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# Theorems about Diagonals of Parallelograms

Lesson 8 of 10

## Objective: SWBAT Write proofs involving theorems about the diagonals of a parallelogram.

#### Do Now

*7 min*

Today's Do Now is once again a review of prior concepts. Students are asked to define the following terms in their own words using a diagram:

- parallel lines
- parallelogram
- perpendicular lines
- rectangles

I project the Do Now on the Smartboard and have the students write the definitions in their notebooks. After about five minutes, we go over the definitions. Even though my students have seen these terms several times, I find it is important to keep reviewing them as we prepare to use write proofs where they are involved.

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#### Mini-Lesson

*10 min*

In today's Mini-Lesson, we will discuss what it means when segments to bisect each other. I find that many of my students assume that if two segments bisect each other, the two segments themselves have to be congruent. I try to address this misconception in today's demonstration. I draw an example that shows two segments bisecting each other, that are obviously not congruent segments.

We then review the properties of a parallelogram that we have proved in the past few lessons:

- The opposite sides of a rectangle are parallel
- The opposite sides of a rectangle are congruent

I list the properties on the board as the students identify them. Students will use these to prove theorems about the diagonals of a parallelogram.

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#### Activity

*23 min*

I start today's activity by displaying given statements and theorems for my students to 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.

Since this is one of the final lessons in the proof unit, I have the students work entirely on their own to prove the diagonals of a parallelogram bisect each other and the diagonals of a rectangle are congruent. They use their knowledge of the properties of parallelograms, congruent triangles, and corresponding parts of congruent triangles to write formal two-column proofs.

At the end of the activity section, I have the students discuss their proofs in pairs and then we go over them as a class (MP3).

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#### Summary

*5 min*

As a summary, I ask my students draw a picture of a rectangle and write down all of the properties they know about rectangles in their notebooks. Since they have graph paper notebooks, it is easy for the students to draw the rectangles precisely (MP6).

We then have a whole class discussion where I question the students about the properties of and theorems about parallelograms in general and specific parallelograms like rectangles and rhombuses. This helps me to assess how well the students know the properties.

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- UNIT 1: Preparing for the Geometry Course
- UNIT 2: Geometric Constructions
- UNIT 3: Transformational Geometry
- UNIT 4: Rigid Motions
- UNIT 5: Fall Interim Assessment: Geometry Intro, Constructions and Rigid Motion
- UNIT 6: Introduction to Geometric Proofs
- UNIT 7: Proofs about Triangles
- UNIT 8: Common Core Geometry Midcourse Assessment
- UNIT 9: Proofs about Parallelograms
- UNIT 10: Similarity in Triangles
- UNIT 11: Geometric Trigonometry
- UNIT 12: The Third Dimension
- UNIT 13: Geometric Modeling
- UNIT 14: Final Assessment

- LESSON 1: Quadrilateral Tangram Investigations
- LESSON 2: Quadrilateral Investigations using Geometer's Sketchpad
- LESSON 3: Sorting Quadrilaterals
- LESSON 4: Attributes of Quadrilaterals
- LESSON 5: Origami Cranes and Geometric Definitions
- LESSON 6: Theorems about Sides of Parallelograms
- LESSON 7: Theorems about Angles of Parallelograms
- LESSON 8: Theorems about Diagonals of Parallelograms
- LESSON 9: Proofs about Parallelogram Assessment Review
- LESSON 10: Proofs about Parallelogram Summative Assessment