1. Ch. 4 Angles and Parallel Lines
Math 10-3

2. Angles and Parallel Lines
Day 1:
Angles and Parallel Lines

3. Terminology Angle Two rays that meet a point called the vertex
Right Angle
900
Straight Angle
1800
Ray 1
Vertex
Ray 2

4. Acute Angle
An angle less than 900
Obtuse Angle
An angle greater than 900 but less than 1800
Reflex Angle
An angle greater than 1800 but less than 3600

5. Full Circle
3600
Quarter Circle
900
Half Circle
1800
*Note: = 1800
¾ Circle
2700
*Note: = 2700

6. All three angles add up to 1800
Triangle
All three angles add up to 1800
A + B + C = 1800
Complementary Angles
Two angles that add up to 900
A + B = 900
Supplementary Angles
Two angles that add up to 1800
A + B = 1800 C B A A B B A

7. Parallel
Two lines that will never cross; they are always the same distance apart
Perpendicular
Two lines at right angles (900)

8. True Bearing (compass)
The angle measured clockwise between true north and an intended path or direction, expressed in degrees N NE NW W E SW SE S

9. Assignment: M103 Angles Day 1 Assignment.doc
Game: Angle Basics Bingo (excel document)

10. Day 2: Estimating and measuring Angles

11. Estimating Angles
One way to estimate angles is to draw dotted lines where 90⁰ and 180⁰ should be
ex1: angle 1 is a little less than 90⁰
An estimate of about 70⁰ is good

12. EX 2:
Angle 2 is a bit more than 90⁰ but less than 180⁰.
An estimate of about 130⁰ is good.

13. How to measure Angles using a Protractor
Step 1: Position the protractor at the vertex of the angle.
Step 2: Line up the straight edge of the protractor (00/1800) along one ray
Step 3: Determine where the other ray reaches. This will be your angle.

14. Ex3. Determine the measure of the following angle:

15. How to Draw Angles Using a Protractor and Ruler
Step 1: Draw a straight line with your ruler, ~5 cm long
Step 2: On one end of your line, position the very center of your protractor on the edge of the line.

16. Step 3: Using the protractor scale, place a faint mark at the desired degree.
Step 4: Remove the protractor, and using your ruler, draw a line that connects the end of your first line with the faint mark you drew in step 3.

17. Ex 4: Construct an angle of 30⁰

18. Ex: 5 construct an angle of 250⁰

19. Assignment:
M103 estimating and constructing Angles.doc

20. Day 3: Describing Angles

21. Adjacent angles are angles that share a common vertex and a common arm.
Complementary angles are angles that add up to 90⁰
Supplementary angles are angles that add up to 180⁰

22. When describing angles using letter, the middle letter is in the vertex position
Ex1: ABC the vertex is at letter B. the angle is between the arms A and C A
B C

23. When given one angle in a pair of complementary / supplementary angles, we can easily calculate the measure of the second angle
We know that  ABC and CBD are complementary (add up to 90⁰)
27⁰ + CBD = 90⁰
CBD = 90⁰ - 56⁰ = 34⁰ A C
27⁰ B D

24. Ex3: Determine the measure of X and Y
We know that y and 56⁰ are complementary.
We know that y and 56 are complementary
56 + y = Y= 90 – 56 = 34°
We also know that a straight line is equal to therefore, x y + 46 = 180
X = x = 180
180 – 136 = 44°
56⁰ Y X
46⁰

25. Angles that are opposite to each other have the same measure
Angles that are opposite to each other have the same measure. We call these Vertically Opposite Angles.
X is vertically opposite 150,
x = 150
Y is vertically opposite 30,
y = 30 X Y
30°
150°

26. Example 5 Determine the measure of x and y
Y is vertically opposite 25, y = 25
Notice how x and 25 are on a straight line?
They are supplementary!
X + 25 = 180
X = 180 – 25 = 155° x y
25°

27. Assignment:
M 103 Describing Angles.doc
Quiz tomorrow!

28. Day 4: Bisecting Angles

29. A bisector is a line that divides an angle or line into two equal parts.
Method 1: measure the angle with a protractor. Divide the measure by 2. Use protractor and ruler to draw the bisecting line

