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Summary Sheet on
Special Quadrilaterals that are not Parallelograms
Transversal Proportionality Theorem (Theorem 6.7, p. 385): If three or
more parallel lines create segments of equal length on one transversal, then they
create segments of equal length on each transversal; however, segments across two
or more transversals are not necessarily congruent. (Indeed, most of the time they
are not congruent.)
TRAPEZOIDS
Definition: A quadrilateral is a trapezoid if and
only if it has exactly one pair of parallel sides
and one pair of non-parallel sides. (p. 410)
Those parallel sides are called bases, and the non-parallel sides are called legs.
The legs and the bases form two pair of base angles—one pair on each base.
The base angles formed on the same leg are always supplementary because they
are same side interior angles formed by the leg as a transversal across the two
parallel bases.
Definition: The two legs of a trapezoid are congruent
if and only if the trapezoid is an isosceles trapezoid.
Isosceles Trapezoid Base Angles Theorem (Theorem 6.19):
The two base angles formed on the same base of a trapezoid are congruent
if and only if the trapezoid is isosceles. Converse is Theorem 6.15
Isosceles Trapezoid Diagonals Theorem (Theorem 6.20): The diagonals
of a trapezoid are congruent if and only if the trapezoid is isosceles.
Isosceles Trapezoid Opposite Angles Theorem: Both pair of opposite angles
of a trapezoid are supplementary if and only if the trapezoid is isosceles.
Definition: The midsegment of a trapezoid (sometimes known as the
median of the trapezoid) is the line that connects the midpoints of the two legs.
Trapezoid Midsegment Theorem (Theorem 6.21):
This segment is parallel to both of the bases and
measures half the sum of the bases, i.e., its length
is the average of the lengths of the bases. (p. 412)
Definition: A midsegment of a triangle is a line segment that
connects the midpoints of two legs. The third side of the triangle
would then be considered the base, relative to the midsegment
and the intersected legs. (p. 301)
Triangle Midsegment Theorem (Theorem 5.1):
The midsegment of a triangle is parallel to the base
and measures half the length of the base. (p. 301)
KITES
Definition:
A kite is a quadrilateral with two pair of adjacent congruent sides.
Theorem 6.18: The diagonals of a kite are perpendicular.
The kite then is a semi-rhombus, with some of the
same distinguishing characteristics, but not all since
the kite is not a parallelogram. Thus, the kite will
partly share some of the qualities of a rhombus.
Characteristics of a Kite
Two pair of adjacent sides are congruent.
The diagonals are perpendicular Theorem 6.22 (p. 415)
The short diagonal divides the kite into two isosceles triangles
The long diagonal divides the kite into two congruent triangles.
The long diagonal bisects the short diagonal, but
the short diagonal does not bisect the long diagonal.
 The long diagonal bisects the pair of opposite angles
whose vertices are connected by the long diagonal.
 The opposite angles whose vertices are connected
by the short diagonal are congruent but generally
not bisected by the short diagonal.
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CONGRUENT QUADRILATERALS
Definition: Two quadrilaterals are congruent if and only if all of their
corresponding parts are congruent. (CPCQC)
Minimum Requirements:
SASAS Theorem: Two quadrilaterals are congruent if any three sides and the
included angles of one quadrilateral are congruent to the corresponding
three sides and included angles of a second quadrilateral.
ASASA Theorem: Two quadrilaterals are congruent if any three angles and the
included sides of one quadrilateral are congruent to the corresponding
three angles and included sides of a second quadrilateral.