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6.5 Trapezoids & Kites
Advanced Geometry
TRAPEZOID
A trapezoid is a quadrilateral with exactly one
pair of parallel sides.
Parts of a Trapezoid
• The parallel sides are the bases
• A trapezoid has two pairs of base angles
• The nonparallel sides are the legs of the
trapezoid
ISOSCELES TRAPEZOID
If the legs of a trapezoid are congruent, then
the trapezoid is an isosceles trapezoid.
THEOREM 6.14
If a trapezoid is isosceles, then each pair of
base angles is congruent.
THEOREM 6.15
If a trapezoid has a pair of congruent base
angles, then it is an isosceles trapezoid.
THEOREM 6.16
A trapezoid is isosceles if and only if its
diagonals are congruent.
Example
CDEF is an isosceles trapezoid with CE = 10
and measure of angle E = 95 degrees. Find DF,
measure of angle C, D, and F.
How?
If we were given the graph of a quadrilateral,
how would we prove that it is a trapezoid?
How would we prove that it is an isosceles
trapezoid?
Example
The vertices of WXYZ are W(-1,2), X(3,0),
Y(4,-3), and Z(-4,1). Show that WXYZ is an
isosceles trapezoid.
MIDSEGMENT OF A TRAPEZOID
The midsegment of a trapezoid is the segment
that connects the midpoints of its legs.
MIDSEGMENT THEOREM FOR TRAPEZOIDS
The midsegment of a trapezoid is parallel to
each base and its length is one half the sum of
the lengths of the bases.
Example
TU is the midsegment of trapezoid QRSP. Find
the length of TU.
Example
MN is the midsegment of trapezoid BCDA.
Find the value of x.
KITE
A kite is a quadrilateral that has two pairs of
consecutive congruent sides, but opposite
sides are not congruent.
THEOREM 6.18
If a quadrilateral is a kite, then its diagonals
are perpendicular.
THEOREM 6.19
If a quadrilateral is a kite, then exactly one
pair of opposite angles are congruent.