1. 3-1 PROPERTIES OF PARALLEL LINES SWBAT: Identify angles formed by two lines and a transversal Prove and use properties of parallel lines

2. Transversal A transversal is a line that intersects two coplanar lines at two distinct points What is the transversal here? Line t Eight angles are formed by a transversal intersecting two coplanar lines

3. Corresponding Angles Angles that lie on the same side of the transversal and in corresponding positions relative to the two intersected lines (Think Same Quadrants in coordinate plane) Which angles here are corresponding?

4. Alternate Interior Angles Alternate Interior Angles are non adjacent interior angles that lie on opposite sides of the transversal. Which angles are alternate interior? <1 and <2 <3 and <4

5. Same Side (Consecutive) Interior Angles Angles that lie on the same side of the transversal between the two lines intersected by the transversal Which angles fit this description? <1 and <4 <3 and <2

6. Alternate Exterior Angles Alternate Exterior Angles are non adjacent exterior angles that lie on opposite sides of the transversal. These angles will lie outside the two lines. Which angles are alternate exterior? <5 and <8 <7 and <6

7. Same Side (Consecutive) Exterior Angles Angles that lie on the same side of the transversal outside the two lines intersected by the transversal Which angles fit this description? <5 and <7 <6 and <8

8. Finding Angle Pairs

9. Special Relationships with Parallel Lines! Parallel Lines: Two coplanar lines that do not intersect. When the two coplanar lines that are intersected by the transversal are parallel, special relationships exist between the different angle types. Let’s explore what these relationships are…

11. In the diagram above, the m<1 = 110° Find the measure of the other 7 angles.

12. Example 1

13. Example 1B

14. Remember… A Postulate is an accepted statement of fact… Theorems are created by using Postulates to prove them.

15. Two Column Proof

16. Proof of Theorem 3-1: Given: a || b Prove: <1 ≅ <3 StatementsReasons 1.) a || b 1. 2 ) <1 ≅ <42. 3.)<4 ≅ <33. 4.)<1 ≅ <34. Given Corresponding Angles Substitution Vertical Angles The above Proof shows that by using the Corresponding Angle Postulate, that Alternate Interior Angles are Congruent!

17. Proof of Theorem 3-2: Given: a || b Prove: m<1 + m<2 = 180 StatementsReasons 1.) a || b 1. 2 ) m<3 + m<2 = 1802. 3.)<3 ≅ <13. 4.)m<1 + m<2 = 1804. Given Supplementary Angles Corresponding Angles Substitution The above Proof shows that by using the Corresponding Angle Postulate, that Same Side Interior Angles are Supplementary!

19. Ex 2: 1)

20. Proof of Theorem 3-3: Given: a || b Prove: <1 ≅ <4 StatementsReasons 1.) a || b 1. 2 ) <1 ≅ <22. 3.)<2 ≅ <43. 4.)<1 ≅ <44. Given Corresponding Angles Substitution Vertical Angles The above Proof shows that by using the Corresponding Angle Postulate, that Alternate Exterior Angles are Congruent!

21. Proof of Theorem 3-4: Given: a || b Proof: m<1 + m<2 = 180 StatementsReasons 1.) a || b 1. 2 ) m<3 + m<2 = 1802. 3.)<3 ≅ <13. 4.)m<1 + m<2 = 1804. Given Supplementary Angles Corresponding Angles Substitution The above Proof shows that by using the Corresponding Angle Postulate, that Same Side Exterior Angles are Supplementary!

23. Example 3

24. Find the Measure of Each Angle

25. Using Algebra to Find Angle Measures

26. Solve for x and y.

27. Solve For x. Find and Explain how to find measure of each angle.