The following are special types of parallelograms, with specific properties about their sides, angles, and/or diagonals that help identify them:

These are three examples of rectangles:

These are two examples of rhombi:

Explore the applet by dragging the vertices of the polygons.

Which polygon(s) are always rectangles? Can you create a rectangle with any of the polygons?

How do you know which polygon(s) form a rhombus versus a square?

Which polygon(s) are always parallelograms? Can you create a parallelogram with any of the polygons?

The following theorems relate to the special parallelograms:

Squares have the same properties as both a rectangle and rhombus.

Note that these theorems are for parallelograms, so if we are only told that a polygon is a quadrilateral, then they may not meet the conditions stated.

List all classifications of quadrilaterals that apply to the figures. Explain your reasoning.

a

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b

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c

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d

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Consider the diagram that illustrates the rhombus diagonals theorem: If a parallelogram is a rhombus, then its diagonals are perpendicular bisectors of one another.

a

Add reasoning to each step of the diagram.

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b

Write a formal proof of the theorem.

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Prove that in the given rectangle ABCD, \overline{AC}\cong \overline{BD}:

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Given:

- ABCD is a rhombus
- m\angle ACD=(5x+8)\degree
- m\angle BCD=(12x+2)\degree

Complete the following:

a

Solve for x.

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b

Solve for m\angle ABC.

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If \overline{AB} \cong \overline{CD}, show that ABCD is not a rhombus.

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Use geometric constructions and properties of rhombuses to verify that parallelogram WXYZ is a rhombus.

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Idea summary

Use the theorems relating to the special parallelogrmas to solve problems:

**Rectangle diagonals theorem**: A parallelogram is a rectangle if and only if its diagonals are congruent.**Rhombus diagonals theorem**: A parallelogram is a rhombus if and only if its diagonals are perpendicular.**Rhombus opposite angles theorem**: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.