Lesson 12

Proofs about Quadrilaterals

12.1: Play with Parallelograms (10 minutes)

In this warm-up, students use 1-inch strips with evenly spaced holes and metal paper fasteners to recall the relationship between parallelograms and rectangles. This prepares students to prove that all rectangles are parallelograms. Before students begin working, they are explicitly asked to determine which mathematical statement(s) they agree with. Making a reasonable conjecture is often an important aspect of making sense of problems (MP1).

Digital

The definition “A parallelogram is a quadrilateral with two pairs of parallel opposite sides.” is already in the reference chart. Ask students to find it.

Ask students to add this definition to their reference charts as you add it to the class reference chart:

A rectangle is a quadrilateral with four right angles. (Definition)

Before opening the applet, ask, “Which statement(s) do you agree with?”

All rectangles are parallelograms.
All parallelograms are rectangles.

Provide access to devices that can run GeoGebra. Encourage students to drag lots of vertices as they answer the questions.

Which figures (if any) are always rectangles? Which figures can be dragged to make a rectangle?
Which figures (if any) are always parallelograms? Which figures can be dragged to make a parallelogram?

For access, consult one of our IM Certified Partners.

The definition “A parallelogram is a quadrilateral with two pairs of parallel opposite sides.” is already in the reference chart. Ask students to find it.

Ask students to add this definition to their reference charts as you add it to the class reference chart:

A rectangle is a quadrilateral with four right angles. (Definition)

Before making any quadrilaterals, ask, “Which statement(s) do you agree with?”

all rectangles are parallelograms
all parallelograms are rectangles

Make several parallelograms with your strips.
Make several rectangles with your strips.

For access, consult one of our IM Certified Partners.

Ask again, “Which statement(s) do you agree with?”

all rectangles are parallelograms (True)
all parallelograms are rectangles (False)

Ask students to add this definition to their reference charts as you add it to the class reference chart:

A rhombus is a quadrilateral with four congruent sides. (Definition)

Invite students to use their strips to hold up first an example and then a non-example of a rectangle and a rhombus. If needed, remind students that the fasteners don’t need to go in the last holes.

12.2: From Conjecture to Proof (15 minutes)

In a previous lesson, students matched diagrams to conjectures. In this lesson, they identify the given information and the statement to prove in the conjectures. For example, “all rhombuses are parallelograms” might turn into “show that in quadrilateral \(ABCD\), with segments \(AB\), \(BC\), \(CD\), and \(DA\) congruent, \(AB\) is parallel to \(CD\) and \(BC\) is parallel to \(DA\).” Deciding what precisely to show in order to prove a conjecture is not trivial. Monitor for students who try to prove other properties of the shapes are true rather than using the definition, such as trying to prove a figure is a rectangle by proving that opposite sides are congruent.

Digital

Arrange students in groups of 2. Demonstrate working on the first conjecture—all rectangles are parallelograms—with students. Instruct students to open GeoGebra Geometry from Math Tools or other dynamic geometry software.

Students have already used the applet to convince themselves the conjecture is true.
Draw a diagram of rectangle \(ABCD\) so all students will use the same labels. Do not label the right angles yet.

Give students 1 minute of quiet work time followed by 1 minute of discussing with their partner.

Ask students what they know and what they want to prove is true. (I know all 4 angles are right angles. I want to prove \(AB\) is parallel to \(CD\) and \(AD\) is parallel to \(BC\).)
Invite students to tell you what given information to mark on the diagram. (Right angles.)
Students may suggest marking that opposite sides are congruent. Remind students we have only proven that for parallelograms; they would need to prove that it is true for rectangles. Ask students if they need to. (No, it is not necessary for this proof.)
Ask students for ideas for a proof. Focus on the idea that lines perpendicular to the same line are parallel because of having congruent alternate interior angles (extend the lines in the diagram to help students see the structure if needed), and therefore, a rectangle has to be a parallelogram.

