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Exploring Parallelograms

ForfatterKathryn Bragwell, Tim Brzezinski, K.O. Ray
In this investigation you will discover some special properties of parallelograms. A parallelogram is a quadrilateral whose opposite sides are parallel.

Sides and Angles of a Parallelogram

Use the interactive image above to answer the following questions.

Sides of a Parallelogram

Click the box next to Opposite Sides 1. What do you notice about the lengths of and (denoted by AC and BD)?

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  • A
  • B
  • C
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Click the box next to Opposite Sides 2. What do you notice about AB and CD?

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  • A
  • B
  • C
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Now move around vertices A, C, and D. Does the previous observations remain true?

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Uncheck the opposite sides boxes by clicking on them again. This will hide the lengths.

Angles of a Parallelogram

Click on the box next to Opposite Angles 1. What do you notes about the measures of and (denoted by and )?

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  • A
  • B
  • C
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Click on the box next to Opposite Angles 2. What do you notice about and ?

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  • A
  • B
  • C
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Now move around vertices A, C, and D. Does the previous observations remain true?

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Properties of Parallelograms.

The previous observations display two properties of parallelograms. What do you think these two properties are?

Diagonals of a Parallelogram

Diagonals

Now we are going to look at the diagonals of a parallelogram. These are the segment and . We will do this using the interactive image above.

Click on the boxes next to Diagonal BC, Diagonal AD, and Show Length. Which of the following seems to be true?

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  • B
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Now move around vertices A, C, and D. Does the previous observations remain true?

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This displays another property of parallelogram. What do you think this property is?

Uncheck the box next to show length.

Diagonal Intersection

Now we will start looking at the angle at which the diagonals intersect. This will lead us to a special kind of parallelogram called a rhombus. Using the same interactive image. Check the box next to diagonal intersection.

Use either vertex A, C, or D to make between and . Look at side lengths AB and BD. Are they equal?

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Use either vertex A, C, or D to make between and . Look at side lengths AB and BD. Are they equal.

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Remember these observations about diagonal intersection as we move on to look at rhombuses.

A Rhombus

Rhombuses

We are going to use this last interactive image to explore a special kind of parallelogram, rhombuses.

Click on the box next to Show Side Lengths. See if you can figure out what the definition of a rhombus is. A parallelogram where all four sides are _____________.

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Now click on the box next to Show Diagonals. What kind of angle is ?

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  • B
  • C
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What does that mean about and ? (Please answer this question in a full sentence.)

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Remember our observation about the intersection of the diagonals in the previous interactive image. Finish this sentence. The diagonals in a parallelogram are perpendicular if and only if it is a ________________.

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