# Lesson 13

## 13.1: Notice and Wonder: Diagonals (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that the diagonals of a parallelogram bisect each other and the diagonals of a rectangle are congruent. Students will write proofs of these conjectures in a subsequent activity. While students may notice and wonder many things about these images, the relationships between the diagonals of a parallelogram and the diagonals of a rectangle are the important discussion points.

This activity is designed to be done digitally because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting and interacting with the applet of the parallelogram and rectangle would be helpful during the launch as students notice and wonder.

### Launch

Tell students their job is 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, followed by a whole-class discussion.

### Student Facing

Here is parallelogram $$ABCD$$ and rectangle $$EFGH$$. What do you notice? What do you wonder?

### Launch

Display 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, followed by a whole-class discussion.

Consider displaying the applet for all to see by navigating to this URL: ggbm.at/cf56pbrv.

### Student Facing

Here is parallelogram $$ABCD$$ and rectangle $$EFGH$$. What do you notice? What do you wonder?

### 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 conjectures about diagonals do not come up during the conversation, ask students to discuss this idea.

## 13.2: The Diagonals of a Parallelogram (15 minutes)

### Activity

This activity invites students to convince themselves, then a friend, and then a skeptic that the diagonals of a parallelogram bisect each other. Students can use transformations or congruent triangles to convince a skeptic that the diagonals of a parallelogram bisect each other.

Stating the goal of the proof in different ways may help students see a different path to the proof. For example, the proof can be restated as “Show that the midpoint of $$AC$$ and the midpoint of $$BD$$ are the point of intersection.” This might suggest a transformation approach based on rotating $$180^{\circ}$$ using the midpoint of $$AC$$ as the center.

Monitor for different approaches to the proof.

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

### Launch

Arrange students in groups of 2. Discuss the conjecture with students briefly before they begin to confirm it by asking how we could re-write the conjecture “the diagonals of a parallelogram bisect each other.” (The intersection of the diagonals is the midpoint of each diagonal. In parallelogram $$ABCD$$ where the diagonals intersect at point $$X$$$$AX = CX$$ and $$BX = DX$$).

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for the proof of the conjecture: The diagonals of a parallelogram bisect each other. Give students time to meet with 2–3 partners to share and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do you know that angles $$ADX$$ and $$CBX$$ are congruent?”, “How do you know that triangles $$AXD$$ and $$CXB$$ are congruent?”, and “How do you know that segment $$B’C’$$ coincides with $$DA$$ after a $$180^\circ$$ rotation using the midpoint of $$AC$$ as the center of rotation?” Invite students to go back and revise or refine their written proofs based on the feedback from peers. This will help students justify their thinking for each step of their proof.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “To prove that the conjecture is always true, I can . . . .”
Supports accessibility for: Language; Conceptual processing

### Student Facing

Conjecture: The diagonals of a parallelogram bisect each other.

1. Use the tools available to convince yourself the conjecture is true.
2. Convince your partner that the conjecture is true for any parallelogram. Can the 2 of you think of different ways to convince each other?
3. What information is needed to prove that the diagonals of a parallelogram bisect each other?
4. Prove that segment $$AC$$ bisects segment $$BD$$, and that segment $$BD$$ bisects segment $$AC$$.

### Anticipated Misconceptions

If students are stuck using triangle congruence, offer these questions:

• To what triangle is triangle $$AXB$$ congruent?
• What do you know about parallelograms? (Encourage students to look at their reference chart.)

If students are stuck using transformations, offer these questions:

• What transformation looks like it would work to take segment $$AX$$ to segment $$CX$$ and segment $$BX$$ to segment $$DX$$?
• Is there a rotation that we know will work? Can we show how that rotation is related to $$X$$?

### Activity Synthesis

Display the proofs of two different groups for all to see. Include one proof that uses triangle congruence and one that does not, if possible.

Ask the class what is the same and what is different. (They both start with the same given information and reach the same conclusion. One uses transformations and the other uses a triangle congruence shortcut.)

If a student wrote a proof without triangle congruence, remind students they proved the triangle congruence shortcuts using transformations so both proofs are based in transformations.

## 13.3: Work Backwards to Prove (10 minutes)

### Activity

Working backwards from what we are trying to show to see how that relates to the given information is a useful skill in geometry. This activity makes that strategy explicit by having partners take turns saying, “I would know that were true if . . . .” This is one of the first cases where students might find and use overlapping triangles in a proof. Students can use triangles $$ADC$$ and $$BCD$$, or they can study the angles in triangles $$CXD$$ and $$DXA$$ and prove that angles $$ADX$$ and $$CDX$$ must be complementary (where $$X$$ is the intersection of $$AC$$ and $$DB$$). This method will require some algebraic manipulation of expressions for the angles.

Monitor for students who are focused on the four non-overlapping triangles formed by the diagonals, and for students who are focused on the two larger overlapping triangles formed by the diagonals.

### Launch

Arrange students in groups of 2.

Conversing: MLR2 Collect and Display. As students work on this activity, listen for the language students use to complete the statement, “I would know _____ if I knew _____.” Capture student language that names specific sides, angles, and triangles and uses terms such as right angles, opposite angles, and congruent. 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 would know the sides are congruent if I knew this was a parallelogram” can be improved by specifying the sides that are congruent and the name of the parallelogram. This will help students use the mathematical language necessary to work backwards from what they are trying to prove to see how that relates to the given information.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Engagement: Develop Effort and Persistence. Provide prompts, reminders, guides, rubrics, and checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, remind students to use the reference chart, to try looking for other triangles, and to redraw the triangles outside of the rectangle.
Supports accessibility for: Attention; Social-emotional skills

### Student Facing

Given: $$ABCD$$ is a parallelogram with $$AB$$ parallel to $$CD$$ and $$AD$$ parallel to $$BC$$. Diagonal $$AC$$ is congruent to diagonal $$BD$$.

