Download Triangles in a Tree without Sketchpad

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