Lesson 10
Practicing Proofs
10.1: Brace Yourself! (5 minutes)
Warmup
Throughout this lesson and in several subsequent lessons, students will make conjectures and prove properties of quadrilaterals. This activity gives students a chance to experience a dynamic tool: 1inch strips cut from card stock with evenlyspaced holes and metal fasteners.
Launch
Explain what diagonal braces mean in the context of carpentry. When carpenters want to ensure that something they are building that is supposed to be rectangular doesn’t slant, sometimes they use diagonal braces, which are pieces of wood or metal that go from opposite corners of the rectangle. Ask students if they’ve ever experienced that, and how the braces help the structure keep its shape. (Four edges are flexible, but the diagonal forces a certain angle, so now the shape is rigid.)
Explain that they will be making different configurations of braces to see what kinds of shapes can be built around them. Demonstrate choosing two strips, attaching them at one point with the metal paper fasteners, arranging them at a desired angle, and then tracing around them to approximate a quadrilateral.
Student Facing
What can you do with the braces and fasteners your teacher will give you?
What different ways can you arrange them?
What different quadrilaterals can you make by changing the braces?
Keep track of your findings.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Students might not realize that braces go diagonally from vertex to vertex, and they may think they go through the midpoints of opposite sides. Clarify by tracing around the endpoints to demonstrate how to visualize the quadrilateral.
Activity Synthesis
Invite a student to name or describe a quadrilateral they made, without describing how they arranged the braces. Give students 1 minute to think, ask questions, and arrange their braces, then have them hold up their idea of how to make a quadrilateral of the same type. Allow the student who suggested the quadrilateral type to give feedback.
10.2: Card Sort: More Practice Seeing Shortcuts (15 minutes)
Activity
Students will sort different images during this activity. A sorting task gives students opportunities to analyze representations closely and make connections (MP2, MP7). Students interpret the model in a situation and then check if their results make sense by applying a triangle congruence theorem.
Students will identify triangles that are congruent based on the triangle congruence theorems. Focusing on triangles \(KNL\) and \(MNL\), monitor for students who:
 think there is not enough information
 use the Isosceles Triangle Theorem or use the AngleSideAngle Triangle Congruence Theorem without finding the measure of the third angle
 use the Pythagorean Theorem to use the SideSideSide Triangle Congruence Theorem
Launch
Encourage students to use the braces and fasteners from the warmup to model.
Arrange students in groups of 2. Tell them that, in this activity, they will sort some cards into categories of their choosing. When they sort the figures, they should work with their partner to come up with categories.
Pause the class after students have sorted the cards into categories of their choosing. Select groups of students to share their categories and how they sorted their cards. Choose as many different types of categories as time allows. Attend to the language that students use to describe their categories and figures, giving them opportunities to describe their figures more precisely. Highlight the use of terms like rigid, congruent, and ambiguous. After a brief discussion, invite students to complete the remaining questions.
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
Supports accessibility for: Conceptual processing
Student Facing

Your teacher will give you a set of cards that show different structures. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into 2 categories in a different way. Be prepared to explain the meaning of your new categories.

Sort the cards by rigid vs. flexible structures.

State at least one set of triangles that can be proved congruent using:

SideAngleSide Triangle Congruence Theorem

AngleSideAngle Triangle Congruence Theorem

SideSideSide Triangle Congruence Theorem

Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
This is the John Hancock Building. What shape do you think surrounds the diagonal braces? List several ways to test your conjecture.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
In the isosceles triangle image, students will need to use either the Triangle Angle Sum Theorem or the Pythagorean Theorem to figure out that if they know two pairs of corresponding angles or sides are congruent in this right triangle, the third pair must be, too. Students may struggle because we don’t have specific measures for the third angle or side, so support them to think about “If we knew these angles were, say, \(90^{\circ}\) and \(60^{\circ}\), could the third pair of angles have different measures?” or “If we knew these sides were, say, 4 cm and 5 cm, could the third pair of sides have different measures?”
Activity Synthesis
Select students to share their different reasoning about triangles \(KNL\) and \(MNL\):
 who thought there was not enough information
 who used the Isosceles Triangle Theorem or used the AngleSideAngle Triangle Congruence Theorem without finding the measure of the third angle
 who used the Pythagorean Theorem to use the SideSideSide Triangle Congruence Theorem
If not mentioned by students, discuss how the Triangle Angle Sum Theorem and Pythagorean Theorem can help us prove that a third side or angle is congruent even if we don’t know the specific measurements.
Ask students, “How could you make the structures that are flexible into rigid ones?” (Add a diagonal brace that would decompose the shape into triangles.)
10.3: Matching Pictures to Proofs (10 minutes)
Activity
In this partner activity, students take turns thinking about which diagrams (on the cards from the previous activity) support different proof statements. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). Monitor for students who confuse the given information with what is to be proven, for example, by matching the rhombus with “A quadrilateral with perpendicular diagonals that bisect each other is equilateral.”
Students will use these diagrams to write proofs in the cooldown.
Launch
Students should still have access to the cards from the previous activity.
Keep students in groups of 2. Tell students that for each statement, one person finds a diagram and explains why they think it could be used to illustrate the given/proven statement. Their partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next statement, the students swap roles. If needed, demonstrate this protocol before students start working. Allow students to use their holepunched strips and metal fasteners, if desired, during this lesson as well.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Take turns with your partner to match a statement with a diagram that could go with that proof. For each match you find, explain to your partner how you know it’s a match. For each match your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
 A quadrilateral with perpendicular diagonals that bisect each other is equilateral.
 If one diagonal of a quadrilateral is the perpendicular bisector of the other, then 2 pairs of adjacent sides are congruent.
 Opposite angles in an equilateral quadrilateral are congruent.
 In a parallelogram, opposite sides are congruent.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Draw students’ attention to figure \(ZIAB\). Check that students understand why it matches “Opposite angles in an equilateral quadrilateral are congruent.” and not “A quadrilateral with perpendicular diagonals that bisect each other is equilateral.”
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
Lesson Synthesis
Lesson Synthesis
Encourage students to make conjectures about what must be true about an equilateral quadrilateral. (It’s a rhombus. It’s a parallelogram. It’s a square. All the angles are congruent. The diagonals are congruent. The diagonals are perpendicular.)
Instruct students to make an equilateral quadrilateral with the shorter strips (because students are in partners, they should have 4 short strips between them). Ask groups if they want to refine any of their conjectures based on their exploration with the strips. (It’s a rhombus. It’s a parallelogram. It’s not necessarily a square. All the angles are not necessarily congruent. The diagonals are not necessarily congruent. The diagonals are perpendicular.)
10.4: Cooldown  Practice Proving (10 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
To prove that segments or angles are congruent, we can look for triangles that those segments or angles are part of. Can the triangles be proven congruent? Are the segments or angles corresponding parts of congruent triangles? Does that help prove the conjecture?
To prove that the triangles are congruent, we can look at the diagram and given information. Think about whether it will be easier to find pairs of corresponding angles that are congruent or pairs of corresponding sides that are congruent. Then check if there’s enough information to use the SideSideSide, AngleSideAngle, or SideAngleSide Triangle Congruence Theorems.
Here is an example: Prove that in a quadrilateral with 4 congruent sides, the opposite sides are parallel.
First, sketch a diagram to see what is given and look for congruent triangles. Since this is about a quadrilateral, adding a diagonal to make triangles will be helpful.
Because all the sides of the quadrilateral are congruent, and the triangles formed by the diagonals share a third side, we can use the SideSideSide Triangle Congruence Theorem to prove that triangles \(ABC\) and \(CDA\) are congruent.