Triangle Inequality Theorems Investigation Lesson Summary: Students will investigate using geometry software the various inequalies that can be written for a triangle. Key Words: Triangle, inequlity Background Knowledge: Students will need to have basic knowledge of triangles, angles, supplementary angles, and experience writing and solving simple inequalities. NCTM Learning Strands: Students will be able to 1) Apply problem-solving strategies and use mathematical reasoning to generalize arguments and solutions. 2) Represent problem situations with geometric models and apply properties of the models to understand and solve the problem situation 3) Estimate and use measurement 4) Demonstrate the ability to identify patterns, note trends, and draw conclusions Learning Objective s: The students will be able to 1) Discover three theorems about triangle inequalities through manipulation of a triangle 2) Apply properties of inequalities to the measures of segments and angles 3) State and apply the Triangle Inequality Theorem, Hinge theorem, and Exterior Angle Inequality Theorem to problems involving triangles 4) Draw valid conclusions from given information Materials: Computers or computer lab with Cabri Geometry II, and lab worksheet Procedure: You may want to pair students together depending on how many computers are available with Cabri Geometry II. Students can also work independently and then compare conclusions in pairs. You may want to do a quick review of their prior knowledge. Make sure students remember how to apply the Segment Addition and Angle Addition Postulates. Students should also be familiar with measuring the lengths of the sides and the angles of a triangle, and use this information to describe or classify the triangle. When discussing all the properties they have learned regarding triangles thus far, you may want to cover the Triangle Angle Sum Theorem last, and then use that as a lead into discovering other relationships between the angles and sides of a triangle. Then let students follow the lab directions and see what they can discover. Review/Assessment: Is left up to the instructor but may include writing proofs, giving students a written assignment (maybe problems from the text or worksheets that correspond to the lesson, giving students a written quiz, or having students present what they found orally, independently or in pairs.
Triangle Inequality Theorem Activity One Team Members Names: File name: Goal: To discover a relationship between the lengths and the sums of the sides of a triangle. Investigate Using Cabri Geometry II* **Remember, to compare to values think <, >, or =. 1. Draw ABC. (use triangle tool) *Make sure you make segment AB the longest side. 2. Measure and label the lengths of the sides. (use measurement and label tools) 3. Compare each side of the triangle to the sum of lengths of the other two sides. 4. What is the relationship between the sum of the two sides and the length of the third side? 5. Grab the vertex at point C and move the vertex around changing the shape of the triangle. (use pointer tool) 6. Does the relationship from #4change as the triangle changes shape? 7. If so, is segment AB still the longest side?
8. What happens when point C approaches segment AB? 9. Repeat Step #5. 10. Compare each side of the triangle to the difference of the lengths of the other two sides. 11. What is the relationship between the difference of the two sides and the length of the third side? 12. Make a conjecture about the relationship between any two sides of a triangle compared to the length of the third side. Theorem:
Extension 1. Is it possible for a triangle to have sides with the given lengths? Explain. a) 2 in., 3 in, 9 in b) 11 cm, 12 cm, 18 cm c) 7 ft, 3 ft, 4 ft d) 2 m, 13 m, 13 m 2. The lengths of two sides of a triangle are 3 cm and 5 cm. Write an inequality to represent the range of values for the third side. a) Construct point O as the center of two concentric circles, one with radius 3cm and the other with radius 5cm. b) Construct radius OA (use numerical edit and measurement transfer) c) Construct radius OB the same way. d) Draw segment AB (use segment tool) e) Move point B around the outside circle (pointer tool) to estimate possible values for the length of side AB (use measurement tool). (Think: What are the limits? When is it no longer a triangle?) b) Algebra: Write and solve three inequalities to find the maximum and minimum values for the third side of the triangle in a). (Express final answer as a combined inequality!)
More Triangle Inequality Theorems Activity Two Team Members Names: File name: Goal: To discover the relationship between angles and sides of a triangle that lie opposite each other. Investigate Using Cabri Geometry II* 1. Draw and label three noncollinear points A, B, and C. (use point tool) 2. Draw segments AB, BC, and AC to form a triangle. (use segment tool) 3. Measure and label the sides of the triangle. (Use measurement and comments tools) *Move measurements off to the side of your picture and hide original measurements. 4. Measure and label angles. (use angle tool) 5. Grab vertex point A and move point A until A has the largest degree measure. 6. What do you notice about the length of BC? How does it compare to the lengths of AB and AC? 7. Move point A until B has the largest measure. 8. How does the length of AC compare to the lengths of AB and BC?
9. Move point A until C has the largest measure. 10. How does the length of AB compare to the lengths of AC and BC? 11. What conjecture can you make about the side opposite the largest angle in a triangle?
Extension Questions: 1. List the sides of each triangle in order from shortest to longest. a) b) ARK, where m A = 90, m R = 40, and m K = 50. 2. List the angles of each triangle in order from largest to smallest. a) b) XYZ, where XY = 5.4 m, YZ = 10.3 m, and XZ = 9.1 m. 3. The Calhouns are driving from Columbus, OH and see a sign that reads, Akron 100 miles, Cincinnati 120 miles. Megan comments that she did not think Akron and Cincinnati were only 20 miles apart. Explain why the distance between the two cities does not have to be 20 miles.
Exterior Angle Inequality Theorem Activity Three Team Members Names: File name: Goal: To discover the relationship between the exterior angle and the two opposite interior angles of a triangle. Investigate Using Cabri Geometry II* **Remember the opposite interior angles are inside the triangle and away from the exterior angle, which is outside the triangle. 1. Draw a line. (use line tool) 2. Draw three collinear points B, C, and D respectively on the line. (use point and label tools) 3. Draw point A somewhere above line BD. (use point, label tools) 4. Connect point A to points B and C to form a triangle. (segment tool) *Point D should be to the right of?bcd. 5. Measure and label ABC, BAC, and ECD. (angle tool) 6. What relationship do you notice when you compare the m ECD to the m ABC or the m BAC? 7. Grab point A and drag it to the right. (use pointer tool) 8. Does the relationship from #6 still exist between the exterior angle ECD and the two opposite interior angles ABC and BAC?
9. Drag point A to the right again. (use pointer tool) 10. What happens to the relationship between the angles as you drag point A to the right? 11. State the relationship between the exterior angle of a triangle and how it compares to the two opposite interior angles of a triangle. *12. Is there another pattern or relationship that you notice about the sum of the two opposite interior angles compared to the exterior angle of a triangle. Explain.
Extension Questions: Use this figure for the Extension Questions 1. m 1 < 2. m 4 > 3. m 3 + m 4 = 4. m 1 + m 2 = 4. If the m 1=53 and m 2=67, then m 4> 5. If the m 4=125 and m 2=70, then m 1= 6. If the m 1=49 and m 2=78, then m 4= m 3=