# Lesson 11

Side-Side-Angle (Sometimes) Congruence

## 11.1: Notice and Wonder: Congruence Fail (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that not every set of three pairs of congruent corresponding parts will guarantee triangle congruence, which will be useful when students explore Side-Side-Angle Triangle Congruence in a later activity. While students may notice and wonder many things about these statements and images, the fact that the triangles are not congruent despite having so many corresponding parts congruent is the important discussion point.

### Launch

First, display the congruence statements without the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner.

Next, display the image for all to see. Give students 1 minute of quiet think time, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

In triangles \(GBD\) and \(KHI\):

- Angle \(GBD\) is congruent to angle \(KHI\).
- Segment \(BD\) is congruent to segment \(HI\).
- Segment \(DG\) is congruent to segment \(IK\).

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If students’ surprise at the images does not come up during the conversation, ask students to discuss this idea.

## 11.2: Dare to Be (Even More) Different (10 minutes)

### Optional activity

In this activity, students are arranged in groups of three, given side lengths and an angle measure, and encouraged to make triangles that do not look like one another’s. Given an angle and two sides that are not both adjacent to the given angle, there are three possible triangles that students can make.

As students negotiate how they will make the triangles, listen for different strategies students have for making different triangles. Monitor for students who:

- changed which side was adjacent to the given angle and which side was opposite
- swung the non-adjacent side around to see if there were multiple triangles that could be made
- found that only one triangle could be made when the longer side was the non-adjacent side

### Launch

Arrange students in groups of 3.

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for the language students use to describe how they created different triangles given two side lengths and an angle measure. Capture student language that attempts to describe the position of the sides relative to the given angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask students to revise and improve how ideas are communicated. For example, a statement such as, “I created a different triangle by switching the sides” can be improved with the statement, “I created a different triangle changing which side was adjacent to the given angle and which side was opposite.” This will help students use the mathematical language necessary to explain how to create different triangles given two sides and an angle.

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

### Student Facing

Use technology to make a triangle using the given angle and side lengths so that the given angle is* not* between the 2 given sides. Try to make your triangle different from the triangles created by the other people in your group.

- Angle: \(40^\circ\)
- Side length: 6 cm
- Side length: 8 cm

### Student Response

For access, consult one of our IM Certified Partners.

### Launch

Arrange students in groups of 3.

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for the language students use to describe how they created different triangles given two side lengths and an angle measure. Capture student language that attempts to describe the position of the sides relative to the given angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask students to revise and improve how ideas are communicated. For example, a statement such as, “I created a different triangle by switching the sides” can be improved with the statement, “I created a different triangle changing which side was adjacent to the given angle and which side was opposite.” This will help students use the mathematical language necessary to explain how to create different triangles given two sides and an angle.

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

### Student Facing

Copy these segments and use them to make a triangle using the given angle so that the given angle is* not* between the 2 given sides. Draw your triangle on tracing paper. Try to make your triangle different from the triangles drawn by the other people in your group.

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

If a group of students decides it is only possible to make one or two different triangles, encourage them to list all the possible ways to order the given pieces and check that they have tried all of them. (Angle, short side, long side or angle, long side, short side)

### Activity Synthesis

Invite students to share how many different triangles their group was able to create. Select and sequence students to share their strategies who:

- changed which side was adjacent to the given angle and which side was opposite
- swung the non-adjacent side around to see if there were multiple triangles that could be made
- found that only one triangle could be made when the longer side was the non-adjacent side

In this activity, students are noticing and conjecturing. If no students observed when one triangle could be made and when two triangles could be made, that’s fine.

Ask students whether they think any two sides and one angle measure is enough information to guarantee that any copies of that triangle will be congruent. Ask them how their ideas are informed by the warm-up and this lesson.

- No, it’s not enough information because that’s what we knew about the warm-up triangle, and those triangles weren’t congruent.
- Sometimes it’s enough information, because the Side-Angle-Side Triangle Congruence Theorem works.
- Sometimes it’s enough information; when we had the longer side opposite the given angle, we could only make one triangle.

*Engagement: Develop Effort and Persistence.*Break the class into small group discussion groups and then invite a representative from each group to report back to the whole class.

