Lesson 2

Adjacent Angles

2.1: Estimating Angle Measures (5 minutes)

Warm-up

The purpose of this warm-up is for students to estimate degree measures (without a protractor) based on angles that are familiar. In the first two rows, an angle that is close to either a right angle or straight angle is given, and students could use this as a reference angle for the other angles in the row.

Asking students to share a wrong estimate first is a good strategy for launching the activity, because students are more confident in sharing wrong estimates, and it can help them start to consider what would be a more correct estimate.

As student discuss with their partner, monitor for students who use phrases such as:

  • “a little more than 90 degrees”
  • “almost a straight line”
  • “a little less than 360 degrees”

Launch

Arrange students in groups of 2. Do not supply protractors or pattern blocks; let students know that in this activity they are estimating the degree measure of each angle.

Before students begin, ask students to think of an estimate that is definitely wrong for angle \(GHI\). Invite a few students to share and explain why it is wrong. Then, ask the same students to come up with an actual estimate.

Give students 2 minutes of quiet work time followed by a partner and whole-class discussion.

Student Facing

Estimate the degree measure of each indicated angle.

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

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students will be tempted to figure out the exact angle measures, encourage students to use estimation to see how close they can get using benchmark angles that they have encountered ($90^\circ$, $180^\circ$, $360^\circ$, etc).

Activity Synthesis

Select previously identified students to share an estimate of the degree measure for each angle; record and display their responses for all to see. Poll the class if they agree or disagree after each one.

After each of the angles is discussed, ask students what tools they might use to check their answers themselves. If/when a protractor is mentioned, ask how they could use it to find the measure of angles like $STU$ or $VWZ$ when the protractors usually only go to 180. (For example, they could find the measure of the angle that is less than 180, and subtract it from 360.)

2.2: Cutting Rectangles (10 minutes)

Activity

The purpose of this activity is to provide a tangible experience with complementary and supplementary angles, which will be formally defined in the next activity. Students cut sheets of paper in two ways to see the decomposition of straight and right angles. In later activities and lessons, students will continue working with the fact that specific angles can be composed to make straight or right angles as a strategy for finding the measure of an unknown angle. In this activity, the language and vocabulary that students use during this task should be allowed to be loose as we will develop it more precisely in the following activities and lessons.

This activity gives students another opportunity to practice using a protractor to measure angles. Especially when students are measuring the angles they cut from the straight angle, it should be readily apparent if they are reading their angle from the wrong side of the protractor. For example, if students think that both of their angles measure 140 degrees, the papers can be positioned on top of each other to show that it is unreasonable to conclude that both angles have the same measure.

As students work on the task, monitor for students whose angle measures sum to exactly 180 degrees (and exactly 90 degrees) and students whose measures sum close to 180 degrees (and close to 90 degrees).

It is recommended not to show this image to students or they may try to copy the image rather than making their cut in different ways, but it is included here to clarify the instructions.

A rectangle, a semi-circle, a segment.

Launch

Distribute two half-sheets of blank paper per student and provide access to straightedges, scissors, and protractors. Emphasize that students should use a straightedge to draw the line they will cut along before they use scissors. Give students 3–5 minutes of quiet work time followed by a whole-class discussion.

Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide handouts of an angle representation for students to draw on or highlight and measure.
Supports accessibility for: Visual - Spatial Processing; Conceptual processing
Reading, Conversing: MLR3: Clarify, Correct and Critique. Display a sample student response that illustrates improper placement of the protractor that results in inaccurate measures or measures that are greater than 180 degrees when summed. Ask students to work with a partner to identify any errors, and to talk possible reasons for them. Invite students write a corrected response based on their conversation. Improved statements should include reference to the proper placement of protractors and how to determine measures with protractors. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Support sense-making; Maximize meta-awareness

Student Facing

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.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may want to make their first cut perpendicular to the side of the paper that they are cutting from. This could make it harder for them to notice the pattern that their two angles sum to 180 degrees. Encourage them to cut at a different angle.

Some students may struggle to position their protractor correctly to measure each cut piece. Prompt them to position the point that represents the center of the protractor on the vertex of their angle and line up the 0 on their protractor with one side of the angle, so that they can measure to the other side.

Some students may get angle measures that do not add up to exactly 180 (or 90) degrees. If the sum is close to 180 (or 90) degrees, this should be allowed during the work time and discussed during the activity synthesis. If the sum is not close to 180 (or 90), ask students to show you how they lined up the protractor to measure their angles.

