Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 4, Part B:
Angles in Polygons

In This Part: Classifying by Measure | Other Classifications | Measuring Angles
Sums of Angles in Polygons

 Problem B7 Angle measurement is recorded in degrees. Using only the polygons (no protractor! no formulas!) and logical reasoning, determine the measure of each of the angles in the polygons, and record your measures in the table. Be able to explain how you determined the angle size.

 One approach would be to use the known measurement of one angle to determine the measures of other angles. For example, Polygon N is an equilateral triangle -- notice that all the angles are equal. If we arrange six copies of the polygon around a center point, the angles completely fill up a circle. So each angle measure must be 60 degrees. Similarly, since two N blocks fit into the vertex angles of Polygon H, the measure of each of the angles in H must be 120 degrees.   Close Tip One approach would be to use the known measurement of one angle to determine the measures of other angles. For example, Polygon N is an equilateral triangle -- notice that all the angles are equal. If we arrange six copies of the polygon around a center point, the angles completely fill up a circle. So each angle measure must be 60 degrees. Similarly, since two N blocks fit into the vertex angles of Polygon H, the measure of each of the angles in H must be 120 degrees.

Polygon

Angle 1

Angle 2

Angle 3

Angle 4

Angle 5

Angle 6

Name of Polygon

 A - - B - - C - - D - - - E - - - F - - - G - - H I - - - J - - - K - - L - - - M - - N - - - O - -

Polygon

Angle 1

Angle 2

Angle 3

Angle 4

Angle 5

Angle 6

Name of Polygon

 A 90° 90° 90° 90° - - square B 90° 90° 90° 90° - - square C 90° 90° 90° 90° - - rectangle D 45° 45° 90° - - - triangle E 45° 45° 90° - - - triangle F 45° 45° 90° - - - triangle G 60° 120° 60° 120° - - parallelo-gram H 120° 120° 120° 120° 120° 120° hexagon I 60° 60° 60° - - - triangle J 30° 30° 120° - - - triangle K 60° 60° 120° 120° - - trapezoid L 30° 60° 90° - - - triangle M 60° 120° 60° 120° - - parallelo-gram N 60° 60° 60° - - - triangle O 30° 150° 30° 150° - - parallelo-gram

 Problem B8 Describe two different methods for finding the measure of an angle in these polygons.

 One method would be to use multiple copies of the angle under investigation to form a circle around a point.    Close Tip One method would be to use multiple copies of the angle under investigation to form a circle around a point.

 Video Segment In this video segment, Jonathan and Lori are trying to figure out the measures of the angles inside different polygons. They use logical reasoning and prior knowledge to find the measures of the unknown angles. How does this hands-on approach help them gain an understanding of angles? If you are using a VCR, you can find this segment on the session video approximately 9 minutes after the Annenberg Media logo.

Problem B9

 a. Based on the data in the table, what is the sum of the angles in a triangle? How might we prove this? b. Trace around four different triangles and cut them out. Label 1, 2, and 3 in each triangle. Then tear off the angles and arrange them around a point on a straight line (i.e., each vertex point must meet at the point on the line), as shown below: Note 3

What do you notice? Is it true for all four of your triangles? Will it be the same for every triangle? Explain. Note 4

Problem B10

 a. Now draw two parallel lines with a triangle between them so that the vertices of the triangle are on the two parallel lines. The angles of the triangle are A, B, and ACB: Earlier you reviewed the relationships between angles (e.g., alternate interior angles). Use some of these relationships to describe the diagram above. b. Use what you know about angle relationships to prove that the sum of the angles in a triangle is 180 degrees.

 Problem B11 Examine the polygons in the table in Problem B7 that are quadrilaterals. Calculate the sum of the measures of the angles in each quadrilateral. What do you notice? How can we explain this sum? Will the measures of the angles in an irregular quadrilateral sum to this amount? Explain.

 Session 4: Index | Notes | Solutions | Video