Tests for Parallelograms Section 6.3 Tests for Parallelograms

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

Example 1: a) Determine whether the quadrilateral is a parallelogram. Justify your answer. b) Which method would prove the quadrilateral is a parallelogram? 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. One pair of oppsite sides both || and .

You can use the conditions of parallelograms to prove relationships in real-world situations. 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. 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, mA + mB = 180° and mC + mD = 180°. By substitution, mA + mD = 180° and mC + mB = 180°.

You can also use the conditions of parallelograms along with algebra to find missing values that make a quadrilateral a parallelogram. Example 3: a) Solve for x and y so that the quadrilateral is a parallelogram. First, solve for x. AB  _________ because…..   DC Opposite sides of a parallelogram are congruent AB = _________ because…..   4x – 1 = _______________ because…. DC Definition of congruent segments 3(x + 2) Substitution 4x – 1 = 3x + 6 Distributive Property x – 1 = 6 Subtract 3x from each side x = 7 Add 1 to each side

Now, solve for y. AD  BC Opposite sides of a parallelogram are congruent AD = BC Definition of congruent segments 3(y + 1) = 4y – 2 Substitution 3y + 3 = 4y – 2 Distributive Property 3 = y – 2 Subtract 3y from each side y = 5 Add 2 to each side

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

We can use the Distance, Slope, and Midpoint Formulas to determine whether a quadrilateral in the coordinate plane is a parallelogram. 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. If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.

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. Step 1 Position quadrilateral ABCD on the coordinate plane such that AB  DC and AD  BC. Begin by placing the vertex A at the origin. 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. The slope of AB is 0. The slope of CD is 0. Since AB and CD have the same slope and AD and BC have the same slope, AB║CD and AD║BC. So, quadrilateral ABCD is a parallelogram because opposite sides are parallel.