Activity Synthesis

Select previously identified students to share their angle measures within the decomposed straight angle. Record each answer displayed for all to see and ask:

  • “What do you notice about the pairs of angle measures?” (They all sum to about 180 degrees.)
  • “Why do you think this is?” (They started out as a straight angle and were cut apart.)
  • “Why do you think some people got measurements that do not sum to exactly 180 degrees?” (measurement error)

Poll the class on the measure of Clare’s second angle. Invite students to share different strategies they used. It is not important to formalize a process for solving supplementary angles at this point, because that will be addressed more in the next activity.

Select previously identified students to share their angle measures within the decomposed right angle. Record each answer displayed for all to see and ask similar questions as before to guide students to articulate that these pairs of angles should sum to 90 degrees.

Ask students to think about how they solved for the measure of Priya’s second angle compared to how they solved for the measure of Clare’s second angle. “What was the same, and what was different?”

  • The process of using one known angle and what they should both add up to was the same.
  • The sums were different: 180 for Clare’s angles and 90 for Priya’s angles.

2.3: Is It a Complement or Supplement? (10 minutes)

Activity

In this activity, students begin to formalize a process for finding the measures of angles that are complements and supplements of angles with known measures. After they have worked on the activity and shared their solutions, they are introduced to the vocabulary terms complementary and supplementary. Complementary describes angles whose measures sum to 90 degrees and supplementary describes angles whose measures sum to 180 degrees.

Monitor for students who use or explain different ways to calculate the unknown angle measure. For example, when finding the measure of angle \(KOM\), some might write \(38+x=180\) and some might write \(180-38\).

Launch

Remind students that in the previous activity they used straight angles and right angles to help figure out unknown angle measures. In this activity, they are doing something similar but must figure out the unknown angle measure without a protractor.

Arrange students in groups of 2. Give students 3–4 minutes of quiet work time, followed by a partner and whole-class discussion.

Representation: Develop Language and Symbols. Create and maintain charts to display definitions and examples of complementary angles and supplementary angles.
Supports accessibility for: Conceptual processing; Memory

Student Facing

  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.

    ​​​​​

  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.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

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.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students get stuck on the first problem, ask them what measurement do angles \(BCA\) and \(ACD\) have to add up to. This should get them started noticing the relationships between all the angles involved.

Activity Synthesis

The goal of this discussion is to introduce students to the terms complementary and supplementary for describing relationships between pairs of angles.

First, have students compare answers and strategies for the last two questions with their partners.

Next, display the last two questions for all to see and ask:

  • “Which other problem in this activity was similar to the third question? How?” (The first problem, about angle \(ACD\), also involved subtracting from 90.)
  • “Which other problem in this activity was similar to the last question? How?” (The second problem, about angle \(KOM\), also involved subtracting from 180.)

Explain to students that the term complementary describes a pair of angles whose measures sum to 90 degrees, and the term supplementary describes a pair of angles whose measures sum to 180 degrees. It is not important at this point to discuss that complementary or supplementary angles do not need to be adjacent, as that will be explored in the next lesson. Ask:

  • “Which angles in this activity were supplementary angles?” (angles \(SOP\) and \(POR\) in the last question, as well as angles \(LOK\) and \(KOM\) from the second question)
  • “Which angles in this activity were complementary angles?” (angles \(CAD\) and \(DAB\) in the third question, as well as from the first question angles \(ACD\) and \(BCA\) or angles \(DAC\) and \(BAC\), or even from the second question angles \(OKN\) and \(OKL\))

Invite students to continue practicing using the words complementary and supplementary throughout the rest of this unit, so they can start to feel more comfortable using them in their vocabulary.

Speaking: MLR8 Discussion Supports. Provide the following sentence frames to support student use of the words complementary and supplementary: “Angles ___ and ___ are complementary angles because . . ." and “Angles ___ and ___ are supplementary angles because . . ." Some students may also benefit from choral repetition of these phrases. This will give all students an opportunity to practice these new words in context.
Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

  • What does it mean for two angles to be supplementary? (Their measures sum to \(180^\circ\).)
  • What does it mean for two angles to be complementary? (Their measures sum to \(90^\circ\).)
  • If you know two angles are supplementary and you know the measure of one angle, how can you find the other? (Subtract the known one from \(180^\circ\).)

Display diagrams and definitions of new vocabulary somewhere in the classroom so that students can refer back to them during subsequent lessons. As the unit progresses, new terms can be added.

2.4: Cool-down - Finding Measurements (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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.