Prove: $$ABCD$$ is a rectangle (angles $$A, B, C,$$ and $$D$$ are right angles).

With your partner, you will work backwards from the statement to the proof until you feel confident that you can prove that $$ABCD$$ is a rectangle using only the given information.

Start with this sentence: I would know $$ABCD$$ is a rectangle if I knew $$\underline{\hspace{1in}}$$.
Then take turns saying this sentence: I would know [what my partner just said] if I knew $$\underline{\hspace{1in}}$$.

Write down what you each say. If you get to a statement and get stuck, go back to an earlier statement and try to take a different path.

### Student Facing

#### Are you ready for more?

Two intersecting segments always make a quadrilateral if you connect the endpoints. What has to be true about the intersecting segments in order to make a(n):

1. rectangle
2. rhombus
3. square
4. kite
5. isosceles trapezoid

### Anticipated Misconceptions

If students get stuck using the small triangles suggest they try looking for other triangles.

If students get mixed up using the overlapping triangles suggest they redraw the triangles outside of the rectangle.

Encourage students who are struggling to come up with reasons to use their reference chart to find statements that justify what they are trying to show.

### Activity Synthesis

Select students to share which triangles they used.

Encourage students who share to use different colors or redraw the triangles on a display for the whole class to see. Discuss which triangles were easier to see, and which turned out to be most useful in the proof. Remind students that the easiest triangles to use might be overlapping or hidden in some way.

## Lesson Synthesis

### Lesson Synthesis

Remind students of the term converse. Ask students what the converse of “If a quadrilateral is a parallelogram, then its diagonals bisect each other.” would be. (If a quadrilateral's diagonals bisect each other, then it is a parallelogram.) We have proven both the statement and its converse. Introduce the phrase if and only if and explain that we use it when the statement and its converse are true. Display these statements for all to see:

1. A quadrilateral is a parallelogram if its diagonals bisect each other.
2. A quadrilateral is a parallelogram only if its diagonals bisect each other.

Ask multiple students to put the second statement in their own words. (There can’t be any parallelograms with diagonals that don’t bisect each other, all quadrilaterals have diagonals that bisect each other, if it’s a parallelogram then the diagonals bisect each other).

Use some more everyday examples and ask students which are examples of if and only if statements (the statement and its converse are true) and which are not. For example:

1. A person is a parent if and only if they are a mother. (This is not an if and only if statement. A father can be a parent too. It’s true that if they’re a mother then they’re a parent, but not true that if they’re a parent then they must be a mother.)
2. A person is vegan if and only if they don’t eat animal products. (This is as close to true as things get in the real world, since it’s the definition of being vegan.)

Invite students to come up with their own examples and non-examples. Ask students why it is useful to know a quadrilateral is a parallelogram if and only if its diagonals bisect each other. (Now we can use either part as the given information so it's like adding two theorems to the reference chart.)

## Student Lesson Summary

### Student Facing

A quadrilateral is a parallelogram if and only if its diagonals bisect each other. The “if and only if” language means that both the statement and its converse are true. So we need to prove:

1. If a quadrilateral has diagonals that bisect each other, then it is a parallelogram.
2. If a quadrilateral is a parallelogram, then its diagonals bisect each other.

To prove part 1, make the statement specific: If quadrilateral $$EFGH$$ with diagonals $$EG$$ and $$FH$$ intersecting at $$Y$$ so that $$EY$$ is congruent to $$YG$$ and $$FY$$ is congruent to $$YH$$, then side $$EF$$ is parallel to side $$GH$$ and side $$EH$$ is parallel to side $$FG$$.

We could prove triangles $$EYH$$ and $$GYF$$ are congruent by the Side-Angle-Side Triangle Congruence Theorem. That means that corresponding angles in the triangles are congruent, so angle $$YEH$$ is congruent to $$YGF$$. This means that alternate interior angles formed by lines $$EH$$ and $$FG$$ are congruent, so lines $$EH$$ and $$FG$$ are parallel. We could also make an argument that shows triangles $$EYF$$ and $$GYH$$ are congruent, so that angles $$FEY$$ and $$HGY$$ are congruent, which means that lines $$EF$$ and $$GH$$ must be parallel.

To prove part 2, make the statement specific: If parallelogram $$ABCD$$ has side $$AB$$ parallel to side $$CD$$ and side $$AD$$ parallel to side $$BC$$, and diagonals $$AC$$ and $$BD$$ that intersect at $$X$$, then we are trying to prove that $$X$$ is the midpoint of $$AC$$ and of $$BD$$.

We could use a transformation proof. Rotate parallelogram $$ABCD$$ by $$180^{\circ}$$ using the midpoint of diagonal $$AC$$ as the center of the rotation. Then show that the midpoint of diagonal $$AC$$ is also the midpoint of diagonal $$BD$$. That point must be $$X$$ since it is the only point on both line $$AC$$ and line $$BD$$. So $$X$$ must be the midpoints of both diagonals, meaning the diagonals bisect each other.

We have proved that any quadrilateral with diagonals that bisect each other is a parallelogram, and that any parallelogram has diagonals that bisect each other. Therefore, a quadrilateral is a parallelogram if and only if its diagonals bisect each other.