1. Objectives: To identify angles formed by two lines and a transversal To prove and use properties of parallel lines

2. Transversal -> a line that intersects two coplanar lines at two distinct points Notice, the transversal creates 8 distinct angles.

3. <1 and <2 are alternate interior angles <1 and <4 are same-side interior angles <1 and <7 are corresponding angles

4. * Postulate 3.1 - Corresponding Angles Postulate * If a transversal intersects two parallel lines, then the corresponding angles are congruent 1 2 <1 ≈ <2

5. * Theorem 3.1 - Alternate Interior Angles Theorem * If a transversal intersects two parallel lines, then alternate interior angles are congruent a b 1 23 4 <1 ≈ <3

6. * Theorem 3.2 - Same-side Interior Angles Theorem * If a transversal intersects two parallel lines, then same-side interior angles are supplementary 1 23 4 m<1 + m<2 = 180 a b

7. * Theorem 3.3 - Alternate Exterior Angles Theorem * If a transversal intersects two parallel lines, then alternate exterior angles are congruent 1 2 3 <1 ≈ <3 a b

8. * Theorem 3.4 - Same-side Exterior Angles Theorem * If a transversal intersects two parallel lines, then the same-side exterior angles are supplementary 1 2 3 a b m<2 + m<3 = 180

9. a b c d Given: a parallel to b c parallel to d Find measures of all angles <1 = ?<5 = ? <2 = ?<6 = ? <3 = ?<7 = ? <4 = ?<8 = ? 8 7 50° 6 2 5 1 3 4

10. a b c d 8 7 6 2 5 1 3 4 m<1 = 50° (corresponding angles) m<2 = 130° (same-side interior angles are supplementary) m<3 = 130° (corresponding angle to <2) m<4 = 130° (vertical angle to <3) m<5 = 50° (alternate interior angle) m<6 = 50° (alternate interior angle) m<7 = 130° (same-side interior angles are supplementary; <6) m<8 = 50 (vertical angle to 50°)

11. * Homework #12 * Due Wednesday Sept 19 * Page 131 – 132 * # 1- 7 odd * # 11 – 25 odd

12. * Objectives: To use a transversal in proving lines parallel

13. * Postulate 3.2 - Converse of the Corresponding Angles Postulate * If two lines and a transversal form corresponding angles that are congruent, then the two lines are parallel. 1 2 P Q P Q

14. * Theorem 3.5 - Converse of the Alternate Interior Angles Theorem * If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. 1 4 2 If <1 = <2, then P parallel to Q P Q ~

15. * Theorem 3.6 - Converse of the Same-Side Interior Angles Theorem * If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. 14 2 P Q If <2 and <4 are supplementary, then P parallel to Q

16. * Ex: Which lines, if any must be parallel if <1 congruent <2? * Justify the answer with a theorem or postulate. E D C K 1 3 4 2 Which lines, if any, must be parallel if <3 congruent <4? Which lines, if any, must be parallel if <3 and <2 are supplementary?

17. * Theorem 3.7 - Converse of the Alternate Exterior Angles Theorem * If two lines and a transversal form alternate exterior angles that are congruent, then the two lines are parallel. P Q 1 3 2 If <1 congruent <2, then P parallel to Q

18. * Theorem 3.8 - Converse of the Same-Side Exterior Angles Theorem * If two lines and a transversal form same-side exterior angles that are supplementary, then the two lines are parallel. P Q 1 3 2 If <1 and <3 are supplementary, then P parallel to Q

19. * Example using algebra 40° (2x + 6)° Find the value of x for which P parallel to Q. Explain how you came to your answer. Use both possible ways. P Q 140°

20. * Homework # 13 * Due Thursday/Friday (Sept 20/21) * Page 137 – 138 * #1-21 odd

21. * Objectives: To relate parallel and perpendicular lines

22. * Theorem 3-9 * If two lines are parallel to the same line, then they are parallel to each other. Given: a II c c II b Therefore a II b

23. * Theorem 3-10 * In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. m n t m II n

24. * Theorems 3-9 and 3-10 gave conditions by which you can conclude that lines are parallel. The next theorem will proved a way to conclude that lines are perpendicular. * Theorem 3-11 * In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. n l m n perpendicular to m

25. * Homework #14 * Due Monday (Sept 24) * Page 143 – 144 * # 1 – 21 odd

26. * Objectives: To classify triangles and find the measures of their angles To use exterior angles of triangles

