Section 6.3Tests for Parallelograms

If a quadrilateral has each pair of opposite sides parallel, it is a parallelogram by definition. 

This is not the only test, however, that can be used to determine if a quadrilateral is a parallelogram.

How do you know if a quadrilateral is a parallelogram.?

Example 1: a) Determine whether the quadrilateral is a parallelogram. Justify your answer.

b) Which method would prove the quadrilateral is a parallelogram?

One pair of oppsite sides both || and .

Each pair of opposite sides has the same measure. Therefore, they are congruent.If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

Example 2: Scissor lifts, like the platform lift shown, are commonly applied to tools intended to lift heavy items. In the diagram, A C and B D. Explain why the consecutive angles will always be supplementary, regardless of the height of the platform.

You can use the conditions of parallelograms to prove relationships in real-world situations.

Since both pairs of opposite angles of quadrilateral ABCD are congruent, ABCD is a parallelogram by Theorem 6.10. Theorem 6.5 states that consecutive angles of parallelograms are supplementary. Therefore, mA + mB = 180° and mC + mD = 180°. By substitution, mA + mD = 180° and mC + mB = 180°.

Example 3:

a) Solve for x and y so that the quadrilateral is a parallelogram.

First, solve for x.

You can also use the conditions of parallelograms along with algebra to find missing values that make a quadrilateral a parallelogram.

AB _________ because…..  

DC

AB = _________ because….. 4x – 1 = _______________ because….

DC

Opposite sides of a parallelogram are congruentDefinition of congruent segments

3(x + 2) Substitution

x – 1 = 6 x = 7

Subtract 3x from each sideAdd 1 to each side

4x – 1 = 3x + 6 Distributive Property

Opposite sides of a parallelogram are congruent

AD BC

AD = BC Definition of congruent segments

3(y + 1) = 4y – 2 Substitution

3 = y – 2 y = 5

Subtract 3y from each sideAdd 2 to each side

3y + 3 = 4y – 2

Distributive Property

Now, solve for y.

Example 3:b) Find m so that the quadrilateral is a parallelogram.

Opposite sides of a parallelogram are congruent, so

4m + 2 = 3m + 8 m + 2 = 8

m = 6 Subtract 3m from each sideSubtract 2 to from each side

Example 4: Quadrilateral QRST has vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula.

We can use the Distance, Slope, and Midpoint Formulas to determine whether a quadrilateral in the coordinate plane is a parallelogram.

If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.

1 3 1slope of

2 2 2ST

1 3 1

slope of 3 1 2

QR

3 1slope of 4

2 3RS

3 1slope of 4

1 2TQ

Since opposite sides have the same slope, || and || .

Therefore, is a parallelogram by definition.

QR ST RS TQ

QRST

Example 5: Write a coordinate proof for the following statement.

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Begin by placing the vertex A at the origin.

Step 1 Position quadrilateral ABCD on the coordinateplane such that AB DC and AD BC.

Let AB have a length of a units. Then B has coordinates (a, 0).

So that the distance from D to C is also a units, let the x-coordinate of D be b and of C be b + a.

Since AD BC, position the endpoints of DC so that they have the same y-coordinate, c.

Step 2 Use your figure to write a proof.

Given: quadrilateral ABCD, AB DC, AD BC

Prove: ABCD is a parallelogram.

Coordinate Proof:

By definition, a quadrilateral is a parallelogram if opposite sides are parallel.

Use the Slope Formula.

0Slope of

0

c cAD

b b

So, quadrilateral ABCD is a parallelogram because opposite sides are parallel.

Since AB and CD have the same slope and AD and BC have the same slope, AB║CD and AD║BC.

The slope of CD is 0.

The slope of AB is 0.

0Slope of

c cBC

b a a b