Math 21 Home. Book 8: Angles. Teacher Version Assessments and Answers Included

Transcription

1 Math 21 Home Book 8: Angles Teacher Version Assessments and Answers Included

2 Year Overview: Earning and Spending Money Home Travel & Transportation Recreation and Wellness 1. Budget 2. Personal Banking 3. Interest 4. Consumer Credit 5. Major Purchases Topic Overview 6. Scale Drawings & Ratios 7. Area & Volume 8. Angles 9. Triangles 10. Slope & Elevation 11. Travel Project 12. Puzzles & Games 13. Understanding Statistics 14. Budgeting Recreation There is a lot of mathematics that can help you understand, design, and create things at home. Measuring angles at home helps you to design and build things such as stairs and furniture. Knowing the language of angles and how to measure them is important to home construction. Suggested Timeframe: 8 Hours Outcomes Theme Specific Outcomes M21.5 [FM20.3] Demonstrate understanding of angles created by parallel, perpendicular, and transversal lines and solve problems within the home theme. 2

3 Contents Topic Overview... 2 Outcomes... 2 Contents... 3 Angles Angle Symbols and Terminology... 6 A. Symbols... 6 B. Types of Angles... 7 Discuss the Ideas Practice Your Skills Classifying Angles Measuring Angles A. Parts of an Angle B. Parts of a protractor C. How to use a protractor: Practice Your Skills Angle Relationships A. Line Terminology B. Complementary and Supplementary Angles B Practice Your Skills Complementary and Supplementary Angles C. Transversal Lines C Practice Your Skills Transversal Lines C Discuss the Ideas Transversal Lines Practice Your Skills: Practice Quiz on Angles Student Evaluation Learning Log Show What You Know Quiz Angles Show What You Know- Project #1: Angles in your Home Show What You Know - Project # 2: House Design Answers

4 Angles What do you think the builder in the picture would have to measure in order to properly cut the stairway railing to fit to size? Many structures you see around the home involve the use of angles. It would be very difficult to build any structures within your home without a knowledge of angles. In this section, you will learn about the various types of angles, how to measure them, and how they relate to one another. 4

5 Check Your Skills Match the following by placing the letter of the correct description in the blank space provided beside each term. 1. point 2. line 3. line segment 4. parallel 5. perpendicular 6. angle 7. degree 8. right angle a. a portion of a line which has two definite endpoints. b. a term used to describe 2 or more lines that never intersect and are the same distance apart at every interval. c. the amount of turn between two straight lines that have a common end point d. is straight, has no thickness and extends infinitely in opposite directions e. a measure for angles - there are 360 of them in a full rotation or circle. f. an angle measure of 90. g. represented with a single dot on a piece of paper although it technically has no length or width. h. a term used to describe 2 lines that meet at 90 angles. 5

6 8.1 Angle Symbols and Terminology A. Symbols Symbol Meaning Example In Words Triangle ΔABC Triangle ABC Angle <ABC Angle ABC Perpendicular AB CD AB is perpendicular to CD Parallel AB CD AB is parallel to CD Degrees <ABC is 45 The measure of angle ABC is 45 degrees Right Angle is 90 A right angle is 90 degrees Line Segment AB Line Ray AB The line between points A and B The infinite line that includes points A and B The line that starts at point A, goes through B, and continues on 6

7 B. Types of Angles Acute Angle Right Angle Obtuse Angle Straight Angle Reflex Angle Full Rotation Less than 90 Equal to 90 Between 90 and 180 Equal to 180 Between 180 and 360 Equal to 360 7

8 Discuss the Ideas 1. There are different types of angles everywhere. In the diagram below, label three acute, right, obtuse, and straight angles. 2. What is something that you notice about straight angles? Are they easy to spot? 3. What type of angle is easiest to identify? Explain. 8

9 8.1 Practice Your Skills Classifying Angles Classify each angle as acute, obtuse, right, or straight

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11 8.2 Measuring Angles You often need to measure angles, not just identify their types. This would be true when you are cutting wood with a mitre saw or designing a pergola for your deck. A. Parts of an Angle B. Parts of a protractor 11

12 C. How to use a protractor: 1. Line up your protractor s zero line with one arm of the angle you are measuring. 2. Match the vertex of your angle with the vertex of your protractor, making sure that the second arm of the angle falls in the measures of the protractor, like above. 3. Start from the zero that your angle arm is on and measure up and toward the other angle arm. If you re not sure which of the two numbers to use, think: Is my angle acute (less than 90 ) or obtuse (greater than 90 )? 8.2 Practice Your Skills Go back to your Practice Your Skills Classifying Angles worksheet. Measure all of the angles that were either acute or obtuse. 12

13 8.3 Angle Relationships Sometimes you know the measure of one angle, and you can calculate others based on the angle relationships that exist. A. Line Terminology Parallel Lines Are equidistant, will never meet > > Perpendicular Lines Meet at a 90 angle Supplementary Angles Add to Complementary Angles Add to Transversal Lines Is a line that crosses at least two other lines In each of the above examples, the red line is the transversal line. 13

14 B. Complementary and Supplementary Angles Complementary and Supplementary Angles let you calculate angle measures. Look at the following picture. The joists are at right angles to one another. 40 Then what are the measures of these two angles? Calculating Angle A: Supplementary angles have a total measure of 180 Angle A = Angle A = 140 Calculating Angle B: Complementary angles have a total measure of 90 Angle B = Angle B = 50 14

