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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