Fundamental Ideas
Polygons
Unlike humans, all quadrilaterals are not created equal. It's not a matter of size I'm alluding to here, but rather a question of features. They may have a pair of parallel sides, two pairs, a right angle ….
Trapezoid
A
trapezoid is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called
bases, and the
nonparallel sides are called
legs. A segment that joins the midpoints of the legs is called the
median of the trapezoid. Any segment that is perpendicular to both bases is called an
altitude
of the trapezoid (Figure
1 ). The length of an altitude is called the
height of the trapezoid.
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AB and CD are bases.
XY is an altitude.
MN is the median.
XY, length of segment XY is the height.
Parallelogram
A
parallelogram is any quadrilateral with both pairs of opposite sides parallel. Each pair of parallel sides is called a pair of
bases of the parallelogram. Any perpendicular segment between a pair of bases is called the an
altitude of the parallelogram. The length of an altitude is the height of the parallelogram. The symbol
is used for the word parallelogram. Figure
2 shows that a parallelogram has two sets of bases and that, with each set of bases, there is an associated height.
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In
ABCD,
XY is an altitude to bases AB and CD
JK is an altitude to bases AD and BC
XY is the height of
ABCD, with
AB
and
CD
as bases, JK is the height of
ABCD, with
AD
and
BC
as bases.
The following are theorems regarding parallelograms:
Theorem 41: A diagonal of a parallelogram divides it into two congruent triangles.
In
ABCD with diagonal
BD
according to
Theorem 41, ΔABD ≅ Δ CDB (Figure
3 ).
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Theorem 42: Opposite sides of a parallelogram are congruent.
Theorem 43: Opposite angles of a parallelogram are congruent.
Theorem 44: Consecutive angles of a parallelogram are supplementary.
In
ABCD (Figure
4 )
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By Theorem 42, AB = DC and AD = BC.
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By Theorem 43, m ∠ A = m ∠ C and m ∠ B = m ∠ D.
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By Theorem 44:
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∠ A and ∠ B are supplementary.
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∠ B and ∠ C are supplementary.
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∠ C and ∠ D are supplementary.
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∠ A and ∠D are supplementary.
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Theorem 45: The diagonals of a parallelogram bisect each other.
In
ABCD (Figure
5 ), by
Theorem 45, AE = EC and
BE = ED.
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