1. H. Lines and Angles Math 10: A and W WA10.9
Demonstrate understanding of angles including:
drawing and sketching
replicating and constructing
bisecting
relating to parallel, perpendicular, and transversal lines
solving problems.

2. Key Terms Find the definition of each of the following terms: Angle
Angle measure
Degree
Parallel lines
Perpendicular lines
Transversal
Angle Referent
True Bearing
Complementary Angles
Supplementary Angles
Angle Bisector
Vertically Opposite Angles
Corresponding Angles
Alternate Interior Angles
Alternate Exterior Angles

3. 1. Measuring and Drawing Angles
WA10.9
Demonstrate understanding of angles including:
drawing and sketching
replicating and constructing

4. 1. Measuring and Drawing Angles
So what is an angle?
An angle is formed when two rays meet at a vertex (a common end point)

5. Angles are measured with tools, such as a protractor, and are measured in degrees.
Visualize an angle that is used to express direction in navigation and mapping, such as east. In this case, the angle is measured relative to true north, which is 0° and may be expressed as a bearing

6. A true bearing describes the # of degrees, measured clockwise, between an imaginary line pointing towards true North and another imaginary line pointing towards a intended direction or along a path
East is represented in land navigations and mapping at a 90° angle from true North.

7. Angle measures can be estimated by using referents, which are common measurements like, 90°, 45°, 30°, 22.5°
A protractor can be used to both draw and measure angles. A compass and straight edge can also be used to draw angle but can not measure angles.

8. Example
Using a protractor create angles with the following measurements:
90°
45°
35°
77°

9. Example
2. a) Draw a 90° angle using a compass and straight edge. b) Replicate any existing angle using compass and straight edge.

10. Activity 5.1 – 5 Angles p. 179
With a partner work through the activity.

12. Mental Example
Look at mental example on page 179 of your text and we will work through together.

14. Example
3. Estimate the following angle.

15. Example 4

16. Activity 5.2 – Creating a Referents Diagram
Working with a partner work through the activity.

17. Example
5. Sort the following angles into pairs of complementary and supplementary angles: ∠1=42° ∠2=107° ∠3=59° ∠4=48° ∠5=121° ∠6=31° ∠7=19° ∠8=73°

18. Activity 5.3 – Using Angles in Weather and Reporting p. 182
Work through with a partner.

19. Example 6

20. Building your Skills
Ex. 5.1 (p.184) #1-7

21. 2. Bisecting Angles and Perpendicular Lines
WA10.9
Demonstrate understanding of angles including:
drawing and sketching
replicating and constructing
bisecting
relating to parallel, perpendicular, and transversal lines
solving problems.

22. 2. Bisecting Angles and Perpendicular Lines
Math on the Job p. 187
Read through together and solve.

24. Bisecting an object involves dividing it into two congruent (equal parts).
When you bisect an angle you divide it into 2 angles of equal measure. For example, when you bisect a 76° angle you get two 38° angles
The line, line segment or rays that separates the 2 halves of a bisected angle in called the angle bisector.

25. A right (90°) angle can be thought of as a bisected straight (180°) angle.

26. Perpendicular lines and line segments form right angles
Perpendicular lines and line segments are drawn using the same techniques that are used to bisect angles.

27. *
Because perpendicular lines and line segments are so common and so frequently made, specialized tools such as carpenters squares (framing squares) have been developed to make them.

28. Finishing carpenters and other wood workers often make miter joints, where the ends of 2 end pieces of wood are cut at angles having the same measure. When the two ends are joined, they form a right angle, with the miter joint acting as a bisector.

29. Example Using a protractor and ruler bisect a line.
Using a compass and straight edge bisect a line.

30. Example 2. Using a protractor and ruler bisect an angle.
Using a compass and straight edge bisect an angle.

31. Activity 5.4 – Kitchen Counter Top Plan p. 190
Work through with a partner.

32. Example
3. Given the following rectangle with the given numbered angles identify the following items:
Adjacent angles that are complementary
Adjacent angles that are supplementary
Pairs of line segments that are perpendicular
The line segments (if any) that are angle bisectors

33. Building your Skills
Ex. 5.2 (p. 192) #1-7, 8 challenge

34. 3. Non-Parallel Lines
WA10.9
Demonstrate understanding of angles including:
bisecting
relating to parallel, perpendicular, and transversal lines
solving problems.

35. 3. Non-Parallel Lines Math on the Job p. 198
Read through together and solve.

37. A variety of objects and materials such as trusses, railroad tracks and fabrics contain intersecting lines.
The measure of certain angles created by intersecting lines and the ability to identify types of angles can indicate whether these lines are parallel or non parallel.

38. When two lines intersect, four distinct angles are created
The angles that share a side are adjacent angles.
The angles the share only a vertex are vertically opposite angles.

39. Which 2 angles are adjacent to ∠1?
What type of angle do they make?
What angle is vertically opposite ∠4?
How are they compared to each other?

40. Adjacent angles are supplementary (add up to 180°)
Vertically Opposite angles are congruent

41. Suppose that there are 2 main lines ( 𝑙 1 𝑎𝑛𝑑 𝑙 2 ) and a third line (t) intersects both of them. The third line (t) is a transversal.
When a transversal intersects 2 lines there are angles formed that fit into categories based on their relative position to each other.

42. 1 2 3 4 5 6 7 8
𝑙 1
𝑙 2

43. Angles that have the same corresponding position at each intersection point are called corresponding angles
Examples
Angles that are inside the 2 lines and on opposite sides of the transversal are called alternate interior angles 1 2 3 4 5 6 7 8
𝑙 1
𝑙 2

44. Angles that are on inside of the 2 lines and same side of the transversal are same side interior angles.
Examples
Angles on the outside of the 2 lines and on opposite sides of the transversal are called alternate exterior angles 1 2 3 4 5 6 7 8
𝑙 1
𝑙 2

45. Angles on the outside of the 2 lines and on the same side of the transversal are called same side exterior angles.
Examples 1 2 3 4 5 6 7 8
𝑙 1
𝑙 2

46. Example 1

47. Example 2

49. Building your Skills
Ex. 5.3 (p. 204) 1-3, 4 challenge

50. 4. Transversals and Parallel Lines
WA10.9
Demonstrate understanding of angles including:
relating to parallel, perpendicular, and transversal lines
solving problems.

51. 4. Transversals and Parallel Lines
Math on the Job p. 209
Read through together and solve.

53. Two lines are parallel if they never intersect each other
Two lines are parallel if they never intersect each other. This only happens when the lines are a constant distance from each other.

54. If two lines are parallel and are intersected by a transversal, the corresponding angles, alternate interior angles, same side interior angles, alternate exterior angles and same side exterior angles all have certain properties.

55. 1 2 3 4 5 6 7 8
𝑙 1
𝑙 2

56. When a transversal cuts two parallel lines:
Corresponding Angles are congruent
Examples
Alternate Interior Angles are congruent 1 2 3 4 5 6 7 8
𝑙 1
𝑙 2

57. Alternate Exterior Angles are congruent Examples
If these angles are not congruent the lines are not parallel 1 2 3 4 5 6 7 8
𝑙 1
𝑙 2

58. Same side Interior Angles are supplementary Examples
Same side Exterior Angles are supplementary
If these angles are not supplementary the 2 lines are not parallel 1 2 3 4 5 6 7 8
𝑙 1
𝑙 2

59. Example 1

60. Example 2

61. Building your Skills
Ex. 5.4 (p.215) #1-7