Invite students to work together to repeat the process for at least one more conjecture.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to help students develop the mathematical language of conjectures with given information and the statement to prove. After partners rewrite the conjecture, draw a diagram of the situation, and write a rough draft of the proof, invite them to create a visual display of their work. Then ask partners to quietly circulate and read at least two other visual displays about the same conjecture. Give students quiet think time to consider what is the same and what is different about their rewritten conjectures, diagrams, and proofs. Next, ask partners to discuss what they noticed. Listen for and amplify the language students use to compare how the given information and statement to prove are represented in the rewritten conjecture and diagram.
Design Principle(s): Cultivate conversation

Action and Expression: Develop Expression and Communication. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example, “If _____ has _____, then it is a . . . ,” “All _____ are . . . .”
Supports accessibility for: Language; Organization

Here are some conjectures:

All rectangles are parallelograms.
If a parallelogram has (at least) one right angle, then it is a rectangle.
If a quadrilateral has 2 pairs of opposite sides that are congruent, then it is a parallelogram.
If the diagonals of a quadrilateral both bisect each other, then the quadrilateral is a parallelogram.
If the diagonals of a quadrilateral both bisect each other and they are perpendicular, then the quadrilateral is a rhombus.

Pick one conjecture and use technology to convince yourself it is true.
Rewrite the conjecture to identify the given information and the statement to prove.
Draw a diagram of the situation. Mark the given information and any information you can figure out for sure.
Write a rough draft of how you might prove your conjecture is true.

For access, consult one of our IM Certified Partners.

Arrange students in groups of 2. Demonstrate working on the first conjecture, all rectangles are parallelograms, with students.

Students have already used the strips to convince themselves the conjecture is true.
Draw a diagram of rectangle \(ABCD\) so all students will use the same labels. Do not label the right angles yet.

Give students 1 minute of quiet work time followed by 1 minute of discussing with their partner.

Ask students what they know and what they want to prove is true. (I know all 4 angles are right angles. I want to prove \(AB\) is parallel to \(CD\) and \(AD\) is parallel to \(BC\).)
Invite students to tell you what given information to mark on the diagram. (Right angles.)
Students may suggest marking that opposite sides are congruent. Remind students we have only proven that for parallelograms, they would need to prove that it is true for rectangles. Ask students if they need to. (No, it is not necessary for this proof.)
Ask students for ideas for a proof. Focus on the idea that lines perpendicular to the same line are parallel because of having congruent alternate interior angles (extend the lines in the diagram to help students see the structure if needed), and therefore a rectangle has to be a parallelogram.

Invite students to work together to repeat the process for at least one more conjecture.

Here are some conjectures:

All rectangles are parallelograms.
If a parallelogram has (at least) one right angle, then it is a rectangle.
If a quadrilateral has 2 pairs of opposite sides that are congruent, then it is a parallelogram.
If the diagonals of a quadrilateral both bisect each other, then the quadrilateral is a parallelogram.
If the diagonals of a quadrilateral both bisect each other and they are perpendicular, then the quadrilateral is a rhombus.

Pick one conjecture and use the strips to convince yourself it is true.
Re-write the conjecture to identify the given information and the statement to prove.
Draw a diagram of the situation. Mark the given information and any information you can figure out for sure.
Write a rough draft of how you might prove your conjecture is true.

For access, consult one of our IM Certified Partners.

If students struggle to identify information they know or could figure out, direct them to their reference charts or remind them that adding auxiliary lines can help them to see structure.

Focus discussion on how students went from the broad statements of the conjectures to specific statements and diagrams that showed what was given and what had to be proved. Invite students to share their thinking. Then ask other students if the statement matches the given information and defines what we want to prove. Support the class to refine one another’s less precise statements, until the conjecture is written as a precise statement.

12.3: Checking a Proof (10 minutes)

Part of being a mathematician is reading and reviewing other people’s proofs. Reviewers look at completed proofs to understand them and how they may be useful, and they look at draft proofs to help improve them. Students continue to practice the skill of reviewing proofs in this activity.

Group students together who chose the same conjecture and if possible are at similar stages with their proof. Remind students of the displayed tips for writing proofs.

Engagement: Internalize Self Regulation. Demonstrate giving and receiving constructive feedback. Use a structured process and display sentence frames to support productive feedback. For example, “Another strategy would be _____ because . . . ,” “Is there another way to say . . . ?,” “How do you know . . . ?,” “Can you give an example?”
Supports accessibility for: Social-emotional skills; Organization; Language

Exchange proofs with your partner. Read the rough draft of their proof. If it convinces you, write a detailed proof together following their plan. If it does not convince you, suggest changes that will make the proof convincing.

For access, consult one of our IM Certified Partners.

Are you ready for more?

