Before class I made handouts with large irregular triangles, quadrilaterals, and pentagons (each on a separate sheet of paper) and taught the students how to use a protractor to measure the angles. Using a protractor is tricky. There are usually two scales, one reading each direction; the center mark is generally not on the edge of the protractor; using a protractor correctly requires manual dexterity to get the vertex mark positioned correctly, then getting the zero mark positioned correctly somewhere else without moving the vertex mark. The technique is to position the vertex mark, then nudge the zero mark, then check and possibly nudge the vertex mark, then check and possibly nudge the zero mark, etc. until both are simultaneously in place. Finally, read the protractor starting from the zero mark and using the scale that counts upward from that point. The angle in the illustration is measured with the inner scale because that is the scale that counts upward from zero at the zero mark. The angle shown is 38°. Check to see that the angle measures less than 90° if it is an acute angle and greater than 90° if it is an obtuse angle.

The sum of the angles theoretically comes out to 180° for any triangle, 360° for any quadrilateral, and 540° for any pentagon (whether the figures are regular or not). Note that a quadrilateral can be split along a diagonal to form two triangles whose angles add up to the angles of the quadrilateral. (360° = 2 x 180°). Similarly, a pentagon can be split along diagonals to form three triangles whose angles add up to the angles of the pentagon (540° = 3 x 180°). The question can then be asked, what is the sum of the angles for a 7-sided figure? (900°) What is the sum of the angles for a 33-sided figure? (5580°) What about a 137 sided figure? (24,300°) Etc. What is a rule that will always work?
(Two less than the number of sides) x 180° or (n-2) x 180

For some students, practice measuring angles it the heart of the lesson. For others, simply adding up the numbers is the challenge. For others, seeing the pattern and extrapolating it out to bizzare lengths is the appropriate challenge. For those in Pre-Algebra, Algebra, or Geometry, this provides an example of deriving a general formula from a geometric insight.