30. Ex. 1 Bisect ABC Step 1: measure the angle Step 2: divide by 2
Step 3: draw a bisecting line at 20° using a ruler

31. Assignment:
M103 Bisecting angles.doc

32. Day 5: replicating Angles

33. Many people who work in the trades may need to replicate angles
Many people who work in the trades may need to replicate angles. Especially carpenters, and construction workers
To replicate an angle, use a protractor

34. Protractor method Using the trapezoid, we will copy CDA
Measure the angle with the protractor
Draw side AD
Use the protractor to mark the correct angle
Draw line CD to form CDA A D B C

35. Percents of a circle Review: How many degrees in a circle? 360°
If you shade an entire circle, what percent would this be?
100%

36. Let’s consider the following habits of Mrs. More per month
Item Amount Percent of total (amount /total)
Shoes $ /1750 = 14%
Clothes $ %
Eating Out $ %
Hockey games $ %
Gym membership $ %
Total $ %

37. How can we change these percents into angles?
Set up a comparison ratio!
Remember, Percent means out of 100!
= x
Cross multiply and divide to determine approximate degrees!

38. Item
Shoes
Clothes
Eating Out
Hockey games
Gym membership
Degree
50°
166°
72°
32°
40°

39. We can now construct an accurate pie chart!

40. Assignment: M103 Replicating Angles.doc

41. Day 6: Classifying Angles and Lines

42. Consider the following rectangle
Which sides are parallel?
Which sides are perpendicular?
Notice how the opposite sides are parallel, and the adjacent sides are perpendicular? Adjacent means “next to” A//C & B //D A,D & D,C & C,B & B,A

43. Can you name all pairs of adjacent supplementary angles?
Many angles are formed by two lines and a transversal – a line that intersects TWO or more lines
Can you name all pairs of adjacent supplementary angles? T 1 2 A 4 3 5 6 B 7 8
1,2 3, , 6 7, 8 1,3 2,4 5,7 6,8

44. Are other ways to describe adjacent pairs of angles
Corresponding angles: two angles formed by two lines and a transversal, located on the same side of the transversal. For example 1 and 5
Opposite angles: non adjacent angles that are formed by two intersecting lines For example  1 and 4
Alternate angles: Two angles formed by two lines and a transversal, located on opposite sides of the transversal For example:  3 and 6 are INTERIOR alternate angles  3 = 6

45. Assignment:
Classifying lines and angles.doc

46. Day 7: Parallel Lines and transversals

47. Parallel lines and transversals have some special properties that you may have already notices.
How can we determine that lines are parallel?
If we draw a perpendicular transversal line between two parallel lines, what will the angles be equal to? Try it!

49. What will the measure of ALL the other angles formed by the transversal be equal to?
90°
This is because of the other rules about angles we already know:
Supplementary angles (angles on a straight line) add up to ( = 1800 )
Vertically opposite angles are equal
A complete circle is equal to 3600 ( = 3600)

50. These properties can help us make more rules about transversal and parallel lines. Consider the following:
With just one angle labeled, we can determine the measure of every other angle.
b: = 75 Corresponding angles formed from a transversal of parallel lines are equal in measure
F = 75
Opposite angles formed from a transversal of parallel lines are equal in measure a b c d e
75° f g

51. g = 105
( = 105) adjacent angles along a transversal line are supplementary
c = 75
Alternate interior angles formed from a transversal of parallel lines are equal in measure. a b c d e
75° g f

52. Assignment:
M103 parallel Lines and transversal.doc
Quiz tomorrow!!

53. Day 8: Calculating angles

54. We can use our knowledge of angles and lines to solve all types of problems.
Ex1: Determine the unknown measures in the following diagram.
216° x y

55. To determine the measure of x, we know that a circle must equal 360
We can then extend the transversal to help create a straight line.
z = 216 – 180 = 36°
Z can be determined by subtracting from 2160.
Angles y and z are corresponding angles of and are thus equal. z x y

56. Ex2: Determine the unknown angles in the following diagram:
We know m is opposite 87° m =87°
87° and f are supplementary
180 – 97 = 93°
We know that a triangle must add up to 180°
M + n + 35 = 180°
180° – 87°-35° = 58° n f
87° m
35°

57. Assignment:
M103 calculating angles.doc