Download 7.3 Trapezoids

Trapezoids20052006.nb
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7.3 Trapezoids
A quadrilateral with exactly one pair of parallel sides is called a trapezoid. The parallel sides
are called the bases and the other sides are called the legs. Look at the the figures below.
base
120°
110°
leg
leg
60°
70°
base
80°
leg
base
100°
base
90°
90°
leg
Notice that we can prove that exactly one pair of opposite sides of the quadrilateral above are
parallel. We have two lines cut by a transversal so that same side interior angles are supplementary. The two trapezoids are shown with different orientations so that we don't automatically
think of the "top" and "bottom" sides as the bases. The bases are the two parallel sides.
A trapezoid with congruent legs is called an isosceles trapezoid. If you cut out an
isosceles trapezoid from a piece of paper and fold it over so that the legs coincide, you will
find the both pair of base angles are congruent.
Trapezoids20052006.nb
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Isosceles trapezoid
This gives us the next theorem.
Theorem: The base angles of an isosceles trapezoid are congruent.
The plan for a proof is shown below. Can you finish the proof?
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Given: Trapezoid ABCD with CD »» AB.
Prove: —A @ —B and —C @ —D
D
A
C
P
Isosceles trapezoid
Q
B
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Plan for proof: Draw DPand CQ so that they are both perpendicular to AB. This will make
DP = CQ (If two lines are parallel, then all points on one line are equidistant from the other
line). Now you can show that DAPD @ DBQC using the HL Theorem. So —A @ —B by
CPCTC. Showing that —D @ —C would require use of the angle addition postulate and substitution.
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The median of a trapezoid is the segment that joins the midpoints of the legs. This is
not the same usage as when we say the median of a triangle. Note the difference so that you
do not make any mistakes. A median of a triangle is the segment that joins a vertex with the
midpoint of the side opposite the vertex. The median of a trapezoid is very similar to the midsegment of a triangle in that both join the midpoints of sides. In fact, the following theroem
shows that the median of a trapezoid "acts" very much like the midsegment of a triangle.
Theorem: The median of a trapezoid
(1) is parallel to the bases;
(2) has a length equal to the average of the base lengths (i.e. 1/2 the sum of the lengths of
the bases)
Example 1: The diagram below shows a trapezoid and its median. Find the value of x.
2x+1
2x+4
3x+2
Solution: Since the length of the median is the average of the length of the bases we have;
2x + 4 = ÅÅÅÅ12 H2 x + 1 + 3 x + 2L
2x + 4 = ÅÅÅÅ12 H5 x + 3L
4x + 8 = 5x + 3
5=x
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Example 2: Two congruent angles of an isosceles trapezoid have measures 3x + 10 and 5x 10. Find the value of x and then give the measures of all angles of the trapezoid.
Solution: Since the two angles are congruent their measures are equal, thus;
3x + 10 = 5x - 10
20 = 2x
10 = x
So the two angles have measure 3(10)+10 = 5(10)-10 = 40. The other two angles are congruent as well and are supplementary to the 40° angles so they must both be 140.
What do you think is the relationship between the diagonals of an isosceles triangle? If you
said they are congruent you would be correct. Look at the figure below.
D
A
C
B
We can show that DABD @ DBAC using the SAS Postulate (—A @ —B because they are base
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angles of an isosceles trapezoid, AD @ BC, and AB @ AB). So DB @ CA by CPCTC.
In summary, a trapezoid is a special quadrilateral in which exactly one pair of opposite sides
is parallel. This, of course, means that a trapezoid is never a parallelogram since a parallelogram must have both pairs of opposite sides parallel. An isosclele trapezoid is a trapezoid in
which the legs are congruent ( which in turn means that the base angle and the diagonals are
also congruent.).