Lesson 2

Adjacent Angles

Let’s look at some special pairs of angles.

2.1: Estimating Angle Measures

Estimate the degree measure of each indicated angle.

Eight angles of varying measure.  Please ask for additional assistance.

 

2.2: Cutting Rectangles

Your teacher will give you two small, rectangular papers.

  1. On one of the papers, draw a small half-circle in the middle of one side.

    A rectangle represents a piece of paper.. A small half-circle is drawn against one of the long sides of the paper.
  2. Cut a straight line, starting from the center of the half-circle, all the way across the paper to make 2 separate pieces. (Your cut does not need to be perpendicular to the side of the paper.)
  3. On each of these two pieces, measure the angle that is marked by part of a circle. Label the angle measure on the piece.
  4. What do you notice about these angle measures?
  5. Clare measured 70 degrees on one of her pieces. Predict the angle measure of her other piece.

  6. On the other rectangular paper, draw a small quarter-circle in one of the corners.

    A rectangle represents a piece of paper. A quarter circle is drawn in one corner.
  7. Repeat the previous steps to cut, measure, and label the two angles marked by part of a circle.
  8. What do you notice about these angle measures?
  9. Priya measured 53 degrees on one of her pieces. Predict the angle measure of her other piece.

2.3: Is It a Complement or Supplement?

  1. Use the protractor in the picture to find the measure of angles \(BCA\) and \(BCD\).

    A polygon is drawn on a polygon.

  2. Explain how to find the measure of angle \(ACD\) without repositioning the protractor.

  3. Use the protractor in the picture to find the measure of angles \(LOK\) and \(LOM\).

    Quadrilateral K L M N sits with a protractor on L M.  Point O, between L and M, coincides with the center of the protractor. Segment K O passes through 37 or 143 on the protractor.

  4. Explain how to find the measure of angle \(KOM\) without repositioning the protractor.

  5. Angle \(BAC\) is a right angle. Find the measure of angle \(CAD\).

    Angle C, A, B is shown. segment D A, lies in the interior of angle C, A, B. Angle D, A, B, is labeled 64 degrees.

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  6. Point \(O\) is on line \(RS\). Find the measure of angle \(SOP\).

    Point O lies on line segment S R. Segment P O is drawn and angle P O R is labeled 76 degrees.


Clare started with a rectangular piece of paper. She folded up one corner, and then folded up the other corner, as shown in the photos.

A piece of decorated paper, the bottom left corner folded up.
A photo of a piece of decorative paper, the bottom left corner folded up, the bottom right corner folded up to meet the first fold.
A photo of a decorative piece of paper which had been folded in the previous photo.  The folds have been indicated by dotted lines, the line where the folds met indicated by a solid line.
  1. Try this yourself with any rectangular paper. Fold the left corner up at any angle, and then fold the right corner up so that the edges of the paper meet.
  2. Clare thought that the angle at the bottom looked like a 90 degree angle. Does yours also look like it is 90 degrees?

  3. Can you explain why the bottom angle always has to be 90 degrees? Hint: the third photo shows Clare’s paper, unfolded. The crease marks have dashed lines, and the line where the two paper edges met have a solid line. Mark these on your own paper as well.

Summary

If two angle measures add up to \(90^\circ\), then we say the angles are complementary. Here are three examples of pairs of complementary angles.

Three images. First, adjacent angles, 30 degrees, 60 degrees. Second, non-adjacent angels formed by two lines, 45 degrees. Third, a triangle, angles 90 degrees, 38 degrees, 52 degrees.

If two angle measures add up to \(180^\circ\), then we say the angles are supplementary. Here are three examples of pairs of supplementary angles.

Three images. First, adjacent angles, 55 degrees, 125 degrees. Second, perpendicular lines, non-adjacent angles marked. Third, distinct angles, 152 degrees, 28 degrees.

Glossary Entries

  • adjacent angles

    Adjacent angles share a side and a vertex.

    In this diagram, angle \(ABC\) is adjacent to angle \(DBC\).

    Three segments all joined at endpoint B. Point A is to the left of B and segment A B is drawn. Point C is above B and segment C B is drawn. Point D is to the right of B and segment B D is drawn.
  • complementary

    Complementary angles have measures that add up to 90 degrees.

    For example, a \(15^\circ\) angle and a \(75^\circ\) angle are complementary.

    complementary angles of 15 and 75 degrees
    Two angles, one is 75 degrees and one is 15 degrees
  • right angle

    A right angle is half of a straight angle. It measures 90 degrees.

    a right angle
  • straight angle

    A straight angle is an angle that forms a straight line. It measures 180 degrees.

    a 180 degree angle
  • supplementary

    Supplementary angles have measures that add up to 180 degrees.

    For example, a \(15^\circ\) angle and a \(165^\circ\) angle are supplementary.

    supplementary angles of 15 and 165 degrees
    supplementary angles of 15 and 165 degrees