Tests for Parallelograms
Objectives

Recognize the conditions that ensure a
quadrilateral is a parallelogram.

Prove that a set of points forms a
parallelogram in the coordinate plane.
Conditions for a Parallelogram

Obviously, if the opposite sides of a
quadrilateral are parallel, then it is a
parallelogram; but there are other tests
we can also apply to a quadrilateral to
test whether it is a parallelogram or not.
Conditions for a
Theorems

Theorem 6.9 – If both pairs of opposite sides
are ≅, then the quad. is a
.

Theorem 6.10 – If both pairs of opposite s are
≅, then the quad. is a
.

Theorem 6.11 – If diagonals bisect each other,
then the quad. is
.

Theorem 6.12 – If one pair of opposite sides is
║ and ≅, then the quad. is a
.
Example 1:
Write a paragraph proof of the statement: If a diagonal
of a quadrilateral divides the quadrilateral into two
congruent triangles, then the quadrilateral is a
parallelogram.
Given:
Prove: ABCD is a parallelogram.
Proof:
CPCTC. By Theorem 8.9, if both pairs of
opposite sides of a quadrilateral are congruent,
the quadrilateral is a parallelogram. Therefore,
ABCD is a parallelogram.
Your Turn:
Write a paragraph proof of the statement: If two
diagonals of a quadrilateral divide the quadrilateral
into four triangles where opposite triangles are
congruent, then the quadrilateral is a parallelogram.
Given:
Prove: WXYZ is a parallelogram.
Your Turn:
Proof:
by CPCTC. By
Theorem 8.9, if both pairs of opposite sides of a
quadrilateral are congruent, the quadrilateral is
a parallelogram. Therefore, WXYZ is a
parallelogram.
Example 2:
Some of the shapes in this
Bavarian crest appear to be
parallelograms. Describe
the information needed to
determine whether the
shapes are parallelograms.
Answer: If both pairs of opposite sides are the same
length or if one pair of opposite sides is a
congruent and parallel, the quadrilateral is a
parallelogram. If both pairs of opposite angles
are congruent or if the diagonals bisect
each other, the quadrilateral is
a parallelogram.
Your Turn:
The shapes in the vest
pictured here appear to be
parallelograms. Describe
the information needed to
determine whether the
shapes are parallelograms.
Answer: If both pairs of opposite sides are the same
length or if one pair of opposite sides is
congruent and parallel, the quadrilateral is a
parallelogram. If both pairs of opposite angles
are congruent or if the diagonals bisect each
other, the quadrilateral is a
parallelogram.
Example 3:
Determine whether the quadrilateral is a parallelogram.
Justify your answer.
Answer: Each pair of opposite sides have the same
measure. Therefore, they are congruent. If both
pairs of opposite sides of a quadrilateral are
congruent, the quadrilateral is a parallelogram.
Your Turn:
Determine whether the quadrilateral is a parallelogram.
Justify your answer.
Answer: One pair of opposite sides is parallel and has
the same measure, which means these sides
are congruent. If one pair of opposite sides of a
quadrilateral is both parallel and congruent,
then the quadrilateral is a parallelogram.
Tests for Parallelograms
Both pairs of opposite sides are parallel.
2. Both pairs of opposite sides are congruent.
3. Both pairs of opposite angles are congruent.
4. The diagonals bisect each other.
5. A pair of opposite sides is both parallel and
congruent.
1.
Example 4a:
Find x so that the quadrilateral is a parallelogram.
A
B
D
C
Opposite sides of a parallelogram are congruent.
Example 4a:
Substitution
Distributive Property
Subtract 3x from each side.
Add 1 to each side.
Answer: When x is 7, ABCD is a parallelogram.
Example 4b:
Find y so that the quadrilateral is a parallelogram.
D
G
E
F
Opposite angles of a parallelogram are congruent.
Example 4b:
Substitution
Subtract 6y from each side.
Subtract 28 from each side.
Divide each side by –1.
Answer: DEFG is a parallelogram when y is 14.
Your Turn:
Find m and n so that each quadrilateral is a
parallelogram.
a.
b.
Answer:
Answer:
Parallelograms on the Coordinate Plane

We can use the Distance Formula and
the Slope Formula to determine if a
quadrilateral is a parallelogram on the
coordinate plane.

Just pick one of the tests… and apply
either or both of the formulas.
Example 5a:
COORDINATE GEOMETRY Determine whether the
figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and
D(1, –1) is a parallelogram. Use the Slope Formula.
Example 5a:
If the opposite sides of a quadrilateral are parallel, then it
is a parallelogram.
Answer: Since opposite sides have the same slope,
Therefore, ABCD is a
parallelogram by definition.
Example 5b:
COORDINATE GEOMETRY Determine whether the
figure with vertices P(–3, –1), Q(–1, 3), R(3, 1), and
S(1, –3) is a parallelogram. Use the Distance and Slope
Formulas.
Example 5b:
First use the Distance Formula to determine whether the
opposite sides are congruent.
Example 5b:
Next, use the Slope Formula to determine whether
and have the same slope, so they are parallel.
Answer: Since one pair of opposite sides is congruent
and parallel, PQRS is a parallelogram.
Your Turn:
Determine whether the figure with the given vertices is
a parallelogram. Use the method indicated.
a. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1); Slope Formula
Your Turn:
Answer: The slopes of
and the
slopes of
Therefore,
Since opposite sides are
parallel, ABCD is a parallelogram.
Your Turn:
Determine whether the figure with the given vertices is
a parallelogram. Use the method indicated.
b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2); Distance and
Slope Formulas
Your Turn:
Answer:
Since the
slopes of
Since one pair of opposite sides is congruent
and parallel, LMNO is a parallelogram.