Geometry Accelerated
Chapter 6 Summary
A.M.D.G.
6-2
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6-3
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6-4
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Properties of Parallelograms
A parallelogram is a quadrilateral whose opposite sides are parallel. The following
theorems apply to all parallelograms:
o If a quadrilateral is a parallelogram, then its opposite sides are congruent
o If a quadrilateral is a parallelogram, then its opposite angles are congruent
o If a quadrilateral is a parallelogram, then its consecutive angles are supplementary
o If a quadrilateral is a parallelogram, then its diagonals bisect each other
Conditions for Parallelograms
In order to prove that a quadrilateral is a parallelogram, you could prove both sets of
sides are parallel – since this is the definition of a parallelogram, it would definitely prove
that you have one. In addition, you could prove parallelograms any of the following
ways:
o If its opposite sides are congruent, then a quadrilateral is a parallelogram
o If its opposite angles are congruent, then a quadrilateral is a parallelogram
o If its consecutive angles are supplementary, then a quadrilateral is a parallelogram
o If the diagonals of a quadrilateral bisect each other, then it is a parallelogram
o If one pair of opposite sides are parallel and congruent, then a quadrilateral is a
parallelogram
Note that these are the converses of the above statements (with the exception of the last
one).
Page 413 has a good summary for recognizing what is a parallelogram
Special Parallelograms
A rectangle is a parallelogram with 4 right angles
o The diagonals are congruent
A rhombus is a parallelogram with 4 congruent sides
o The diagonals are perpendicular
o The diagonals bisect opposite angles
A square is a parallelogram with 4 right angles and 4 congruent sides
o It is both a rectangle and a rhombus, so it has all the properties of both (see above)
6-5
Conditions for Special Parallelograms
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All of these are on pages 430 to 431, in section 6-5 of the book. You should find them all
and write them below.
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Using these conditions, you can prove whether something is a rectangle, rhombus, or
square. Often, we will prove these with coordinate proofs.
6-5
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Properties of Kites and Trapezoids
Kites and trapezoids are quadrilaterals that are not parallelograms, but they still have
some unusual properties.
A kite is a quadrilateral that has exactly two pairs of adjacent and congruent sides (see
section 6-6 for an illustration)
o In a kite, the diagonals are perpendicular.
o In a kite, exactly one pair of opposite angles is congruent.
o In a kite, the diagonal connecting the congruent angles is always bisected by the
other diagonal.
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
o The parallel sides are called bases and the non-parallel sides are called legs
o Base angles are the consecutive angles which have a base as a common side.
o An isosceles trapezoid has legs that are congruent
 Similar to an isosceles triangle, its base angles are also congruent
The midsegment of a trapezoid connects the midpoints of the legs of the trapezoid
o It is parallel to the bases
o Its length is equal to the average of the lengths of the bases