Join us for conversations that inspire, recognize, and encourage innovation and best practices in the education profession.
Available on Apple Podcasts, Spotify, Google Podcasts, and more.
Print several copies of the sample polygons (PDF) and cut out the polygons. These are based on Power Polygons, a set of 15 different plastic polygons, each marked with a letter.
There are numerous ways of classifying angles. One way is according to their measures.
Identify a polygon that has at least one of the following angles:
a. Acute angle (an angle between 0 and 90 degrees)
b. Right angle (an angle equal to 90 degrees)
c. Obtuse angle (an angle between 90 and 180 degrees)
a. Without using a protractor, find two obtuse angles. Are they in the same polygon? How did you identify them? What do you notice about the other angles in the polygon(s) that has or have an obtuse angle?
b. Find a polygon with two or more acute angles.
c. Find a polygon with two or more obtuse angles.
d. Find a polygon with two or more right angles.
e. Can a triangle have two obtuse angles? Why or why not?
f. Can a triangle have two right angles? Why or why not?
Try to draw a triangle with two right angles or two obtuse angles. What happens?
Which polygons are equilateral triangles (all three sides are equal), isosceles triangles (two sides are equal), or scalene triangles (no sides are equal)? What can you say about the angles in each of these triangles?
Another way to classify angles is by their relationship to other angles. As you work on the types of classifications in Problems B4 and B5, think about the key relationships between angles.
The types of angles you will be looking at in Problems B4 and B5 are easily shown on two parallel lines cut by a transversal. You can use several copies of the same polygon and place them together to form parallel lines that are cut by a transversal.
Problem B4
Use two or more polygons to illustrate the angles below:
Problem B5
Use two or more polygons to illustrate the angles below, and explain how you would justify that some of the angles are congruent:
Problem B6
Find one or more polygons you can use to see examples of the following angles:
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.
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 | – | – |
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.
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? You can find this segment on the session video approximately 9 minutes after the Annenberg Media logo. |
Problem B9
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
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.
We determined that the sum of the measures of the angles of a triangle is 180 degrees. Notice in this diagram that the diagonal from one vertex of a quadrilateral to the non-adjacent vertex divides the quadrilateral into two triangles:
The sum of the angle measures of these two triangles is 360 degrees, which is also the sum of the measures of the vertex angles of the quadrilateral. Note 5
Problem B12
Video Segment
In this video segment, the participants explore the sum of the angles in different polygons. Laura demonstrates a method that will work for any polygon. Can the measure of individual angles be determined based on dividing the polygon into triangles? Why or why not? You can find this segment on the session video approximately 13 minutes and 10 seconds after the Annenberg Media logo. |
Note 3
For this problem, be sure to tear, not cut, the three angles off the triangle. This way you will be sure which point is the angle under consideration. If you cut them, you will end up with three clean-cut points (angles), and it is easy to become confused about which are the actual angles of the triangle.
Note 4
This approach for justifying that the sum of the angles in a triangle is 180 degrees is an informal justification or proof. A more formal proof can be written using what you know about the angles formed when parallel lines are cut by a transversal, which will be explored next.
Note 5
You may want to draw different quadrilaterals to show visually how a quadrilateral can only be divided into two triangles (from any one vertex)
Problem B1
Problem B2
Problem B3
Equilateral triangle polygons are I and N. They are also isosceles. Equilateral triangles have three angles that are equal in measure.
Isosceles triangle polygons are D, E, F, and J. All these triangles have two equal side lengths, opposite the equal angles. Isosceles triangles have at least two equal angles.
The only scalene triangle polygon is L. All of its sides have different lengths. Scalene triangles have no equal angles.
Problem B4
Problem B5
One way to review these relationships is to use either the G, M, or O polygon. Place multiple copies of one of these polygons together as illustrated. Next, identify segments that are parallel and segments that are transversals.
Problem B6
Problem B7
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
One method is to tile the polygon until a complete circle is formed, then divide 360 degrees by the number of polygons required to complete the circle. A second method is to figure out angles for the regular polygons, then lay other polygons on top to compare the angles.
Problem B9
Problem B10
Problem B11
The sum of the four angles in any quadrilateral is 360 degrees. One way to explain this is to draw an interior diagonal in the quadrilateral (a line connecting opposite vertices). This divides the quadrilateral into two triangles. Since we know each triangle’s angles add up to 180 degrees, the two triangles’ angles must add up to 360 degrees.
Problem B12