27. * Theorem 3-12 -> Triangle Angle-Sum Theorem * The sum of the measures of the angles of a triangle is 180. m<A + m<B + m<C = 180 A B C

28. G F J H 65 39 21 x yz Find the value of x, y, and z.

29. * In chapter 1, we classified angles by their measures (acute, right, obtuse). We can also classify a triangle by its angles and sides. Equiangular All angles congruent Acute all angles acute Right one right angle Obtuse one obtuse angle Equilateral all sides congruent Isosceles two sides congruent Scalene no sides congruent

30. * Exterior angle of a polygon -> an angle formed by a side and an extension of an adjacent side * Remote interior angles -> the two nonadjacent interior angles for each exterior angle 1 <– Exterior angle 3 2 Remote interior angles

31. * Theorem 3-13 -> Triangle Exterior Angle Theorem * The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. 1 2 3 m<1 = m<2 + m<3

32. * Homework # 15 * Due Tuesday (Sept 25) * Page 150 * # 1-20 all * Quiz Thursday/Friday * Section 3.1 – 3.5

33. * Objectives: To classify polygons To find the sums of the measures of the interior and exterior angles of polygons

34. * Polygon -> a closed plane figure with at least three sides that are segments. The sides intersect only at their endpoints, and no adjacent sides are collinear. * Naming polygons -> start at any vertex and list the vertices consecutively in a clockwise or counterclockwise direction.

35. Polygon Not a polygon: not a closed figure Not a polygon: two sides intersect between endpoints A B C D E The polygon above can be named in a few ways: ABCDE, EDCBA, CDEAB, BCDEA, etc… The polygon includes five segments Segment: AB, BC, CD, DE, EA The polygon includes five angles <A, <B, <C, <D, <E

36. * Classifying Polygons * Can be classified by the number of sides * 3 – Triangle, 4 – quadrilateral, 5 – pentagon, 6 – hexagon, 7 – heptagon, 8 – octagon, 9 – nonagon, 10 – decagon, 12 – dodecagon * Can be classified by shape * Convex -> has no diagonal with points outside the polygon * Concave -> has at least one diagonal with points outside the polygon Convex Concave

37. * Classify the following polygons by its sides. Identify each as convex or concave

38. * Theorem 3-14 -> Polygon Angle-Sum Theorem * The sum of the measures of the, angles of an n-gon is (n - 2)180, where n is the number of sides of the figure.

39. * Ex: * Find the sum of the measures of the angles of a 15-gon Ex: The sum of the measures of the angles of a given polygon is 720. How many sides did the polygon have?

40. * Theorem 3-15 -> Polygon Exterior Angle-Sum Theorem * The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. 1 2 3 4 5 For the above pentagon: m<1 + m<2 + m<3 + m<4 + m<5 = 360 Equilateral polygon -> all sides are congruent Equiangular polygon -> all angles are congruent Regular polygon -> both equilateral and equiangular

41. * Homework #16 * Due Wed (Sept 26) * Page 161 * # 1 – 25 all * Quiz Thursday/Friday

42. * Objectives: To graph lines given their equations To write equations of lines

43. * Slope-intercept form -> y = mx + b * m = the slope of the line * b = the y-intercept * The y-intercept is the y-coordinate of the point where a line crosses the y-axis. The x- intercept is the x-coordinate of the point where a line crosses the x-axis.

44. * By postulate 1-1 (two points determine a line), you need only two points to graph a line. * The y-intercept gives one point, the slope can be used to plot another. * Standard form of a linear equation -> Ax + By = C, where A, B, and C are real numbers and A and B are not both zero. * To graph an equation written in standard form, you can readily find two points for the graph by finding: * The x-intercept * The y-intercept Ex: Graph 6x + 3y = 12 Graph -2x + 4y = -8

46. * Homework #17 * Due Tuesday (Oct 2) * Page 169 * #1 – 27 odd

47. * Objectives: To relate slope and parallel lines To relate slope and perpendicular lines

48. * Slopes of Parallel Lines * If two non-vertical lines are parallel, their slopes are equal. Likewise, if the slopes of two distinct non-vertical lines are equal, the lines are parallel. * Any two vertical lines are parallel.

51. * Homework #18 * Due Wednesday (Oct 3) * Page 177 – 178 * #1 – 31 odd