What Is the Sum of My Angles?

Objective:
  • To investigate the sum of the measure of the interior angles of a triangle using several methods.
  • To make a mathematical conjecture from observations.

    • Materials Needed
  • Protractor
  • Ruler
  • Scissors
  • A Balloon (optional)
  • Printout of this activity. Printout of the triangles. Cut out one set of triangles before you start the activity.
    1. Write a definition of a triangle. Include drawings. Clearly show the Interior Angles and the Exterior Angles. (Note that Interior and Exterior have different meanings depending on the geometric figure.)



    2. Notice that the interior angles of the triangles on your printout are labeled 1, 2, and 3. Cut out one set of triangles. You will be tearing these triangles into pieces. To keep track of which triangle is A, B, and C, color each triangle with a different color. Keep the second set of triangles to use later in the activity.

    3. Take triangle A and rip off the angles labeled 1, 2, and 3. Tape the angles along the lines below so that the angles have common sides. Repeat for each this for each triangle.



      4. Triangle A



      Triangle B



      Triangle C

    4. What do you notice? Write a conjecture about the sum of the interior angles of a triangle.





    5. Draw (or construct) a line through A parallel to line BC. Label this line with points D on the left of A and E on the right of A.

    6. Trace angle ABC and angle ACB using tracing paper. Compare these angles to angle DAB and angle EAC respectively. What do you notice? Can you make a conjecture about the sum of the interior angles of a triangle from this drawing?

    7. Vocabulary: Angles ABC and DAB are called alternate interior angles. Notice they are congruent!

    8. Vocabulary: Angles ACB and EAC are called alternate interior angles. Notice they are congruent!


    9. Next, take a protractor and measure the interior angles of each triangle. Write the sum of the measurements and use mental math to add the angles together.

      10. Triangle A ______________________________

      Triangle B ______________________________

      Triangle C ______________________________

    10. What do you notice? What can you conjecture?


    11. Using interactive geometry software, set up a triangle. Have the angle measures showing. Drag point A to several different positions so that you change the angles of the triangle. Write the sum of the angles for at least five different triangles. What can you conjecture about the sum of the interior angles of a triangle from this investigation?

      12. Triangle 1 ______________________________

      Triangle 2 ______________________________

      Triangle 3 ______________________________

      Triangle 4 ______________________________

      Triangle 5 ______________________________

    12. Do any of the above activities actually prove that the sum of the angles of a triangle is 180 degrees? Write a summary and a reaction to the different ways of exploring the angles of a triangle.



    Extension

    1. Sum of the Exterior Angles of a Triangle
      Now, use a protractor to find the sum of the Exterior Angles of the triangles on your printout. You will have to draw in the appropriate rays on your printout.

      2. Triangle A ______________________________

      Triangle B ______________________________

      Triangle C ______________________________

    2. What can you conjecture? Write two other ways of investigating the exterior angles.

    3. Blow up a balloon. Draw a triangle on the balloon with a marker. What is the sum of the interior angles of this triangle? Investigate other triangles on a balloon!

    Top

    Back to Investigating Polygons

    Copyright 1998 by Margo Lynn Mankus