Tests for ParallelogramsTests 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 .

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.

Prove: ABCD is a parallelogram.

Given:

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.

Example 1:

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.

Prove: WXYZ is a parallelogram.

Given:

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.

Your Turn:

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.

Example 2:

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.

Your Turn:

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.

Example 3:

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.

Your Turn:

Tests for Parallelograms

1. 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.

Find x so that the quadrilateral is a parallelogram.

Opposite sides of a parallelogram are congruent.

A B

CD

Example 4a:

Substitution

Distributive Property

Add 1 to each side.

Answer: When x is 7, ABCD is a parallelogram.

Subtract 3x from each side.

Example 4a:

Find y so that the quadrilateral is a parallelogram.

Opposite angles of a parallelogram are congruent.

F

D E

G

Example 4b:

Subtract 6y from each side.

Substitution

Subtract 28 from each side.

Divide each side by –1.

Answer: DEFG is a parallelogram when y is 14.

Example 4b:

Find m and n so that each quadrilateral is a parallelogram.

Answer: Answer:

a. b.

Your Turn:

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.

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 5a:

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:

Answer: Since one pair of opposite sides is congruent and parallel, PQRS is a parallelogram.

Next, use the Slope Formula to determine whether

and have the same slope, so they are parallel.

Example 5b:

Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.

Slope Formulaa. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1);

Your Turn:

Answer: The slopes of and the

slopes of Therefore,

Since opposite sides are parallel, ABCD is a

parallelogram.

Your Turn:

Distance and Slope Formulas

b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2);

Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.

Your Turn:

Answer: Since the

slopes of

Since one pair of opposite sides is congruent

and parallel, LMNO is a parallelogram.

Your Turn: