Lesson 4

Congruent Triangles, Part 2

4.1: Make That Triangle (5 minutes)

Warm-up

From middle school, students should be familiar with the idea that only some information is needed to uniquely determine a triangle, and this warm-up emphasizes that not every piece of information you can measure about a triangle is needed to draw it. This activity allows students to practice using the tools they will need in subsequent activities.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Student Facing

Draw triangle $$ABC$$ with these measurements:

• Angle $$A$$ is 40 degrees.
• Angle $$B$$ is 20 degrees.
• Angle $$C$$ is 120 degrees.
• Segment $$AB$$ is 5 centimeters.
• Segment $$AC$$ is 2 centimeters.
• Segment $$BC$$ is 3.7 centimeters.

Highlight each piece of given information that you used. Check your triangle to make sure the remaining measurements match.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The goal of this discussion is for students to be clear that not every piece of given information was used to draw a unique triangle, and so some information wasn’t needed to be sure that all the triangles are congruent. This discussion gives an initial opportunity to begin to connect these ideas:

• Not every piece of information you can measure about one triangle is needed to make an exact copy of that triangle.
• Not every piece of information you can measure about two triangles is needed to prove the triangles are congruent.

If needed, demonstrate how to use protractors and rulers precisely. Discuss the level of precision possible with the given tools (the triangle built with $$AB=5, AC=2$$, and $$m \angle A =40^\circ$$ actually has $$BC=3.699, m \angle C=119.66^\circ$$, and $$m \angle B=20.34^\circ$$). Since the goal is to write a proof without numbers, we can focus on the number of measurements needed to draw a congruent triangle, so rounding and drawings that are close enough will be just fine.

4.2: Info Gap: Too Much Information (20 minutes)

Activity

This Info Gap activity gives students an opportunity to determine and request the information needed to construct a triangle congruent to the given triangle.

The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Monitor for students who asked for limited information either successfully (Angle-Side-Angle, Side-Angle-Side, Side-Side-Side) or unsuccessfully (Angle-Angle-Angle, Side-Side).

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Here is the text of the cards for reference and planning:

Launch

Tell students they will continue to think about what is the least amount of information they need to construct a triangle that is congruent to their partner’s. Explain the info gap structure, and consider demonstrating the protocol if students need more practice. If you choose to demonstrate it, here is an additional sample Data Card for you to use:

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Pause the class if needed to ask a few groups to share something they tried that didn’t lead to making congruent triangles, and something that did. “Did anyone make congruent triangles with only four pieces of information? Only three? Two?”

Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to construct a triangle congruent to the given triangle. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
4. Read the problem card, and solve the problem independently.
5. Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

3. Explain to your partner how you are using the information to solve the problem.
4. When you have enough information, share the problem card with your partner, and solve the problem independently.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Elena wonders whether she could play the Info Gap with area included as an extra piece of information in the data cards. She draws a card with this information and asks Han to play.

1. If Han asks for 2 sides and the area, do you think this will be enough information for Han to draw a congruent triangle?
2. If Han asks for 2 angles and the area, do you think this will be enough information for Han to draw a congruent triangle?

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

If students are struggling to create their drawings, invite a group that is having success to demonstrate how they are using the protractors, rulers, and dried linguine pasta.

Activity Synthesis

The goal of this conversation is to focus on building the visual intuition for why Angle-Side-Angle, Side-Angle-Side, and Side-Side-Side are unambiguous.

Select groups to share who tried the following combinations, in this sequence:

• Angle-Side-Angle
• Side-Angle-Side
• Side-Side-Side

Invite the group sharing to demonstrate the steps they took to construct the matching triangle. Leave the work displayed throughout the lesson, if possible.

Then, select a group to share a combination that was not enough information, such as Angle-Angle-Angle, or Side-Side. Ask students, “Do you think if all I knew about two triangles was that information, that I could write an accurate proof that the two triangles are congruent?” (No.)

