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Math Investigations
Quadrilaterals
Name
Trapezoids
Trapezoids have on pair of parallel sides; these are called the bases.
The median is the line segment which joins the two non parallel sides. It is parallel to the bases and has
length equal to the average of the lengths of the bases.
b1
median
b2
Why is this true? Hint: Extend the nonparallel sides of the trapezoid until they intersect.
Parallelograms
A quadrilateral is a parallelogram if its opposite sides are parallel.
The following properties are true of parallelograms:
1. Opposite sides are congruent if and only if the quadrilateral is a parallelogram.
2. Opposite angles are congruent if and only if the quadrilateral is a parallelogram.
3. The diagonals bisect each other if and only if the quadrilateral is a parallelogram.
Try to figure out why each of the statements above is true below:
L Marizza A Bailey
Art of Problem Solving: Geometry
BASIS Scottsdale
Math Investigations
Quadrilaterals
Name
Rhombus
A rhombus is a parallelogram with all sides congruent to each other. Since it is already a parallelogram,
then it inherits all the properties of parallelograms.
Theorem: The diagonals of a rhombus are perpendicular to each other.
1. Label all the vertices of the rhombus above, and label the intersection of the diagonals.
2. Find congruent triangles. Don't forget to name the congruency theorems.
3. What is true about all four angles created by the intersection of the diagonals?
* The area of a rhombus is half the product of its diagonals.
Rectangles
A rectangle is a parallelogram with four right angles.
Lemma: The diagonals of a rectangle are congruent.
L Marizza A Bailey
Art of Problem Solving: Geometry
BASIS Scottsdale
Math Investigations
Quadrilaterals
Name
Exercises:
1. Prove that if you connect the midpoints of all the sides of a rhombus, you get a rectangle.
2. Prove that the sum of the squares of the diagonals of a parallelogram equals the sum of the squares of
the sides of the parallelogram.
Challenge Problems from Art of Problem Solving.
L Marizza A Bailey
Art of Problem Solving: Geometry
BASIS Scottsdale