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.
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
This is the first Math Talk activity in the course. See the Launch for extended instructions for facilitating this activity successfully.
The purpose of this Math Talk is to review angle pairs in figures made up of one to three triangles to prepare students for figures made up of several triangles. It encourages students to think about complementary, supplementary, and vertical angle pairs, as well as the Triangle Angle Sum Theorem, and to rely on what they know about these angle relationships to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students find unknown angles in more complex figures.
To find unknown angles, students need to look for and make use of structure (MP7).
Launch
This is the first time students do the Math Talk instructional routine in this course, so it is important to explain how it works before starting.
Explain that a Math Talk has four problems, revealed one at a time. For each problem, students have a minute to quietly think and should give a signal when they have an answer and a strategy. The teacher then selects students to share different strategies (likely 2–3, given limited time) and might ask questions, such as “Who thought about it in a different way?” The teacher then records the responses for all to see, and might ask clarifying questions about the strategies before revealing the next problem.
Consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. (Start with unknown angle measure , then angle , angle , and angle ). For each problem:
Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity Synthesis
The goal of this discussion is to review students’ strategies for finding unknown angles when they encounter common angle pairs and special triangles. To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
If not brought up in students’ explanations, discuss with students the meaning of “complementary,” “supplementary,” “vertical angles,” and “isosceles triangles.” Remind students what the tick marks mean on the legs of the isosceles triangle and that the base angles in an isosceles triangle are congruent.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
“First, I because .”
“I noticed , so I .”
Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
Math Community
After the Warm-up, display the class Math Community Chart for all to see and explain that the listed “Doing Math” actions come from the sticky notes students wrote in the first exercise. Give students 1 minute to review the chart. Then invite students to identify something on the chart they agree with and hope for the class or something they feel is missing from the chart and would like to add. Record any additions on the chart. Tell students that the chart will continue to grow and that they can suggest other additions that they think of throughout today’s lesson during the Cool-down.
We now have many methods to help us learn more information about triangles. Ask students:
“What are some tools we used to find unknown angles?” (complementary angles, supplementary angles, vertical pairs, linear pairs, Triangle Angle Sum Theorem, base angles are the same in isosceles triangles, angles are congruent in an equilateral triangle)
“What are some tools we can use to find out information about the sides of a triangle?” (Triangle Inequality Theorem, Pythagorean Theorem for right triangles, short sides are across from small angles and long sides are across from large angles)
Student Lesson Summary
Suppose you need to find the unknown angles in a complicated figure and figure out which side is the longest. Here is a figure with several unknown angles.
One way to start is to consider the equilateral triangle . Since all sides are congruent, all angles in that triangle are congruent. Since the angles must add up to 180 degrees, each angle in triangle is 60 degrees, because .
Next, consider the angles with a vertex at . Using rules for supplementary and vertical angles, we can find these unknown angles. Angle is supplementary to angle , so the measure of angle is 120 degrees because . Since angle forms a vertical pair with angle , these angles are congruent. Similarly, angle forms a vertical pair with angle , so the measures of angles and are equal.
Triangle is isosceles, so we can use the Isosceles Triangle Theorem to find the measure of the two base angles. The measures of angles and are both 30 degrees because .
Finally, we can use the Triangle Angle Sum Theorem to find the two unknown angles. The measure of angle is 45 degrees, because , and the measure of angle is 25 degrees, because .
With this knowledge, we can say that is the longest side in triangle and is the longest side in triangle . This is not enough information to know which side is the longest overall or help us find the exact lengths of any of the sides.
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In this activity, students practice using the idea that, in a triangle, larger angles are opposite longer sides. Students also critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).
This is the first time Math Language Routine 3: Critique, Correct, Clarify is suggested in this course. In this routine, students are given a “first draft” statement or response to a question that is intentionally unclear, incorrect, or incomplete. Students analyze and improve the written work by first identifying what parts of the writing need clarification, correction, or details, and then writing a second draft (individually or with a partner). Finally, the teacher scribes as a selected second draft is read aloud by its author(s), and the whole class is invited to help edit this “third draft” by clarifying meaning and adding details to make the writing as convincing as possible to everyone in the room. Typical prompts are: “Is anything unclear?” and “Are there any reasoning errors?” The purpose of this routine is to engage students in analyzing mathematical writing and reasoning that is not their own, and to solidify their knowledge and use of language.
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
Allow students 3–5 minutes of quiet work time to complete the task. Allow 5–10 minutes for the Activity Synthesis.
Activity
None
Student Task Statement
Here is triangle . List the angles in order from smallest to largest.
In triangle (not drawn), side is the longest. Which angle is the largest? Explain how you know.
Student Response
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Building on Student Thinking
Activity Synthesis
The goal of this activity is for students to practice justifying their reasoning. Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response for the second question by correcting errors, clarifying meaning, and adding details.
Display this first draft:
Angle is the largest.
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
Give students 2–4 minutes to work with a partner to revise the first draft.
Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
Engagement: Provide Access by Recruiting Interest. Revisit math community norms to prepare students for the whole-class discussion. Supports accessibility for: Social-Emotional Functioning, Attention
None
Activity Narrative
Students look for and make use of structure (MP7) in a figure made up of several triangles to find unknown angles. Then, using what they know about the relationship between sides and angles in a triangle, students find the longest and shortest sides in each triangle. Students will have the opportunity to discuss their work with a partner. Monitor for students that change their answers after discussing with a partner.
Launch
Give students 5 minutes of quiet work time. Tell students to work individually as much as possible but to ask for help from a partner if they get stuck. Encourage partners to compare answers and correct any mistakes in their angles before they begin filling out the table. If students disagree on an answer, encourage them to discuss their reasoning and come to an agreement on the correct answer. Select students who changed their answers to discuss during the Activity Synthesis.
MLR8 Discussion Supports. Display sentence frames to support whole-class discussion. Examples:
“First, I because .”
“I noticed , so I .”
Advances: Speaking, Representing
Activity
None
Student Task Statement
Given that triangle and triangle are isosceles, triangle is equilateral, and the measure of angle is 90 degrees, label all unknown angle measures in the figure.
For each triangle listed in the table, name the longest side and shortest side.
triangle
longest side(s)
shortest side(s)
Student Response
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Building on Student Thinking
If students have trouble filling in the unknown angles, consider saying:
“Tell me more about the angles in equilateral or isosceles triangles.”
“How could vertical, supplementary or complementary angle pairs help you find unknown angles?”
Activity Synthesis
Display the figure from the second question for all to see. Ask previously selected students to share their answers and reasoning. As students share, fill in the unknown information on the figure and in the table.
Standards Alignment
Building On
Addressing
Building Toward
HSG-CO.C.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
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.