*Supports accessibility for: Attention; Social-emotional skills*

## 11.3: Ambiguously Ambiguous? (20 minutes)

### Optional activity

In the previous activity, students may have noticed that once they decided which side would be adjacent to the given angle, they could sometimes make two triangles and sometimes make one triangle. This may lead to questions about whether knowing an angle, a side, and another side (moving clockwise around the triangle) could be enough information to guarantee you can make a congruent copy of the triangle. This activity formalizes that question and answer by giving students instructions to make different triangles and exploring which instructions always lead to a single unique triangle and which are ambiguous.

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

### Launch

Identify a way for students to compare all the examples of a given triangle. For example, invite students to place all the triangles labeled \(ABC\) in a single visual display.

Arrange students in 8 groups. Provide each group with tools to create a visual display. Assign a different card to each group.

*Representation: Internalize Comprehension.*Activate strategies students have for making different triangles. Allow students to use dynamic geometry software to ensure inclusive participation in the activity.

*Supports accessibility for: Memory; Conceptual processing*

### Student Facing

Your teacher will give you some sets of information.

- For each set of information, make a triangle using that information.
- If you think you can make more than one triangle, make more than one triangle.
- If you think you can’t make any triangle, note that.

When you are confident they are accurate, create a visual display.

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Triangle \(ABC\) is shown. Use your straightedge and compass to construct a new point \(D\) on line \(AC\) so that the length of segment \(BD\) is the same as the length of segment \(BC\).

Now use the straightedge and compass to construct the midpoint of \(CD\). Label that midpoint \(M\).

- Explain why triangle \(ABM\) is a right triangle.
- Explain why knowing the angle at \(A\) and the side lengths of \(AB\) and \(BC\) was not enough to define a unique triangle, but knowing the angle at \(A\) and the side lengths of \(AB\) and \(BM\) would be enough to define a unique triangle.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Display the prompt: “When you are given that two pairs of corresponding sides are congruent, and a pair of corresponding angles that are *not* between the sides are congruent, that is enough to guarantee triangle congruence if \(\underline{\hspace{.5in}}\), but not enough information if \(\underline{\hspace{.5in}}\).”

Invite students to do a gallery walk and determine how to fill in the blanks. (The longer side is opposite the angle...the shorter side is opposite the given angle.)

Note that triangles \(GHI\) and \(STU\) have the same side lengths and angle measures, just with a different ordering. Comparing these two cases might help students who are struggling to see the difference.

“Only one triangle can be made, and triangle congruence is guaranteed, when you know that the longer of the two given sides is opposite the given angle.” Add this theorem to the display of triangle congruence theorems.

*Speaking, Representing: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. As students share how to fill in the blanks of the statement, provide the class with the following sentence frames to help them respond: "I agree because _____” or "I disagree because _____.” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “The longer side must be opposite the angle” can be restated as the question, “If the longer side is opposite the angle, is this enough to guarantee triangle congruence?”

*Design Principle(s): Support sense-making; Cultivate conversation*

## Lesson Synthesis

### Lesson Synthesis

The goal of this discussion is for all students to understand this statement: “When you are given that two pairs of corresponding sides are congruent, and a pair of corresponding angles that are *not* between the sides are congruent, that is enough to guarantee triangle congruence if the longer side is opposite the given angle, but not enough information if the shorter side is opposite the given angle.”

Encourage students to sketch pictures of the ambiguous case, and the non-ambiguous case, and label them in ways that help them understand. Invite students to share their sketches with the class, and work as a class to annotate and understand the sketches. Make sure to discuss the case of the right triangle and how we know the hypotenuse must be the longest side.

A possible sketch might look like this:

## 11.4: Cool-down - Are They Ambiguous? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

Imagine we know triangles have 2 pairs of corresponding, congruent side lengths, and a pair of corresponding, congruent angles that is not between the given sides. What can we conclude?

Sometimes this is not enough information to determine that the triangles made with those measurements are congruent. These triangles have 2 pairs of congruent sides and a pair of congruent angles, but they are not congruent triangles.

If the longer of the 2 given sides is opposite the given angle though, that does guarantee congruent triangles. In a right triangle, the longest side is always the hypotenuse. If we know the hypotenuse and the leg of a right triangle, we can be confident they are congruent.