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Theorems Dealing with
Trapezoids and Kites

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trapguy
def
A trapezoid is a quadrilateral with at least one pair of parallel sides.
The median of a trapezoid is a segment joining the midpoints of the legs of the trapezoid.
(At the right, mn2 is the median for trapezoid ABCD.)
thmed

The theorems will be stated in "if ...then" form. Both the theorem and its converse (where you swap the "if" and "then" expressions) will be examined.

Click PROOFsmall in the charts below to see each proof.
While one method of proof will be shown, other methods are also possible.

Definition and Theorems pertaining to a trapezoid:
DEFINITION: A trapezoid is a quadrilateral with at least one pair of parallel sides.
trap1
THEOREM: The median of a trapezoid is parallel to the bases and half the sum of the lengths of the bases.
trapmedianstuff
PROOFsmall
trap2

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trapisosguy
def
A isosceles trapezoid is a trapezoid with congruent base angles.

Note:
The definition of an isosceles triangle states that the triangle has two congruent "sides".
But the definition of isosceles trapezoid stated above, mentions congruent base "angles", not sides (or legs). Why?
If an "inclusive" isosceles trapezoid is defined to be "a trapezoid with congruent legs", a parallelogram will be an isosceles trapezoid. If this occurs, the other properties that an isosceles trapezoid can possess can no longer hold, since they will not be true for a parallelogram.
If, however, we define an isosceles trapezoid to be a "trapezoid with congruent base angles", the legs can be proven congruent, a parallelogram will NOT be an isosceles trapezoid, and all of the commonly known properties of an isosceles trapezoid will remain true.

Definition and Theorems pertaining to an isosceles trapezoid:
DEFINITION: An isosceles trapezoid is a trapezoid with congruent base angles.
isostrap1
THEOREM: If a quadrilateral (with one set of parallel sides) is an isosceles trapezoid, its legs are congruent.
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isostrap2
THEOREM: (converse) If a trapezoid has congruent legs, it is an isosceles trapezoid.
PROOFsmall
isostrap2
THEOREM: If a quadrilateral is an isosceles trapezoid, the diagonals are congruent.   
PROOFsmall
isostrap3
THEOREM: (converse) If a trapezoid has congruent diagonals, it is an isosceles trapezoid.
PROOFsmall
isostrap3
THEOREM: If a quadrilateral is an isosceles trapezoid, the opposite angles are supplementary.
PROOFsmall
isostrap4
THEOREM: (converse) If a trapezoid has its opposite angles supplementary, it is an isosceles trapezoid.
PROOFsmall
isostrap4

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kiteguy
def
A kite is a quadrilateral whose four sides are drawn such that there are two distinct sets of adjacent, congruent sides.

Note: Kites discussed on this page are convex kites.
concaveVexKites

Definition and Theorems pertaining to a kite:
DEFINITION: A kite is a quadrilateral whose four sides are drawn such that there are two distinct sets of adjacent, congruent sides.
thkite1

THEOREM: If a quadrilateral is a kite, the diagonals are perpendicular.

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thkite5

THEOREM: If a quadrilateral is a kite, it has one pair of opposite angles congruent.

PROOFsmall

thkite2

THEOREM: If a quadrilateral is a kite, it has one diagonal forming two isosceles triangles.

PROOFsmall

thkite6

THEOREM: If a quadrilateral is a kite, it has one diagonal forming two congruent triangles.

PROOFsmall

thkite1

THEOREM: If a quadrilateral is a kite, it has one diagonal that bisects a pair of opposite angles.

PROOFsmall

thkite4

THEOREM: If a quadrilateral is a kite, it has one diagonal that bisects the other diagonal.

PROOFsmall

thkite3

THEOREM: If one of the diagonals of a quadrilateral is the perpendicular bisector of the other, the quadrilateral is a kite.

PROOFsmall

thkite10

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