Unit 7, Lesson 3
Nonadjacent Angles
Grade 7
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The purpose of this Warm-up is for students to use structure to reason about equivalent equations. In this unit, students will write equations to represent how angles are related to each other, and this Warm-up helps prepare for that work.
All of the given statements could be true so students may be quick to say each of them must be true. Ask these students if there is a case in which that particular statement would not be true for possible values for
Arrange students in groups of 2.
Ask students, “If we know for sure that
Give students 2 minutes of quiet work time followed by 1 minute to discuss their responses with a partner. Follow with a whole-class discussion.
If
Invite students to share their reasoning for each statement. If students disagree, allow students to discuss until they come to an agreement. Consider asking some of the following questions while students discuss:
If students claim that
If students claim that
MLR8: Discussion Supports
To Gather
Geometry toolkits
In this activity, students see that angles do not need to be adjacent to each other in order to be considered complementary or supplementary. Students are given two different polygons and are asked to find complementary and supplementary angles, using any tools in their geometry toolkit (MP5). The most likely approaches are:
As students work, monitor for students who use either approach listed or some other strategy. Also, encourage students to use precise vocabulary and language that they learned in previous activities and lessons (MP6).
Invite students to share their definitions of complementary and supplementary, and consider displaying the meanings for all to see through the remainder of the class. Highlight ideas that involve angle measurements or sums, and leave ideas implying that they must be touching or adjacent as an open question for now.
Arrange students in groups of 2. Provide access to geometry toolkits. Give students 3–4 minutes of quiet work time, followed by partner and whole-class discussions.
Identify any pairs of angles in these figures that are complementary or supplementary.
Invite previously selected students to share their strategies for finding a pair of complementary and supplementary angles. Ensure that correct use of a protractor to find the measure of an angle is clearly and carefully demonstrated. This will help all students prepare for the next activity, in which everyone will be using a protractor.
MLR2: Collect and Display
To Gather
Geometry toolkits
The purpose of this activity is for students to learn about vertical angles. Each student draws two intersecting lines and measures the four resulting angles. Then, students examine multiple examples to come up with a conjecture for any relationships they noticed (MP8).
Arrange students in groups of 2–4. Provide access to geometry toolkits.
Display this image for all to see. Tell students that this is a pair of intersecting lines, or lines that cross at a point.
Ask students to identify 2 angles that are supplementary to angle
Tell students they will draw their own diagram with a different set of intersecting lines. Remind them to draw arcs to label the degree measures of their angles.
Give students 2–3 minutes to draw and measure the figure. Remind them to draw arcs to label the degree measures of their angles. Each person in the group should draw a different diagram. Follow with small-group and whole-class discussions.
Use Collect and Display to direct attention to words collected and displayed from an earlier activity. Collect the language that students use to describe the relationships they notice between angles. Display words and phrases, such as “adjacent,” “right,” “straight,” “complementary,” and “supplementary,” as well as descriptions such as “across from each other,” “mirrors of each other,” or “add up to a straight angle.”
Direct students’ attention to the reference created using Collect and Display. Ask students to share their observations. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “The angles across from each other look the same” already on it, which can be updated with the more precise phrase “The opposite angles have the same angle measure.”)
If there are students who use supplementary angles to explain why vertical angles have equal measures, ask them to share their observations last.
Define vertical angles as a pair of angles, formed by two intersecting lines, that are opposite each other.
Display the image, and ask students to identify four pairs of vertical angles. In particular, students may have trouble seeing that angles
Although students don’t need to know a proof that vertical angles always have the same measure, it may be helpful to show one way to understand why they are. If time allows, share that in the image. . .
Optional
None
This activity gives students an opportunity to practice recognizing complementary, supplementary, and vertical angles and using what they know about those types of angles to find unknown angle measures. Some students may feel comfortable writing equations to show their reasoning, but it is not important that all students use this strategy at this point, as it will be the focus of future lessons. Encourage students to continue using the new vocabulary.
Arrange students in groups of 2. Make sure students know how to play a row game. Give students 5–6 minutes of partner work time followed by a whole-class discussion.
Find the measurement of the angles in one column. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error and correct it.
column A
column B
Find the value of
Find the value of
In right triangle
column A
column B
Angle
Find the value of
Two angles are complementary. One angle measures 37 degrees. Find the measure of the other angle.
Two angles are supplementary. One angle measures 127 degrees. Find the measure of the other angle.
Ask students, “Were there any rows on which you and your partner did not get the same answer?” Invite students to share how they came to an agreement on the final answer for the problems in those rows.
Consider asking some of the following questions:
When two lines cross, they form two pairs of vertical angles. Vertical angles are across the intersection point from each other.
Vertical angles always have equal measure. We can see this because they are always supplementary with the same angle. For example:
This is always true!
That means
Help us improve by sharing suggestions or reporting issues.
Some students may struggle to use a protractor to measure angles when the rays are not drawn long enough to reach the edge of the protractor. Prompt them to extend the sides of the angle, using a straightedge.
Use a straightedge to draw two intersecting lines. Use a protractor to measure all four angles whose vertex is located at the intersection.
Compare your drawing and measurements to the drawings and measurements created by others in your group. What patterns do you notice about the relationships between angle measures at an intersection?
Some students may label the angle measures toward the end of the rays, where they read the number from the protractor. This is not precise enough, because two different angles share each ray. Remind students about drawing arcs to clarify which angle they measured.
If students struggle to see relationships of angles in figures, prompt students to look for complementary, supplementary, or vertical angles.