The purpose of this activity is for students to recall prior work with supplementary angles and to connect vertical angles and 180-degree rotations of intersecting lines. As students find the angle measures, listen to their conversations, specifically for the use of vocabulary such as “supplementary angles,” “vertical angles,” and “rotations.”
Launch
Provide access to geometry toolkits, including protractors and tracing paper. If needed, display the image from the problem and invite a student to state the name of the angle (). Consider tracing the segments from to , then to , as the angle is being named to help students visualize the naming convention for angles where the middle letter denotes the angle’s vertex.
Student Lesson in Spanish
Activity
None
Find the measure of angle . Explain or show your reasoning.
Find and label a second angle in the diagram. Find and label an angle congruent to angle .
Student Response
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Building on Student Thinking
Activity Synthesis
Display the image for all to see. Invite students to share their responses, adding onto the image as needed to help make clear student thinking.
If no students use supplementary angles or the property that a straight line is , ask students how they could determine the measure of angle without a protractor. Highlight 2 supplementary angles, such as and , and write the term “supplementary” on the display near those angles. Highlight two vertical angles such as and and write the term “vertical angles” on the display.
14.2
Activity
Standards Alignment
Building On
Addressing
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations:
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
In this task, students explore the relationship between angles formed when two parallel lines are cut by a transversal line. Students investigate whether knowing the measure of one angle is sufficient to figure out all the angle measures in the diagram.
Monitor for students who use these different strategies to find the angle measures:
Measure every angle using a protractor
Measure some angles and use tracing paper to identify congruent angles
Use what they know about vertical and supplementary angles to identify some angles
Select these students to share during the whole-class discussion.
Launch
A transversal (or transversal line) for a pair of parallel lines is a line that meets each of the parallel lines at exactly one point. Introduce this idea, and provide a picture such as this picture where line is a transversal for parallel lines and :
Arrange students in groups of 2. Provide access to geometry toolkits. Give students 2–3 minutes of quiet think time, then 5–8 minutes of partner work time, followed by a whole-class discussion.
MLR2 Collect and Display. Collect the language students use to find angle congruence. Display words and phrases, such as “congruent,” “translate,” and “rotate.” During the Activity Synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed. Advances: Conversing, Reading
Activity
None
Lines and are parallel. They are cut by transversal .
Line A C contains point B. Line D F contains point E. Line J H contains points B and E. Angle A B J is labeled question mark. Angle A B E is labeled 63 degrees. Angle E B C is labeled question mark. Angle C B J is labeled question mark. Angle D E H is labeled question mark. Angle H E F is labeled question mark. Angle F E B is labeled question mark. Angle B E D is labeled question mark.
With your partner, find the seven unknown angle measures in the diagram. Explain your reasoning.
What do you notice about the angles with vertex and the angles with vertex ?
Using what you noticed, find the measures of the four angles at point in the diagram. Lines and are parallel.
Line A C contains point B. Line D F contains point E. Line H G contains points B and E. Angle A B H is labeled with a question mark. Angle A B E is labeled with a question mark. Angle E B C is labeled with a question mark. Angle C B H is labeled with a question mark. Angle G E F is labeled 34 degrees.
Student Response
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Building on Student Thinking
Students may fill in congruent angle measurements based on the argument that they look the same size. Ask students how they can be certain that the angles don't differ in measure by 1 degree. Encourage them to explain how we can know for sure that the angles are exactly the same measure.
Activity Synthesis
The goal of this discussion is for students to describe the relationships they notice between the angles formed when two parallel lines are cut by a transversal.
Display the image from the problem for all to see and invite groups to share what they noticed. Encourage students to use precise vocabulary, such as "supplementary angles" and "vertical angles," when describing how they figured out the different angle measurements (MP6). After students point out the matching angles at the two vertices, define the term alternate interior angles: Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.
Ask a few students to identify the pairs of alternate interior angles from the activity.
14.3
Activity
Standards Alignment
Building On
Addressing
8.G.A.1
Verify experimentally the properties of rotations, reflections, and translations:
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
The goal of this task is for students to connect their work with rigid transformations with the property that alternate interior angles are congruent. In this activity, students use properties of 180-degree rotations to find corresponding angles in the figure. As students describe their rigid transformations to their partner and listen to their partner’s reasoning, they construct and critique the argument that these angles are congruent (MP3).
Listen for different strategies students use to show that the angles are congruent and select these students to share their strategies during the discussion. Approaches might include:
A 180-degree rotation about
First translating to and then applying a 180-degree rotation with center
Launch
Arrange students in groups of 2. Give 2–3 minutes of quiet work time, followed by 2–3 minutes of partner discussion, then a whole-class discussion.
