Lesson 13

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.

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.

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).

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 \(BX = DX\)). 

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 \(X\)?

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.

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.

Arrange students in groups of 2.

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.

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.