Theorems about Sides of Parallelograms
Lesson 6 of 10
Objective: SWBAT write proofs involving theorems about the sides of parallelograms.
Today's Do Now is designed to help students review prior concepts. Students are asked to find the measure of angles formed by two parallel lines intersected by a transversal. Then, they identify the relationships between pairs of angles.
I begin class by projecting the Do Now on the Smartboard. I ask students to draw the diagram, and, to answer the questions in their notebooks. After about five minutes we will go over the answers. Students will use the relationships discussed at the end of the Do Now to write proofs later in the lesson.
For today's Mini-Lesson, I plan to review the definition of a parallelogram and discuss how the definition links to the Do Now. We then discuss the fact that the “Opposite sides of a parallelogram are congruent (G.CO.11)." I ask my students to write this theorem in their notebooks and draw and label a parallelogram showing this theorem. In the upcoming activity, students will write a proof of this theorem.
For the activity, I project the given statements and the theorems students will prove. Most students are able to draw the diagrams and write the proofs in their notebooks. However, I print out the presentation for some students who may have difficulty seeing the presentation or taking notes from the board.
In addition to proving the opposite sides of a parallelogram are congruent, students are given a second parallelogram with a diagonal that bisects opposite angles and are asked to prove that all four sides of the parallelogram are congruent. I have the students work in pairs to write a formal two-column proof for each problem. At this point in the year, students have written several two-column proofs and should be familiar with the format. If they are not, I instruct them to look back in their notes and use ideas from previous proofs to help with these proofs.
After about 12 minutes, the pairs share their proofs with another pair and compare their results. If both pairs have different results, students critique the other pair’s proofs and decide which of their proofs is correct. Sometimes the pairs have different proofs, but both proofs are correct. The students discuss how the proofs differ and how they are both correct. If both pairs have written their proofs in the same way, they have to justify how they know they are both correct.
After the students have discussed their proofs in groups, I call on two different groups to present their proofs to the class. I then ask the students to write an alternative “prove statement” for the second proof. Based on prior knowledge, they should see that they have proved the second parallelogram is a rhombus and can write down this definition in their notebooks.
As a closing reflection I plan to ask my students to write a paragraph explaining how the definition of a parallelogram can be used to prove the sides of a parallelogram are congruent. Since we have explored the properties of angles formed by parallel lines intersected by a transversal several times, this writing activity helps me to assess how much students understand how to apply their knowledge.
After about five minutes, I call on a few students to read their paragraphs. The other students can add any relevant information they might not have written to their paragraphs. I ask them to skip a line so I can see which is their original information, and which is the added information.