SCMP PD Day October 12, 2010
Angle Sum in a Triangle
Demonstrate measuring angles in a triangle then work through the following activity.
In this activity, we discover the sum of the 3 angles in a triangle 3 different ways:
· By measuring the 3 angles of several triangles with a protractor and calculating the sum
· By tearing off the 3 corners of one triangle and rearranging them
· By cutting out 3 identical triangles and rearranging them
Measuring Angles
Decide in your table group what type of triangle each of you will draw. Each person should draw a different one (e.g. acute, right, obtuse, scalene isosceles, equilateral). On a piece of plain white paper, draw a triangle. Use a protractor to measure each angle of the triangle. Now trade your paper with your neighbor. Measure your neighbor’s angles and see if your angle measures on his/her triangle are the same angle measures he or she got. Trade papers back so that you now have your own paper.
Add all of the angle measures together. Compare your total with your neighbor. Did you get the same angle sum? If so, were your angles the same measure? If you did not get the same angle sum, discuss why not.
Tearing Corners
On a piece of plain white paper, draw a triangle and cut it out. Label the interior of each angle. Now tear off each corner of the triangle and rearrange the 3 “angles” so that their vertices meet at one point with no overlap. What does this tell you about sum of the angles in the triangle?
Three Identical Triangles
Cut out 3 identical triangles (stack 3 sheets of paper). Label the interior of each angle. Place one triangle on a line and the second triangle directly next to it in the same orientation. Rotate and place the third triangle in the space between the 2 triangles that are next to each other. What does this tell you about the sum of the angles in one of the triangles
Angle Sum in a Polygon
In this activity, we will discover the sum of the angles in a polygon with n sides.
Triangle / Quadrilateral / Pentagon / Hexagon / Heptagon / OctagonNumber of sides
Number of triangles
Sum of interior angles
For each of the polygons listed in the table:
1. Draw an example of it on blank paper using a ruler. It’s up to you to decide whether your polygons are convex or concave, and whether they are regular or not.
2. Choose one vertex on your polygon and draw as many diagonals as possible from that one vertex. Your polygon should now consist of a bunch of non-overlapping triangles. Record the number of triangles in the table.
3. We already concluded that the sum of the angles in any triangle is 180°. Use this fact to find the sum of the (interior) angles in each of the polygons you drew.
When you have finished filling in your table, compare your table with your neighbor’s and answer the following questions.
1. Did you notice any patterns as you were completing the table? Explain your answer.
2. What is the sum of the interior angles in a nonagon (nine sides)? A decagon (10 sides)? A 25-gon?
3. State a formula that you could use to find the sum of the interior angles in an n-gon (n sides).
Now assume that each of the polygons listed in the following table are regular.
Regular Triangle / Regular Quadrilateral / Regular Pentagon / Regular Hexagon / Regular Heptagon / Regular OctagonNumber of sides
Sum of interior angles
Measure of each angle
Find a formula that could be used to find the measure of each interior angle in a regular n-gon.
Extension: For each of the polygons you drew, find one set of exterior angles then find the sum of the exterior angles.
Adapted from an activity on Illuminations.nctm.org
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