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Geometry in the Trees Triangles in a Tree Worksheet Materials: Ruler, protractor, calculator This is a three part investigation. First you will investigate and make a conjecture about the sum of the measures of the angles in a triangle. Second you will discover the relationship between an exterior angle of a triangle, and third you will discover relationships among the measures of the sides and angles of a triangle (triangle inequality). Draw and Investigate: (Triangle Sum) 1. Draw an acute triangle and label it ABC. Measure its three angles. 2. Calculate the sum of the angle measures. 3. Draw a right triangle, label the angles DEF, measure its three angles and calculate the sum of the angles measures. 4. Draw an obtuse triangle, label the angles GHI, measure its three angles and calculate the sum of the angles measures. 5. Make a conjecture about the sum of the angles in any triangle? _____________ Based on activities from GSP and Key Press Curriculum Continue to draw and Investigate: (Exterior angles of a Triangle) 6. Using triangle ABC, draw ray AC to extend side AC. 7. Label point J on ray AC, outside of the triangle. 8. Measure exterior angle BCJ. 9. Measure the remote interior angles ABC and CAB. 10. Using triangle DEF, draw ray DF to extend side DF. Label point K on ray DF, outside the triangle. Measure the exterior angle and the two remote interior angles. Do the same for triangle GHI. Look for a relationship between the measures of the remote interior angles and the exterior angle. 11. How are the measures of the remote interior angles related to the measure of the exterior angle? Write a conjecture. Sketch and Investigate one more time: (Triangle Inequalities) 12. Measure the lengths of the sides of triangle ABC. AB = _____, BC = _____, AC = _____ 13. Calculate the sum of the two shortest side lengths. _____. How does this sum compare to the length of the longest side? ___________________________ 14. Measure the lengths of the sides of triangle DEF. DE = _____, EF = _____, DF = _____ 15. Calculate the sum of the two shortest side lengths. _____. How does this sum compare to the length of the longest side? ___________________________ 16. Measure the lengths of the sides of triangle GHI. GH = _____, HI = _____, GI = _____ 17. Calculate the sum of the two shortest side lengths. _____. How does this sum compare to the length of the longest side? ___________________________ 18. Is it possible for the sum of the lengths in a triangle to be equal to the third side length? Explain. 19. Do you think it is possible for the sum of the lengths of any two sides of a triangle to be less than the length of the third side? Explain. 20. Summarize your findings as a conjecture about the sum of the lengths of any two sides of a triangle. 21. 22. 23. 23. Using triangle ABC, measure angle ABC _____, angle BAC _____ and angle ACB _____. Using triangle DEF, measure angle DEF _____, angle EFD _____, and angle FDE _____. Using triangle GHI, measure angle GHI _____, angle HIG _____, and angle IGH _____. Using the previous information, fill in the chart below. In triangle ABC, _____ is the longest side and _____ is the largest angle. In triangle ABC, _____ is the shortest side and _____ is the smallest angle. In triangle DEF, _____ is the longest side and _____ is the largest angle. In triangle DEF _____ is the shortest side and _____ is the smallest angle. In triangle GHI, _____ is the longest side and _____ is the largest angle. In triangle GHI _____ is the shortest side and _____ is the smallest angle. 24. Make a conjecture about the location of the largest angle and the longest side in any triangle. 25. Make a conjecture about the location of the smallest angle and the shortest side in any triangle. Based on activities from GSP and Key Press Curriculum Geometry in the Trees Triangles in a Tree Investigation (Suggested Answers) This is a three part investigation. First you will investigate and make a conjecture about the sum of the measures of the angles in a triangle. Second you will discover the relationship between an exterior angle of a triangle, and third you will discover relationships among the measures of the sides and angles of a triangle (triangle inequality). Draw and Investigate: (Triangle Sum) 1. Draw an acute triangle and label it ABC. Measure its three angles. 2. Calculate the sum of the angle measures. 180 3. Draw a right triangle, label the angles DEF, measure its three angles and calculate the sum of the angles measures. 180 4. Draw an obtuse triangle, label the angles GHI, measure its three angles and calculate the sum of the angles measures. 180 5. Make a conjecture about the sum of the angles in any triangle? The sum of the three angles of any triangle is 180. Continue to draw and Investigate: (Exterior angles of a Triangle) 6. Using triangle ABC, draw ray AC to extend side AC. 7. Label point J on ray AC, outside of the triangle. 8. Measure exterior angle BCJ. 9. Measure the remote interior angles ABC and CAB. 10. Using triangle DEF, draw ray DF to extend side DF. Label point K on ray DF, outside the triangle. Measure the exterior angle and the two remote interior angles. Do the same for triangle GHI. Look for a relationship between the measures of the remote interior angles and the exterior angle. 11. How are the measures of the remote interior angles related to the measure of the exterior angle? Write a conjecture. The measure of the exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Sketch and Investigate one more time: (Triangle Inequalities) 12. Measure the lengths of the sides of triangle ABC. AB = _____, BC = _____, AC = _____ 13. Calculate the sum of the two shortest side lengths. _____. (Answers will vary) How does this sum compare to the length of the longest side? The sum is greater than the length of the longest side. 14. Measure the lengths of the sides of triangle DEF. DE = _____, EF = _____, DF = _____ 15. Calculate the sum of the two shortest side lengths. _____. (Answers will vary) How does this sum compare to the length of the longest side? The sum is greater than the length of the longest side. 16. Measure the lengths of the sides of triangle GHI. GH = _____, HI = _____, GI = _____ 17. Calculate the sum of the two shortest side lengths. _____. (Answers will vary) How does this sum compare to the length of the longest side? The sum is greater than the length of the longest side. 18. Is it possible for the sum of the lengths in a triangle to be equal to the third side length? Explain. No. No vertices would be formed because the two segments would lie on top of the longest segment. 19. Do you think it is possible for the sum of the lengths of any two sides of a triangle to be less than the length of the third side? Explain. No. It would not form a triangle. 20. Summarize your findings as a conjecture about the sum of the lengths of any two sides of a triangle. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Based on activities from GSP and Key Press Curriculum (21 – 23 Answers will vary) 21. Using triangle ABC, measure angle ABC _____, angle BAC _____ and angle ACB _____. 22. Using triangle DEF, measure angle DEF _____, angle EFD _____, and angle FDE _____. 23. Using triangle GHI, measure angle GHI _____, angle HIG _____, and angle IGH _____. 23. Using the previous information, fill in the chart below. In triangle ABC, _____ is the longest side and _____ is the largest angle. In triangle ABC, _____ is the shortest side and _____ is the smallest angle. In triangle DEF, _____ is the longest side and _____ is the largest angle. In triangle DEF _____ is the shortest side and _____ is the smallest angle. In triangle GHI, _____ is the longest side and _____ is the largest angle. In triangle GHI _____ is the shortest side and _____ is the smallest angle. 24. Make a conjecture about the location of the largest angle and the longest side in any triangle. The largest angle of a triangle is opposite the longest side of the triangle. 25. Make a conjecture about the location of the smallest angle and the shortest side in any triangle. The smallest angle of a triangle is opposite the shortest side of a triangle. Based on activities from GSP and Key Press Curriculum