more compact than two-column proofs.

11. b. Prove the Vertical Angles Theorem using a flowchart proof. Given: a and b are intersecting lines. Prove: ∠1 and ∠3 are congruent. d. Prove Theorem 5-4: If two lines are perpendicular, then they form congruent adjacent angles. Given: Lines s and t are perpendicular. Prove: Angles 1 and 2 are congruent adjacent angles.

12. Page 200 Lesson Practice (Ask Mr. Heintz) Page 201 Practice 1-30 (Do the starred ones first)