Section 3.3: Proving Lines are Parallel Review of the Parallel Lines Postulate & Theorems. Converses of Parallel Lines Postulate & Theorems Proof of the Converse of the Alt Int Angles Theorem Two more ways to prove lines are parallel Example 1 Example 2 Parallel & Perpendicular Through a Point Theorems HW: Pg. 87 #1-15 odd, 19, 25, 27, 29 MK: 3.3 Makeup Homework from website H Q S

1. Review: || Line Postulate & Theorems b t When you know the lines are parallel… Corr Post: Alt int Thm: Ss int Thm: If the lines are || then corr are If the lines are || then alt int are If the lines are || then ss int are supplementary. H Q S

2. Converses of Parallel Lines Postulate & Theorems b t When you don’t know the lines are parallel… Converse of Corr Post: If corr are then the lines are ||. Converse of Alt int Thm: If alt int are then the lines are ||. Converse of Ss int Thm: If ss int are supplementary then the lines are ||. H Q S

3. Proof of the Converse of the Alt Int Angles Theorem If alternate interior angles are congruent, then the lines are ||. a b t 1 3 2 Given: Prove: a || b Statements: Reasons: Proof: 1. 1. Given 2. 2. Vertical Angles Theorem 3. 3. Transitive Property of Congruence 4. a || b 4. Converse of Corr Angles Post. H Q S

4. Two more ways to prove lines are parallel b c to Same Line Theorem (3.7): If 2 lines in a plane are to the same line, then those lines are ||. k p w k || w & p || w : || to Same Line Theorem (3.10): If 2 lines are || to the same line, then those lines are ||. H Q S

5. Example 1 a Find the value of x that would make a || b. b 46 b Find the value of x that would make a || b. 3 4 1. Angles of interest: 2. They are ss int 3. If ss int are supplementary then the lines are ||. so 46 + (4x+10) = 180 4x + 56 = 180 4x = 124 x = 31 H Q S

6. Example 2 a 1 2 b c d which lines are || ? If Put a dot on both sides of each angle. Highlight all lines with a dot. The transversal has 2 dots; the lines each have one. Since corr c || d. H Q S

7. Parallel & Perpendicular Through a Point Theorems || Thru a Point Theorem (3.8): Through a point not on a line, there exists exactly one line || to the given line. Thru a Point Theorem (3.9): Through a point not on a line, there exists exactly one line to the given line. H Q S

3.3 Summary The 5 ways to prove that lines are parallel: Show a pair of corresponding angles are congruent (11) Show a pair of alternate interior angles are congruent (3.5) Show a pair of same-side interior angles are supplementary (3.6) Show that both lines are perpendicular to a 3rd line (3.7) Show that both lines are parallel to a 3rd line (3.10) Summary does not include other theorems. H Q S

3.3 Homework Index HW: Pg. 87 #1-15 odd, 19, 25, 27, 29 MK: 3.3 Makeup Homework from website 1 - 16 18 - 19 25 - Proof 27 - 29 H Q S

3.3 Homework, p. 87 (1 - 16) Use the information given to name the || segments. If there are no || segments, write none. 1. 2. 3. 4. 5. 6. 7. 8. 9. 13. are compl. & 5, 7, & 8 are NONE 10. 14. 11. 15. 12. are suppl. 16. H Q S

3.3 Homework, p. 87 (18 - 19) Find the values of x & y that make the red lines parallel & the blue lines parallel. 18. 19. (x - 40) (x + 40) 3x 105 y 2y x H Q S

3.3 Homework, p. 87 (25) 25. C D B A E 1 2 3 Given: Prove: Proof: Statements Reasons 1. 2. 3. 4. 5. 6. Given 2. 3. 4. 5. 6. H Q S

3.3 Homework, p. 87 (27 - 29) 27. 28. 110 T S X R Y 120 X R 40 S 70 T Y 29. Find the values of x & y that make the lines shown in red parallel. 30 2x (x - y) 5y H Q S

CP Geometry Homework Quiz Section: Period: Date: Name: Answers: 1 2 Box 1 Box 2 Box 3 3 4 Your answer to question 4 5 6 Instructions: All red fields are required. Name must be FIRST & LAST. One point deduction if anything missing. Boxes are for showing work. (If calculations are required, write the formula first.) Put all answers in the spaces to the right. If an answer does not fit, put it in a Box & draw an arrow to it (as showing in the example above.) Do not copy the problem or drawing from the board onto your HWQ form. IF YOU WERE ABSENT: Fill in all red fields; write “ABSENT on <date you were absent>” in Box 1. If you do not have one, ask for a Makeup Form. Answers A, D, B, F, C, E H Q S

CP Geometry Homework Quiz 3.3A Questions 1-4. Write the letter that indicates which segments must be || if… Questions 5 & 6. Find the values of x & y that make the red & blue lines || D 2. B 3. C 4. C 5. x=16 6. y=17 1. Letter 2. Letter 3. Letter 4. Letter 5. x= 6. y = 6x 96 4y x H Q S

CP Geometry Homework Quiz 3.3B Questions 1-4. Write the letter that indicates which segments must be || if… Questions 5 & 6. Find the values of x & y that make the red & blue lines || C 2. B 3. D 4. B 5. x = 24 6. y = 15 1. Letter 2. Letter 3. Letter 4. Letter 5. x= 6. y = 4x 96 4y x H Q S

CP Geometry Homework Quiz 3.3C Questions 1-4. Write the letter that indicates which segments must be || if… Questions 5 & 6. Find the values of x & y that make the red & blue lines || B 2. D 3. C 4. B 5. x = 32 6. y = 13 1. Letter 2. Letter 3. Letter 4. Letter 5. x= 6. y = 3x 96 4y x H Q S