2. Warm Up 9/19/12  Examine the diagrams below. For each pair of angles marked on the diagram, decide what relationship their measures have. Your responses should be limited to one of three relationships: congruent, complementary, and supplementary. SupplementaryCongruent Complementary Congruent

3. Transversal  Definition: A transversal is a line that intersects two coplanar lines at two distinct points.

4. Transversal  A transversal forms eight angles. The diagram below shows the eight angles formed by a transversal t and two lines l and m.

5. Special Angles Pairs of the eight angles have special names as suggested by their positions

6. Identifying Angles Alternative Angles: Same-Side Angles: Corresponding Angles:

8.  Marco was walking home after school thinking about special angle relationships when he happened to notice a pattern of parallelogram tiles on the wall of a building. Marco saw lots of special angle relationships in this pattern, so he decided to copy the patter into his notebook

9.  The beginning of Marco’s diagram is shown at right. This type of pattern is sometimes called a tiling. In a tiling, a shape is repeated without gaps or overlap to fill an entire page. In this case, the shape being tiled is a parallelogram

10.  Consider the angles inside a single parallelogram. Are any angles congruent?

11.  Since each parallelogram is a translation of another, what can be stated about the angles in the rest of Marco’s tiling?  Let’s watch the tiling take place on the next slide

12. Click to start tessellation Click to see angle being tessellated Marco’s Tile Pattern How can you create a tile pattern with a single parallelogram? Click to move on …

13.  What about relationships between lines? Can you identify any lines that must be parallel?  Mark all the lines with the same number of arrows to show which lines are parallel

14. Let’s take a closer look at Marco’s tiling…  Remember that a line that crosses two or more other lines is called a transversal.  In this diagram, which line is the transversal? Which lines are parallel?  Thinking about our parallelogram tiling, what is the relationship between angles x and b?

15. Corresponding Angles Postulate  What do we call angles x and b?  When two parallel lines are cut by a transversal, corresponding angles are congruent! b ≅ xb ≅ x

16. Corresponding Angles Postulate  Name all the other pairs of congruent corresponding angles a ≅ wa ≅ w b ≅ xb ≅ x c ≅ yc ≅ y d ≅ zd ≅ z

17. Properties of Parallel Lines

18. Let’s look again  Suppose m  b = 60  Use what you know about vertical, supplementary, and corresponding angles to find the measures of all the other angles

19. More Postulates  When a transversal intersects two parallel lines, we have two other interesting angle properties

22. Find the measure of angles 1 and 2. How do you know?

23. Using a two-column proof  You can display the steps that prove a theorem in a two-column proof

24. Use a two-column proof to prove Alt. Int. Angles are Congruent

25. StatementReasons

26. Since a||b, m  1 = 50 because corresponding angles are congruent (Corr. Angles Postulate) Since c||d, m  2 = 130 because same-side interior angles are congruent (Same Side Angles Theorem)

30. Classwork and Homework  Classwork  Lesson 3-1 Practice (whole page)  Homework  Practice 3-1 (half page)