Complimentary and Supplementary Angles

These pictures show two parallel lines (in black) cut by a third line (in orange) in the plane of the screen (the drawing is flat as you look down upon it; it is not a perspective drawing). In each picture, the two labelled angles are "congruent" (equal) and are related as "alternate interior angles". Angles which are

  1. similar in appearance because they are in a repeating pattern of parallel lines (A and B, or C and D),
  2. on opposite sides of the crossing point in a letter X figure (C and E), or
  3. on opposite inside corners of a letter Z figure or a letter N figure (E and F)
will be congruent. This is useful information because sometimes the angle you are given in the diagram of a system is not the angle that you need for use in an equation, but you can show that it has the same value. The angles adjacent to one another in the figures (A and C, B and D) add together to equal 180o and are called supplementary angles.

The figure on the right shows a right triangle. The corner symbol represents the 90o angle, and the other two angles are labeled with letters A and B. The three angles add up to 180o (true for any triangle, even if not a right triangle), and angles A and B add up to 90o (true for right triangles only). The angles A and B are said to be complimentary.

Now let's apply these ideas. The figure on the left shows a stick being supported in two places: at the upper-right end where it is attached to a wall, and along its length where it is being pulled by a horizontal wire. The angle between the stick and the wall is 30o but it is the angle q between the wire and the stick that is wanted for calculations.

In this next figure to the right, we see a red right triangle drawn into the picture is a useful tool for deducing that the angle q is 60o. The angle labeled A is complimentary to the 30o angle while also alternate interior to q.

Try these three practice problems: use the given angles (red and green) along with the rules described above to determine the unknown angle (blue).

Answers:    a. 18o     b. 100o     c. 112o

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