Estimating Angle Measures
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1 1 Estimating Angle Measures Compare and estimate angle measures. You will need a protractor. 1. Estimate the size of each angle. a) c) You can estimate the size of an angle by comparing it to an angle that you know, such as 45,, or º about about 90 b) d) about 80 about 45 90º 2. Which angles that you know did you use to help you estimate the angles in Question 1? Give reasons for your choices. a) I used a angle c) because the angle in part a) looks like it is greater than 45. I used a 90 angle because the angle looks like it is very close to 90. b) I used a angle d) I used a 45 angle because the angle looks like it is greater because the angle looks like it is very than the angle in part a). close to Measure the angles in Question 1. How close were your estimates? a) 50 c) 87 b) 70 d) 39 All my estimates were within 10 of the actual angle measurements. Copyright 2006 Nelson Answers Chapter 7: 2-D Geometry 63
2 2 Investigating Properties of Triangles Investigate angle and side relationships of triangles. You will need a ruler and a protractor. 1. a) Without using a protractor, label these angles on the triangles:, 80, 20, 80,, and. D M In a triangle, the largest angle is opposite the longest side and the smallest angle is opposite the shortest side cm 5.0 cm F E O 1.7 cm b) Explain how you know the angle sizes for both triangles. I know that an equilateral triangle has all angles the same size. So the angles must be. I know that an isosceles triangle has two angles that are the same size and the third angle is different. So the two angles that are the same size are 80. The third angle must be 20. N In equilateral or isosceles triangles, the sides opposite the equal angles are also equal. equilateral isosceles c) Measure the angles with a protractor to check your answer. 2. The angles in triangle PQR are 90, 35, and 55. The side lengths are 5.8 cm, 7.0 cm, and 4.0 cm. a) Without using a ruler or protractor, label the angle sizes and side lengths. P 35 b) Measure the angles and side lengths to check your answers. 5.8 cm 7.0 cm R cm 55 Q 64 Answers Chapter 7: 2-D Geometry Copyright 2006 Nelson
3 3 Communicate About Triangles Communicate and explain geometric ideas. 50º 110º A C 20º a) Add the angle measures for each triangle. A: = 180 B: = 180 C: + + = 180 D: = 180 b) Make a hypothesis about the sum of all the angles in a triangle. Use the Communication Checklist. When I add all the angles in a triangle, the sum is always 180. I think this is always true. 45º 80º 90º a) Add the angle measures on each line. A: = 180 B: = 180 C: = 180 D: = 180 D B 50º 50º 45º A hypothesis is a statement that you think you can test. For example, a hypothesis for the triangles below might be: When I add the lengths of two sides of a triangle, the sum is always greater than the length of the other side. I think this is always true. B 4 cm 3 cm 3 cm 4 cm A hypothesis must be checked with other examples to see if it is still true. Use the Communication Checklist. Communication Checklist Did you use math language? Did you explain your thinking? Did you include diagrams? 25º 35º 30º 45º 40º 90º 30º 75º 25º 50º 105º 100º A B C D A C 10º b) Make a hypothesis about the sum of the angles on a straight line. Use the Communication Checklist. When I add all the angles on a straight line, the sum is always 180. I think this is always true. Copyright 2006 Nelson Answers Chapter 7: 2-D Geometry 65
4 4 Constructing Polygons Construct polygons based on angle measures and side lengths. You will need a ruler and a protractor. 1. Draw each polygon. Label all side lengths and angle measures. a) equilateral triangle with side lengths of 3 cm and angle measures of To draw a scale diagram of a polygon, use appropriate tools. When angle measures and side lengths are given, use a ruler and a protractor. 3 cm 3 cm 3 cm b) scalene triangle with side lengths of d) parallelogram with angle measures 3 cm, 4 cm, and and one angle of 120 and and side lengths measure of 90 of 4 cm and 4 cm cm 3 cm c) rectangle with side lengths e) regular hexagon with side lengths of of 3 cm and and angle measures of cm º cm 3 cm Answers Chapter 7: 2-D Geometry Copyright 2006 Nelson
5 5 Sorting Polygons Sort polygons by line symmetry. You will need a ruler. 1. a) Name each polygon. Draw all the lines of symmetry you can find. Many polygons have lines of symmetry. Different polygons have different numbers of lines of symmetry. For example: scalene triangle equilateral triangle isosceles triangle You can check for lines of symmetry using a transparent mirror. quadrilateral parallelogram trapezoid rectangle square kite rhombus pentagon hexagon b) Sort the polygons using a Venn diagram. Choose categories from the property list. Sorting Polygons At least four lines of symmetry square pentagon hexagon At least two equal sides equilateral triangle isosceles triangle parallelogram rectangle kite rhombus Property list number of lines of symmetry number of equal sides number of equal angles number of sides number of angles scalene triangle quadrilateral trapezoid Copyright 2006 Nelson Answers Chapter 7: 2-D Geometry 67
6 6 Investigating Properties of Quadrilaterals Sort and classify quadrilaterals by their properties. You will need a ruler and a protractor. 1. a) Name each quadrilateral. The diagonals of quadrilaterals have certain properties. quadrilateral square rectangle Squares have diagonals that are equal and meet at 90. Kites and rhombuses have unequal diagonals that meet at 90. Rectangles have diagonals that are equal lengths that do not meet at 90. Parallelograms and some trapezoids have unequal diagonals that do not meet at 90. rhombus kite parallelogram trapezoid b) Draw all the diagonals in each quadrilateral above. Mark any right angles you find where the diagonals meet. 2. Sort the quadrilaterals using a Venn diagram. Choose categories from the property list. Sorting Quadrilaterals Equal diagonals rectangle square Diagonals that meet at 90 kite rhombus Property list equal diagonals unequal diagonals diagonals that meet at 90 diagonals that do not meet at 90 parallelogram irregular quadrilateral trapezoid 68 Answers Chapter 7: 2-D Geometry Copyright 2006 Nelson
7 Test Yourself Page 1 Circle the correct answer. 1. Which angles that you know would you use to estimate these angles? A. 180, 90, 45 B. 45, 120, 90 C. 45, 90, D. 90,, 2. The side lengths of the triangle are 10 cm and 14 cm. Which side lengths are correct? A. XZ 14 cm, XY 10 cm, YZ 10 cm X 90º B. XZ 10 cm, XY 14 cm, YZ 10 cm C. XZ 14 cm, XY 14 cm, YZ 10 cm Z 45º 45º Y D. XZ 10 cm, XY 10 cm, YZ 14 cm 3. Which polygons have more than two lines of symmetry? a A. b, c, f B. a, b, f C. c, d, e, f D. c, e, f 4. What information do you need to construct a regular polygon? A. all side lengths and all angle measures B. one side length and one angle measure C. two side lengths and one angle measure D. two side lengths and two angle measures b c d e f Copyright 2006 Nelson Answers Chapter 7: 2-D Geometry 69
8 Test Yourself Page 2 Use the polygons below to answer Questions 5 to 7. a b c d e f g 5. Which quadrilaterals have equal diagonals? A. d, e, f B. b, c, g C. b, c, f, g D. a, d, e 6. Which quadrilaterals have diagonals that do not meet at 90? A. a, b, d B. a, b, c, d C. c, e, g D. c, e, f, g 7. Which quadrilaterals have diagonals that are unequal but meet at 90? A. b, c, d B. c, e C. a, b D. a, f, g 70 Answers Chapter 7: 2-D Geometry Copyright 2006 Nelson
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