Chapter 5 Section 3

1. Chapter 5 Section 3 Theorems Involving Parallel Lines

2. Warm-Up: Thursday, March 27th Part 1: Parallelograms Methods of Proof Part 2: Algebraic Connections What values must x and y have to make the quadrilateral a parallelogram • State the 4 ways to prove a quadrilateral is a parallelogram. 8x - 6 3y 42

3. 1.) Definition of parallelogram 2.) Both pairs of opposite sides  3.) One pair of sides both || and  4.) Diagonals bisect each other 5.) Both pairs of opposite angles  6.) Answers vary 7.) Diagonals bisect each other 14.) See next slide 19.) x =18, y=14 20.) x=20, y=6 or -5 21.) x=10, y=2 22.) x=11, y=5 p. 174 #’s 1 – 7, 14, 19 -22 Homework: Wednesday, April 2

4. 14.) Given: Parallelogram ABCD M and N are midpoints of AB and DC Prove: AMCN is a parallelogram D N C A M B

5. Parallel Lines…What do you know? You should know lots… I hope 

6. Tell me EVERYTING You know about… Parallel Lines Parallelograms

7. Theorem 5.9: • If parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. (not just limited to 3 lines) Given: AX || BY||CZ AB  BC Prove: XY  YZ A X B Y C Z

8. Application of Theorem 5.9 • Given: AR||BS||CT; RS  ST • RS = 12, ST = • AB = 8; BC = • AC = 20; AB = • AC = 10x; BC = A R B S C T

9. Theorem 5.10: • A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. A Given: M is the midpoint of AB MN ||BC Prove: N is the midpoint of AC M N B C

10. Proof of Theorem 5.10: D Given: M is the midpoint of AB MN ||BC Prove: N is the midpoint of AC A Paragraph Proof: Let AD be the line through A parallel to MN. M N Then AD, MN , and BC are three parallel lines that cut off congruent segments on transversal AB. By Theorem 5.9 they also cut off congruent segments on AC. Thus AN  NC and N is the midpoint of AC. B C

11. Theorem 5.11: • The segment that joins the midpoints of two sides of a triangle: • (1) is parallel to the third side • (2) is half as long as the third side A Given: M is the midpoint of AB N is the midpoint of AC Prove: (1) MN || BC (2) MN = ½BC M N B C

12. Application of Theorem 5.11 B • Given: R, S, and T are midpoints of the sides of ABC. R S A T C

13. Mixing it all Together Applications of Theorems 5.9, 5.10, and 5.11

14. Algebraic Connections: Systems Practice If AB = 15, BC = 2x – y, and CD = x + y. Calculate x and y. A E B F C G D H

15. Algebraic Connections: Perimeter F • P, Q, and R are midpoints of the sides of DEF. • What kind of figure is DPQR? • What is the perimeter of DPQR? 12 Q R 10 E P D 8

16. Cool Down: Thursday, March 27th Self Test: p. 182 #’s 1 – 8 Remember: Quiz on Sections 5.1 – 5.3 Friday!!