15 8.3B Practice Your Skills Complementary and Supplementary Angles Find the measure of angle x x x x 37 x x 145 x X 89 x x 114 x 15

16 C. Transversal Lines Transversal Lines can be found everywhere. The most common occurrences are when you have two parallel lines and a transversal line that runs through them (windows, building designs, railways, roads). Working with these angles can help us work backwards to create parallel lines, too. Vertically Opposite Angles X-pattern (a = b) - Are equal These are angles that are across from each other Corresponding Angles F-pattern (a = b) - Are equal when lines are parallel These are angles in matching corners Consecutive Interior Angles C-pattern (a + b = 180 ) - Add to 180 when lines are parallel These are angles on the same side of the transversal, in the inside space made by the two lines Alternate Interior Angles Z-pattern (a = b) - Are equal when lines are parallel These are angles on opposite, interior sides of the transversal 16

17 8.3C Practice Your Skills Transversal Lines Calculate the measure of each angle indicated. 17

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19 8.3C Discuss the Ideas Transversal Lines 1. Highway Number 1 is a major highway that begins and ends at the Pacific and Atlantic Oceans. Why do you think Highway 1 has been nicknamed the Trans Canada Highway? 2. Label the arrow that acts as the transversal. Using the angle indicated, label the measure of all other angles

20 Practice Your Skills: Practice Quiz on Angles A. Draw angle A to the nearest degree. 20

21 B. Find the measure of the angle to the nearest degree. 21

22 c. Classify each angle as acute, obtuse, right or straight. D. Find the measure of angle x. 22

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24 Student Evaluation Insufficient Evidence (IE) Student has not demonstrated the criteria below. Developing (D) Growing (G) Proficient (P) Exceptional (E) Student has rarely demonstrated the criteria below. Student has inconsistently demonstrated the criteria below. Student has consistently demonstrated the criteria below. Student has consistently demonstrated the criteria below. In addition they have shown their understanding in novel situations or at a higher level of thinking than what is expected by the criteria. Proficient Level Criteria IE D G P E M21.5 [FM20.3] Demonstrate understanding of angles created by parallel, perpendicular, and transversal lines and solve problems within the home theme. a. I can observe and group pairs of lines as perpendicular, parallel, or neither and justify the reasoning (e.g. tiling, ceiling tiles, flooring, framing, cutting a window frame). b. I can generalize, create, explain, and use relationships between pairs of angles formed by parallel lines and a transversal, including: Corresponding angles Vertically opposite angles Alternate interior angles Alternate exterior angles Interior angles on the same side of the transversal Exterior angles on the same side of the transversal c. I can provide real and picture examples that show that there are no angle relationships (excluding vertically opposite angles) when two lines that are not parallel are crossed by a transversal. 24

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26 Learning Log Date Starting Point Ending Point 26

27 Show What You Know Quiz Angles A. Draw angle A to the nearest degree. 1) 2) B. Find the measure of each angle to the nearest degree. c. Classify each angle as acute, obtuse, right or straight. 27

28 D. Find the measure of angle x. 28

29 Show What You Know- Project #1: Angles in your Home In construction of homes and buildings, angles are created when two lines meet. Pick two of the following items from within your home or school environment: Porch Veranda Railings on stairs Trusses Fence Take a photo of the item and highlight with different coloured markers, lines which are parallel, perpendicular, or neither. Also highlight lines that are transversals. *If you cannot take your own pictures, you can use images from online. 1. How do you know if the lines are parallel? Justify your answer. 2. How do you know if the lines are perpendicular? Justify your answer. 3. What objects represent transversals? 29

30 Show What You Know - Project # 2: House Design Use a computer application (Paint will work) to design the front of a house. Your house should have the following: Door 2 or more windows At least 3 acute angles At least 3 obtuse angles At least 5 right angles BONUS: At least 2 examples of transversal lines 30

31 Answers 8.1 Check Your Skills 1. g 2. d 3. a 4. b 5. h 6. c 7. e 8. f 8.2 Practice Your Skills Classifying Angles 1. Acute 2. Obtuse 3. Obtuse 4. Straight 5. Acute 6. Right 7. Straight 8. Acute 9. Right 10. Obtuse 11. Right 12. Right 13. Straight 14. Obtuse 15. Acute 16. Straight 17. Acute 8.3 Practice Your Skills

32 8.4B Practice Your Skills Complementary and Supplementary Angles 1. X = X = X = X = X = X = X = X = X = X = C Practice Your Skills Transversal Lines 1. x = y = k = b = a = x = z = m = x = 70 y = a = 45 b = k = 140 m = a = 105 b = k = 112 j = s = 100 t = e = 66 f = 66 g = a = 115 b = 115 c = 65 d = 65 e = 115 f = 65 g =

33 Practice Your Skills: Practice Quiz on Angles Answers A. Draw angle A to the nearest degree. 33

34 B. Find the measure of the angle to the nearest degree c. Classify each angle as acute, obtuse, right or straight. D. Find the measure of angle x

35 Show What You Know: Angles Quiz Answers A. Draw angle A to the nearest degree. B. Find the measure of the angle to the nearest degree c. Classify each angle as acute, obtuse, right or straight. D. Find the measure of angle x