Proving that a Quadrilateral Is a Parallelogram

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Proving that a Quadrilateral Is a Parallelogram
GEOMETRY LESSON 6-3
Find values of x and y for which ABCD must be a
parallelogram.
If the diagonals of quadrilateral ABCD bisect
each other, then ABCD is a parallelogram by
Theorem 6-5. Write and solve two equations to
find values of x and y for which the diagonals
bisect each other.
If x 18 and y 89, then ABCD is a
parallelogram.
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Proving that a Quadrilateral Is a Parallelogram
GEOMETRY LESSON 6-3
Determine whether the quadrilateral is a
parallelogram. Explain.
a. All you know about the quadrilateral is that
only one pair of opposite sides is congruent.
a.
Therefore, you cannot conclude that the
quadrilateral is a parallelogram.
b.
b. The sum of the measures of the angles of a
polygon is (n 2)180, where n represents the
number of sides, so the sum of the measures of
the angles of a quadrilateral is (4 2)180
360.
If x represents the measure of the unmarked
angle, x 75 105 75 360, so x 105.
Theorem 6-8 states If both pairs of opposite
angles of a quadrilateral are congruent, then the
quadrilateral is a parallelogram. Because both
pairs of opposite angles are congruent, the
quadrilateral is a parallelogram by Theorem 6-8.
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Proving that a Quadrilateral Is a Parallelogram
GEOMETRY LESSON 6-3
The captain of a fishing boat plots a course
toward a school of bluefish. One side of a
parallel rule connects the boat with the school
of bluefish. The other side makes a 36 angle
north of due east on the charts compass. Explain
how the captain knows in which direction to sail
to reach the bluefish.
Because both sections of the rulers and the
crossbars are congruent, the rulers and crossbars
form a parallelogram.
Therefore, the angle shown on the charts compass
is congruent to the angle the boat should travel,
which is 36 north of due east.
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Proving that a Quadrilateral Is a Parallelogram
GEOMETRY LESSON 6-3
Find the values of the variables for which GHIJ
must be a parallelogram.
1.
2.
a 34, b 26
x 6, y 0.75
Determine whether the quadrilateral must be a
parallelogram. Explain.
3.
4.
5.
No both pairs of opposite sides are not
necessarily congruent.
Yes the diagonals bisect each other.
Yes one pair of opposite sides is both
congruent and parallel.