Draw 2 circles (of different sizes) that intersect in 2 places. Label the centers \(A\) and \(B\) and the points of intersection \(C\) and \(D\). Prove that segment \(AB\) must be perpendicular to segment \(CD\).

For access, consult one of our IM Certified Partners.

Inform students that while all the conjectures are true, only one of them will be used for future work.

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

If a parallelogram has (at least) one right angle, it is a rectangle. (Theorem)

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. To help students share the reasons they were convinced or not convinced by the proof, provide sentence frames such as: “I am convinced by this proof because _____” or “I am not convinced by this proof because _____.” If students are not convinced by a proof, press for details by asking how they would change the proof to make it more convincing. Prompt students to think about different possible audiences for the proof, and about the level of specificity or formality needed for a classmate versus a mathematician.
Design Principle(s): Support sense-making; Optimize output (for explanation)

Lesson Synthesis

The key idea in this synthesis is to focus students’ attention on the work they did to go from general claims to specific proofs.

Arrange students in groups of 2. Display these three statements that students made to re-write the conjecture. Use a Think Pair Share routine to have students critique the claims. Each claim has something to critique.

Conjecture: All rhombuses are parallelograms.

Andre: If \(ABCD\) is a rhombus, then it is a parallelogram.
Clare: If \(ABCD\) has four congruent sides, then the opposite sides are congruent, so it must be a parallelogram.
Diego: If \(AB \cong BC \cong CD \cong DA\) then \(AB \parallel CD\).

Possible critiques:

Andre doesn’t give us any new specific information to help us figure out what to prove.
For Clare: We know that all parallelograms have opposite sides congruent, but we don’t know if all shapes with opposite sides congruent have to be parallelograms, so proving this wouldn’t prove the shape was definitely a parallelogram. We don’t know that the converse of “in a parallelogram, opposite sides are congruent” must be true.
Diego is telling us we have to prove one pair of opposite sides parallel, but parallelograms have to have both pairs of opposite sides parallel.

12.4: Cool-down - A Proof In Time Saves Nine (5 minutes)

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Student Lesson Summary

Why did we spend so much time learning about when triangles are congruent? Because we can decompose other shapes into triangles. By looking for triangles that must be congruent we can prove other shapes have many properties. For example we could learn more about these types of quadrilaterals:

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
A rectangle is a quadrilateral with 4 right angles.
A rhombus is a quadrilateral with 4 congruent sides.
A square is a quadrilateral with 4 right angles and 4 congruent sides.
A kite is a quadrilateral which has 2 sides next to each other that are congruent and where the other 2 sides are also congruent.

Knowing how to decompose quadrilaterals into triangles using their diagonals lets us prove how the different quadrilaterals’ definitions lead to their diagonals having different properties. We can also look at whether arranging the diagonals to have certain properties gives us enough information to prove which type of quadrilateral must be formed. For example, we might conjecture that if one diagonal is the perpendicular bisector of the other, the figure is a kite. But how do we turn that into a statement that we can prove?

Here is a specific statement that shows what we mean by “one diagonal is the perpendicular bisector of the other” and “the figure is a kite.” In quadrilateral \(ABCD\) with diagonals \(AC\) and \(BD\), the diagonals intersect at \(P\). Segment \(AP\) is congruent to segment \(PC\), and \(AC\) is perpendicular to \(BD\). Prove that segment \(AB\) is congruent to segment \(BC\), and that segment \(CD\) is congruent to segment \(DA\). This specific statement lets us draw and label a diagram, which might give us some ideas about how to prove the statement is true.

Quadrilateral A B C D, with diagonal line segments A C and B D drawn, intersecting at Point P. Segment A C is perpendicular to B D.

Lesson 12

12.1: Play with Parallelograms (10 minutes)

Warm-up

Launch

Student Facing

Student Response

Launch

Student Facing

Student Response

Activity Synthesis

12.2: From Conjecture to Proof (15 minutes)

Activity

Launch

Student Facing

Student Response

Launch

Student Facing

Student Response

Anticipated Misconceptions

Activity Synthesis

12.3: Checking a Proof (10 minutes)

Activity

Launch

Student Facing

Student Response

Student Facing

Are you ready for more?

Student Response

Activity Synthesis

Lesson Synthesis

Lesson Synthesis

12.4: Cool-down - A Proof In Time Saves Nine (5 minutes)

Cool-Down

Student Lesson Summary

Student Facing