In this lesson, students prove two statements about the diagonals of parallelograms.
- The diagonals of a parallelogram bisect each other.
- If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
Students learn a new strategy for looking for structure (MP7) by working backwards from the statement they are trying to prove to the given statements. Students also encounter a situation where they could use overlapping triangles, which can be challenging. Students learn techniques for redrawing or marking diagrams to help them see more subtle triangles which might be used in congruence proofs. As students prove theorems about parallelograms, they are explicitly practicing proof techniques and looking for structure.
One of the activities in this lesson works best when each student has access to internet-enabled devices because students will benefit from seeing the relationship in a dynamic way.
- Justify (orally) and prove (in writing) that the diagonals of a parallelogram bisect each other.
- Prove (orally and in writing) that if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
- Let’s prove theorems about parallelograms.
Acquire internet-enabled devices that can run the applet in Notice and Wonder: Diagonals, one for every 2-3 students. If technology is not available there is a paper and pencil alternative, but consider displaying the applet for all to see.
- I can prove theorems about the diagonals of a parallelogram.
A quadrilateral with four right angles.
A quadrilateral with four congruent sides.