Provide access to geometry toolkits. Tell students that in this activity, we will try to figure out why we saw all the matching angles we did in the last activity.
Action and Expression: Develop Expression and Communication. To help get students started, display sentence frames such as “First, I because . . .”, “I noticed so I . . .”, “I agree/disagree because . . . ,” or “Why did you . . .?” Supports accessibility for: Language, Organization
Activity
None
Activity Synthesis
Invite previously selected students to share their explanations. Ask students to describe and demonstrate the transformations they used to show that alternate interior angles are congruent. If any students connect this diagram to earlier work with 180-degree rotations or their justifications that vertical angles are congruent, invite them to share their observations.
Consider displaying this image for all to see as students share their thinking:
If any students use a translation to take to or vice versa then claim that vertical angles are congruent, encourage more precision in their description by asking what rigid transformation tells them that vertical angles are congruent.
MLR8 Discussion Supports. For each response that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language. Advances: Listening, Speaking
14.4
Activity
Optional
Standards Alignment
None
Building On
Addressing
Building Toward
Instructional Routines
None
Materials
None
Activity Narrative
This activity is optional because it provides additional opportunities for students to find supplementary and vertical angles on a diagram. Students also compare their reasoning about alternate interior angles and transversals with a diagram that does not include parallel lines. This allows students to conclude that when two lines which are not parallel are cut by a third line, there are no alternate interior angles formed.
Launch
Provide access to geometry toolkits.
Activity
None
Lines and are not parallel in this image.
Line A C contains point B. Line D F contains point E. Line J H contains points B and E. Angle A B J is labeled question mark. Angle A B E is labeled 63 degrees. Angle E B C is labeled question mark. Angle C B J is labeled question mark. Angle D E H is labeled question mark. Angle H E F is labeled 108 degrees. Angle F E B is labeled question mark. Angle B E D is labeled question mark.
Find the missing angle measures around point and point .
What do you notice about the angles in this diagram?
Point is the midpoint of line segment .
Can you find a rigid transformation that shows angle is congruent to angle ? Explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this discussion is for students to articulate that for pairs of alternate interior angles to be formed, a transversal must cut two parallel lines. If the lines are not parallel, then we cannot use rigid transformations to show these pairs of angles are congruent.
Display both of these images for all to see:
Here are some questions for discussion:
"What differences do you see between these diagrams?" (One diagram has a set of parallel lines, the other does not have any. One diagram has the same angle measures around point and , but the other has different angle measures around and ).
"Which of these diagrams has a transversal and alternate interior angles?" (The diagram on the left with the parallel lines.)
"If you know the angle measure of , what other angles can you determine on each diagram?" (On the left diagram, you can determine all of the angles around point and then use alternate interior angles to determine the angles around point . On the right diagram, you can only know the angles around point .)
Lesson Synthesis
The goal of this discussion is for students to articulate which angles are congruent to one another and give an example of a rigid transformation that explains why.
Display the image of two parallel lines cut by a transversal. Tell students that in cases like this, translations and rotations can be particularly useful in figuring out angle measurements since they move angles to new positions, but the angle measure does not change.
Invite students to identify pairs of alternate interior, vertical, and supplementary angles in the image.
Here are some questions for discussion:
What is the value of , and how do you know? ( because it is the measure of an angle forming an alternate interior angle with the given 60-degree angle.)
How do you know the values of and ? and both equal 120 because they are also alternate interior angles, each supplementary to a 60-degree angle.
Student Lesson Summary
When two lines intersect, vertical angles are congruent, and adjacent angles are supplementary, so their measures sum to 180. For example, in this figure angles 1 and 3 are congruent, angles 2 and 4 are congruent, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.
Two intersecting lines. Angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees.
When two parallel lines are cut by another line, called a transversal, two pairs of alternate interior angles are created. (“Interior” means on the inside, or between, the two parallel lines.) For example, in this figure angles 3 and 5 are alternate interior angles and angles 4 and 6 are also alternate interior angles.
Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees. At the second intersection, angle 5 is marked 70 degrees. Angle 6 is marked 110 degrees. Angle 7 is marked 70 degrees. Angle 8 is marked 110 degrees.
Alternate interior angles are equal because a rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point halfway between the two intersections. Can you see how rotating about takes angle 3 to angle 5?
Using what we know about vertical angles, adjacent angles, and alternate interior angles, we can find the measures of any of the eight angles created by a transversal if we know just one of them. For example, starting with the fact that angle 1 is we use vertical angles to see that angle 3 is , then we use alternate interior angles to see that angle 5 is , then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is since . It turns out that there are only two different measures. In this example, angles 1, 3, 5, and 7 measure , and angles 2, 4, 6, and 8 measure .
Standards Alignment
Building On
7.G.B.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
