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7.1 Try These e.g., Triangles Try These You will need • string • scissors • plain paper • a millimetre ruler • a protractor 1. e.g., Make a paper triangle. Draw a dot at each vertex. Cut the triangle so that each vertex is separate. Show that the sum of the angles is 1808. Cut three pieces of string that you can use to make a triangle. How many different triangles can you make? 1 Place your string on paper to make a triangle. Mark the vertices with a pencil. Join the vertices. 2 175 mm, and 184 mm What are the side lengths? e.g., 128 mm,C07-F14-AW12.ai 3 What are the angle measures? e.g., 668, 748, and 408 4 Two triangles are different if they are not congruent. Are any no different triangles possible with your side lengths? 5 Compare your triangle with other students’ triangles. Could anyone make more than one triangle? no Reflecting Suppose that the sum of the lengths of the two shortest sides is less than the length of the longest side. Can these three pieces of string make a triangle? Explain. C07-F114-AW12.ai property a characteristic that is shared by all the members of a group Example 1 The bamboo stems in this photograph create an isosceles triangle. An isosceles triangle AW12 has 0176519637 two equal sides called legs. The interior Figure Number the C07-F14-AW12.ai angles opposite legs are also equal. Company B MPS Do all isosceles triangles have these Technical properties? 1st pass Pass Approved Solution Not Approved A. Find the midpoint of side AC. Label it M. Draw MB. A M C B. What are the side lengths, in millimetres? nABM: 19 mm, 50 mm, and 47 mm nCBM: 19 mm, 50 mm, and 47 mm 164 Apprenticeship and Workplace 12 C07-F15-AW12.ai NEL AW12.ai 07_AW12_Ch07.indd 164 02/03/12 12:25 PM C. Is nABM congruent to nCBM? How do you know? Yes. e.g., They are congruent because only one triangle is possible with these sides. OR They are the same size and shape. D. e.g., D. Kate said that this property is a property of all isosceles E triangles. Do you agree with Kate? Explain. Include a diagram. • The angles opposite the equal legs are equal. e.g., Yes, I agree. If you draw a centre line, you get two congruent triangles. So the corresponding angles are equal. 20 mm D 20 mm 45° M 45° 35 mm F Example 2 Pavlo is a carpenter. He uses triangular brackets for shelving. The sides of each bracket extend past the vertices to create exterior angles. What types of triangles have this property? • Each exterior angle is 908 or greater. Solution A. Measure the interior and exterior angles in the acute triangle below. Record the angle measures on the diagram. Acute triangle: 90° C07-F15a-AW12.ai150° 120° Obtuse triangle: e.g., 105° 152° 75° 35° 70° 110° 145° 28° 132° 48° 20° 160° AW12 B. Draw an obtuse triangle in0176519637 Part A. Extend one side at each Figure Number C07-F15a-AW12.ai vertex to create three exterior angles. Measure the interior Company MPS and exterior angles. Record the measures. Are any exterior Technical angles acute? yes Pass 1st pass Approved C. Is the following a propertyNot ofApproved all triangles? Explain. • Each exterior angle is 908 or greater. No. e.g., One exterior angle than 908. C07-F18-AW12.ai AW12SB on0176519637 the obtuse FN CO Technical D. What triangles have the property in Part C? acute triangles and right NEL 07_AW12_Ch07.indd 165 Why is showing that something is not a property easier than showing that it is a property? triangle is less C07-F17-AW12SB C07-F19-AW12.ai CrowleArt Group Pass Approved triangles Not Approved Reflecting 2nd pass Hint Use the triangular bracket above as an example of a right triangle. Chapter 7 Polygons 165 02/03/12 12:25 PM Practice Reflecting Does it matter which side of a triangle you extend to make an exterior angle? Explain. 1. Use your triangles from Example 2. a) The sum of the interior angle plus the exterior angle is the same at each vertex. What is this sum? 1808 b) Why does it make sense that each vertex has the same sum? When you extend one side, you create two angles that form a straight line . Angles that form a straight line have a sum of 1808 . c) Is this a property for all triangles? Explain. • The sum of the interior angle plus the exterior angle is 1808. Yes. e.g., You always create an exterior angle by extending a side. The interior angle and exterior angle will always form a straight line. Hint Use the diagrams and definitions of different types of triangles in Getting Started. 2. Cables on the Esplanade Riel Bridge in Winnipeg illustrate many types of triangles. Circle the types of triangles that have each property. a) Some sides are equal. equilateral triangle isosceles triangle scalene triangle b) Some exterior angles are equal. equilateral triangle isosceles triangle scalene triangle c) No interior angles are equal. equilateral triangle isosceles triangle scalene triangle d) All three exterior angles are 908 or greater. acute triangle obtuse triangle right triangle e) Each exterior angle is equal to the sum of the interior angles at the other two vertices. acute triangle obtuse triangle right triangle 3. a) What is one property of isosceles triangles that is not a property of all triangles? e.g., Isosceles triangles have exactly two equal sides. b) What is one property of isosceles triangles that is a property of all triangles? 166 e.g., The sum of the interior angles is 1808. Apprenticeship and Workplace 12 07_AW12_Ch07.indd 166 NEL 02/03/12 12:25 PM 4. Use the angle measures to calculate the unknown angles in each triangle. Include interior angles and exterior angles. Record the measurements on the diagrams. 75° 105° 43° 1 60° 120° 35° 125° 2 30° 137° 32° 20° 30° 150° 145° 160° 3 55° 150° 148° 5. Use the triangles in Question 4. Complete this chart. Triangle Sum of 3 interior angles Sum of 3 exterior angles 1 1808 1808 1808 3608 3608 3608 2 3 C07-F20-AW12.ai 5408 5408 C07-F22-AW12.ai 5408 Marcel wonders about this question. • Does drawing a line parallel to the base of any triangle create a second triangle with angles that are equal to those in the original triangle? 0176519637 Figure Number C07-F20-AW12.ai Company MPS Technical 1st pass Pass 60° 60° Not Approved 60° 60° AW12 idea. a) Test Marcel’s 0176519637 • Draw a triangle. Draw a line through your triangle so Figure Number C07-F22-AW12.ai that the line is parallelMPS to the base. Company C07-F21-AW12.ai Technicalin the small triangle equal to the angles in MPSthe angles Are the largePass triangle? yes 1st pass 1st pass Approved Do you think that the sum of the interior angles and the exterior angles is the same for all triangles? Explain. C07-F21-AW12.ai 6. Marcel’s crew builds A-frame cabins in Tofino. • The balcony is parallel to the base of a cabin. • The front of this cabin is an equilateral triangle. • The section above the balcony is also an equilateral triangle. AW12 Reflecting Sum of 3 interior angles 1 sum of 3 exterior angles b) CompareApproved your results with a classmate’s results. Not Approved Did your classmate get the same results? yes c) Will adding a line that is parallel to the base always create a smaller triangle with the same angles? Explain. 60° Hint One way to draw parallel lines is to draw along both sides of a ruler. 6. a) e.g., 65° 25° 25° Yes. e.g., One angle is shared by both triangles. The other two angles are corresponding angles, formed by transversals that meet the parallel lines at the same angle. So each angle in the small triangle has a matching equal angle in the large triangle. AW12SB NEL 07_AW12_Ch07.indd 167 Chapter 7 Polygons 0176519637 FN CO Technical 167 C07-F24-AW12SB CrowleArt Group 02/03/12 12:25 PM

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