Special Parallelograms
Lesson 6-4A : Rhombuses
Warm-Up
Find the value of x in the accompanying diagram.
The Definition of a Rhombus
A rhombus is a parallelogram that has two consecutive congruent sides.
For example, in rhombus ABCD, AB = BC.
The Special Properties of the Rhombus
1) All sides of a rhombus are congruent.
Since a rhombus is a parallelogram, opposite sides are also congruent.
With consecutive and opposite sides congruent, all four sides are.
2) The diagonals of a rhombus are perpendicular to each other.
The diagonals of a rhombus form right angles with each other.
3) The diagonals of a rhombus bisect its angles.
Each diagonal divides the opposite angles into two congruent angles.
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Model Problems
1) WXTV is a rhombus. Find:
a) XT
b)
2)
3) Find the measure of each numbered angle in the rhombus.
Exercise
1) ABCD is a rhombus. Find:
a) AB
b) m <ABC
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2) The diagonals of a rhombus are 9 inches and 12 inches. Find:
a) the length of a side of the rhombus
b) the perimeter of the rhombus
3) Find the measure of each numbered angle in the rhombus.
Definition of a Square
A square is a rectangle that has two consecutive congruent sides.
In other words…
A square is a rectangle that is also a rhombus.
Special Properties of a Square
From the rhombus: 1. A square is equilateral.
2. A square has perpendicular diagonals.
3. The diagonals of a square bisect its angles.
From the rectangle: 1. A square has four right angles
2. A square has congruent diagonals.
The square also has all the properties of the parallelogram.
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Parallelogram Family Summary
Parallelogram Family
Opposite sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
Rectangles also have…
Four right angles
Congruent diagonals
Rhombuses also have…
Congruent sides
Perpendicular diagonals
Diagonals bisect the angles
Squares have everything!
Model Problem
Graph the following quadrilateral ABCD.
1) Determine whether it is a parallelogram.
2) If yes, is it also a rectangle?
3) Is it also a rhombus?
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Exercise
Exit Ticket
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Homework
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Proving that a Quadrilateral is a Rhombus
There are three ways to prove that a quadrilateral is a rhombus:
1) If a parallelogram has two consecutive congruent sides, then it is a rhombus.
Ex. If WXYZ is a parallelogram and WX = XY, then WXYZ is also a rhombus.
2) If a parallelogram has perpendicular diagonals, then it is a rhombus.
Ex. If WXYZ is a parallelogram and
, then WXYZ is also a rhombus.
3) If a quadrilateral is equilateral, then it is a rhombus.
Ex. If WX = XY = YZ = ZW, then WXYZ is a rhombus.
You do not have to show it is also a parallelogram.
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4) If a parallelogram has a diagonal that bisects the angles it touches, then it is a
rhombus.
Ex. If the figure is a parallelogram and the diagonal shown divides bisects the pair of
opposite angles, then it is a rhombus.
Model Problems
Determine if the given conclusion is valid. If it is NOT valid, tell what additional
information is needed to prove the conclusion.
1)
Describe what is being given:
1)
2)
3)
Is this enough to prove a rhombus? _________________________
If not, what else is needed? _________________________________________________
2) Given: The diagram at right.
Conclusion: The figure is a rhombus.
Describe what is being given:
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Is this enough to prove a rhombus? _________________________
If not, what else is needed? _________________________________________________
Exercise
Determine if the given conclusion is valid. If it is NOT valid, tell what additional
information is needed to prove the conclusion.
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