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.](https://cms-im.s3.amazonaws.com/iyTRCu9DNBXMa2TqBavgERvd?response-content-disposition=inline%3B%20filename%3D%227-7.6.A3.Image.01.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.A3.Image.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T024246Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0eeaa19b4ff3987694313a09093af13257e736906cc1971051435924e23d64f4)
2.2: Cutting Rectangles
Your teacher will give you two small, rectangular papers.
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On one of the papers, draw a small half-circle in the middle of one side.
- 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.)
- On each of these two pieces, measure the angle that is marked by part of a circle. Label the angle measure on the piece.
- What do you notice about these angle measures?
- Clare measured 70 degrees on one of her pieces. Predict the angle measure of her other piece.
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On the other rectangular paper, draw a small quarter-circle in one of the corners.
- Repeat the previous steps to cut, measure, and label the two angles marked by part of a circle.
- What do you notice about these angle measures?
- 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?
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Use the protractor in the picture to find the measure of angles \(BCA\) and \(BCD\).
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Explain how to find the measure of angle \(ACD\) without repositioning the protractor.
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Use the protractor in the picture to find the measure of angles \(LOK\) and \(LOM\).
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Explain how to find the measure of angle \(KOM\) without repositioning the protractor.
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Angle \(BAC\) is a right angle. Find the measure of angle \(CAD\).
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Point \(O\) is on line \(RS\). Find the measure of angle \(SOP\).
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.](https://cms-im.s3.amazonaws.com/mCnJbJNjZLdwjqtCfpi4VEur?response-content-disposition=inline%3B%20filename%3D%227-7.7.ext.paper1.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.ext.paper1.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T024246Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b9487c195e97231f473810914d7272aa70459a2d87a26cfc9227187793d33140)
![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.](https://cms-im.s3.amazonaws.com/smwEspEhSaHn6Ha78vSZFHvR?response-content-disposition=inline%3B%20filename%3D%227-7.7.ext.paper2v2.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.ext.paper2v2.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T024246Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=99a5bc311a63edd90b967d36b1bfa0eedaa95dcdad7e05ec5bebb1dc9e370bfe)
![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.](https://cms-im.s3.amazonaws.com/gqmXdj9boRzWHwh6QFLiSqR3?response-content-disposition=inline%3B%20filename%3D%227-7.7.ext.paper3.png%22%3B%20filename%2A%3DUTF-8%27%277-7.7.ext.paper3.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T024246Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cad7b69142d028a39263e7460bd8b6456c6b2de0d20660f5eb94b25fe0169f5d)
- 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.
- Clare thought that the angle at the bottom looked like a 90 degree angle. Does yours also look like it is 90 degrees?
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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.](https://cms-im.s3.amazonaws.com/qqGNcFbJ8WxHa9GqBJhD9ctX?response-content-disposition=inline%3B%20filename%3D%227-7.6.A3.Image.09.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.A3.Image.09.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T024246Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b2bc71e32c9a5cf5ce65c2fcb5a3a6e583be0f67270c0530dbe065188dcb5ff1)
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.](https://cms-im.s3.amazonaws.com/3nq92xyFPM8inrDnSZTzbwHG?response-content-disposition=inline%3B%20filename%3D%227-7.6.A3.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.A3.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T024246Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=161acfc1f070848ceb34addda6ed10a54fb64209410dcf46704821175bba6983)
Glossary Entries
- adjacent angles
Adjacent angles share a side and a vertex.
In this diagram, angle \(ABC\) is adjacent to angle \(DBC\).
- complementary
Complementary angles have measures that add up to 90 degrees.
For example, a \(15^\circ\) angle and a \(75^\circ\) angle are complementary.
- right angle
A right angle is half of a straight angle. It measures 90 degrees.
- straight angle
A straight angle is an angle that forms a straight line. It measures 180 degrees.
- supplementary
Supplementary angles have measures that add up to 180 degrees.
For example, a \(15^\circ\) angle and a \(165^\circ\) angle are supplementary.