4.3: Too Little Information? (10 minutes)

Activity

While this activity does draw on the ambiguous Side-Side-Angle case, the main concept in the activity is that the position of the given angles and sides relative to one another matter. Students might at first think that Tyler put the sides in the wrong order relative to the given angle, which can help them see that being specific about the location of the sides and angles relative to one another matters.

Monitor for students who:

• use materials to recreate what Tyler should have gotten; they might generate valuable insights into why Side-Side-Angle does not work
• compare what Tyler asked for to the examples that worked from the previous activity

Launch

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, highlight the obtuse angle of both triangles the same color.
Supports accessibility for: Visual-spatial processing

Student Facing

Jada and Tyler were playing the Info Gap, using Card 3.

Tyler asked, “Can I have 2 sides and an angle?”

Jada told Tyler that one angle was $$16^{\circ}$$, one side was 5 cm, and one side was 4 cm. Here is the triangle Tyler made:

1. Is Tyler’s triangle congruent to the triangle on the Data Card?
2. Did Tyler do anything that didn’t match Jada’s instructions?
3. How could Tyler have made a more specific request for 2 sides and an angle so that his triangle was guaranteed to match Jada’s?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The main idea in this discussion is to distinguish Side-Angle-Side from the more general two sides and one angle.

Select a group to share that physically explored Tyler’s construction. If students struggle to explain, display this diagram and ask students how it relates. (There are 2 places the circle with radius 4 intersects the ray.​​​​)

If a group compared Tyler’s request to the successful two sides and one angle criteria from the previous activity, invite them to share their thinking next. If no groups discussed this idea, re-display the Side-Angle-Side example from the previous discussion and ask the class to compare it to Tyler’s request. Invite multiple students to explain the difference between what Tyler asked for and the successful triangle congruence criteria.

Speaking: MLR8 Discussion Supports. To help students produce statements that distinguish Side-Angle-Side from the more general two sides and one angle, provide sentence frames such as: “The difference between what Tyler asked for and the strategy you used is _____,” “In the Side-Angle-Side example from the Info Gap activity, I noticed that _____,” and “In Tyler’s construction of the triangle, I noticed that _____.”
Design Principle(s): Support sense-making; Optimize output (for comparison)

Lesson Synthesis

Lesson Synthesis

Explain to students that today’s activities were all focused on making conjectures about how much information is needed to prove that two triangles are congruent. Remind students of the three examples shared that seemed to work to establish that any triangles with these measurements would be exact copies of one another—congruent triangles.

Create a display labeled “Triangle Congruence Criteria Conjectures.” This display should be posted in the classroom for the remaining lessons within this unit. It should look something like:

• Angle-Side-Angle “Two angles and the side between them.”
• Side-Angle-Side “Two sides and the angle between them.”
• Side-Side-Side “All three sides.”

Remind students of Tyler’s triangle and ask them to name and illustrate some sets of information that they are convinced are not valid triangle congruence criteria. They might list:

• Side-Side-Angle
• Angle-Angle
• Other sets of information involving just one or two sides or angles

4.4: Cool-down - Angles All the Way (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.

Student Lesson Summary

Student Facing

If we know that every pair of corresponding parts is congruent, then we know the 2 triangles are congruent. But we don‘t need that much information. If we know the angles of a triangle are 30 degrees and 60 degrees, we can figure out the third angle is 90 degrees. So when we start drawing a triangle, the triangle is complete before we measure every angle. Figuring out which sets of measurements are enough to draw a complete triangle tells us which sets of measurements are enough to prove triangles are congruent. Here are 3 sets of measurements that appear to be enough information to prove that the 2 triangles will be congruent:

• Two pairs of corresponding sides are congruent and the angles between those sides are congruent.
• Two pairs of corresponding angles are congruent and the sides between those angles are congruent.
• Three pairs of corresponding sides are congruent.