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Ch. 4 Angles and Parallel Lines

Ch. 4 Angles and Parallel Lines. Math 10-3. Day 1: . Angles and Parallel Lines. Terminology . Ray 1. Vertex. Ray 2. C. B. A. A. B. B. A. N. NE. NW. W. E. SW. SE. S. Assignment: M103 Angles Day 1 Assignment.doc Game: Angle Basics Bingo (excel document).

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Ch. 4 Angles and Parallel Lines

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

  2. Day 1: Angles and Parallel Lines

  3. Terminology Ray 1 Vertex Ray 2

  4. C B A A B B A

  5. N NE NW W E SW SE S

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

  7. Day 2: Estimating and measuring Angles

  8. 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

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

  10. 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.

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

  12. 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.

  13. 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.

  14. Ex 4: Construct an angle of 30⁰

  15. Ex: 5 construct an angle of 250⁰

  16. Assignment: • M103 estimating and constructing Angles.doc

  17. Day 3: Describing Angles

  18. 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

  19. 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

  20. 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

  21. 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 = 90 Y= 90 – 56 = 34° We also know that a straight line is equal to 180. therefore, x + 56 + y + 46 = 180 X + 56 + 34 + 46 = 180 x + 136 = 180 180 – 136 = 44° 56⁰ Y X 46⁰

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

  23. 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°

  24. Assignment: • M 103 Describing Angles.doc • Quiz tomorrow!

  25. Day 4: Bisecting Angles

  26. 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

  27. 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

  28. Assignment: • M103 Bisecting angles.doc

  29. Day 5: replicating Angles

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

  31. 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

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

  33. Let’s consider the following habits of Mrs. More per month Item Amount Percent of total (amount /total) Shoes $250 250/1750 = 14% Clothes $800 46% Eating Out $350 20% Hockey games $150 9% Gym membership $200 11% Total $ 1750 100%

  34. How can we change these percents into angles? • Set up a comparison ratio! • Remember, Percent means out of 100! 14 = x 100 360 Cross multiply and divide to determine approximate degrees!

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

  36. We can now construct an accurate pie chart!

  37. Assignment: M103 Replicating Angles.doc

  38. Day 6: Classifying Angles and Lines

  39. 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

  40. 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, 4 5, 6 7, 8 1,3 2,4 5,7 6,8

  41. 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

  42. Assignment: • Classifying lines and angles.doc

  43. Day 7: Parallel Lines and transversals

  44. 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? 900. Try it!

  45. 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 (900 + 900 = 1800 ) • Vertically opposite angles are equal • A complete circle is equal to 3600 (900 + 900 + 900 + 900 = 3600)

  46. 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

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