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Properties of Special Parallelograms

On this web page you will discover some properties of rhombuses, rectangles, and squares.

Sketch

This sketch shows parallelogram LOVE formed by two, intersecting pairs of parallel lines. Both pairs of parallel lines are the same distance apart. Drag vertex E to change the position and angle of intersection of the lines, and use the red segment to change the distance between the pairs of parallel lines. If you drag a point off the screen, press Start Over to return it.

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Investigate

  1. Use the control points to change the parallelogram. How do the lengths of the four sides compare?
  2. Formulate the Double-Edged Straightedge Conjecture: If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a _____.

Sketch

This sketch shows rhombus RHMB and its diagonals, which intersect at point X. Drag vertices R, H, or M to change the size and shape of the rhombus. If you drag a point off the screen, press Start Over to return it.

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Investigate

  1. Only the measure of angle RXH is given. How can you determine the measures of other angles formed by the intersection of the two diagonals?
  2. Based on the measure of angle RXH, what is true about the diagonals? Change the rhombus to see whether this is always true.
  3. You already know that the diagonals of a parallelogram bisect each other. Do the diagonals of this rhombus bisect each other? Why?
  4. Formulate the Rhombus Diagonals Conjecture: The diagonals of a rhombus are _____, and they _____.

    The diagonals and the sides of the rhombus form two angles at each vertex. Press Show Angles At Vertices to see the measures of these angles.

  5. How does each pair of angles compare? Change the size and shape of the rhombus to see whether this is always true.
  6. Formulate the Rhombus Angles Conjecture: The _____ of a rhombus _____ the angles of the rhombus.

Sketch

This sketch shows rectangle RECT and its diagonals. Drag vertices R, E, or C to change the size and shape of the rectangle. Press Start Over if you drag a vertex too far.

Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled. Note: Java-compatible browsers are Internet Explorer and Opera.

Investigate

  1. Compare the lengths of diagonals RC and ET. What do you notice? Change the rectangle to see whether this is always true.
  2. Recall that a rectangle is also a parallelogram, so its diagonals also have the properties of a parallelogram's diagonals. So, what else can you say about the diagonals of this rectangle?
  3. Formulate the Rectangle Diagonals Conjecture: The diagonals of a rectangle are _____, and _____.
  4. A square is a parallelogram, as well as both a rectangle and a rhombus. Use what you know about the properties of these three quadrilaterals to formulate the Square Diagonals Conjecture: The diagonals of a square are _____, _____, and _____.