Theorems about Angles of Parallelograms

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Objective

SWBAT write proofs involving theorems about angles of parallelograms.

Big Idea

Students use parallel lines, linear pairs and supplementary angles to prove theorems about the angles of parallelograms.

Do Now

7 minutes

Students will need to use the properties of linear pairs of angles later in the lesson, so I use a Do Now to begin today's lesson that activates students' prior knowledge. Students are asked to define the term linear pair of angles and use the definition to find the value of variables in angle measures. I project the Do Now on the board and have the students answer the questions in their notebooks. After about five minutes, I have a class share out to go over the answers.

Mini-Lesson

5 minutes

During today's Mini-Lesson, I will clarify student misunderstandings about linear pairs and supplementary angles, making sure that all students are using precise language. I do this while we review the relationships between angles formed by parallel lines intersected by transversals.

As I present on the board, I have my students draw a sketch showing two parallel lines intersected by a transversal in their notebook. I ask them to identify linear pairs of angles that are therefore supplmentary. I also ask them to label and make note of pairs of angles that are supplementary, but not are not linear pairs.

Later in the lesson, students will use these relationships to prove theorems about the angles of a parallelogram. 

 

Activity

28 minutes

For today's main activity, students will complete a pair of proofs. I begin by projecting the given statements and the theorems students will prove on the board.

  • Opposite angles of a parallelogram are congruent
  • Adjacent angles of a parallelogram are supplementary

I expect most of my students will be 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.

Before the students write formal two-column proofs, I have them brainstorm their procedures in pairs. I also encourage students to number the angles instead of just using letters to identify angles. Using numbers to speak about the diagrams usually improves the clarity of the students' conversations.

As my students are talking, I circulate and listen to their discussions. I expect that some students will have difficulty seeing the relationships between the angles because there are two sets of parallel lines. For these students, I suggest choosing an angle measurement and using it to fill in the rest of the angles. They then translate the numbers into a description about the relationships.

After about five minutes of brainstorming, I will encourage the students to begin to work independently to write formal proofs. At the end of the activity, I call on students to present their proofs. If students have different results, I call on more students to explain their proofs. 

Summary

5 minutes

To bring this lesson to closure, I ask my students to draw and label a parallelogram in their notebooks. Then, I ask them to summarize the properties we have discussed and proven in the past two lessons. This brief activity helps reinforce their knowledge of the properties and helps me to assess how much they remember about the properties.