In this target we are going to learn how to prove a quadrilateral is a parallelogram. There are five ways to do that. These five ways are connected to the four properties of a parallelogram and are listed below along with their diagrams.
Five Ways to Prove a Quadrilateral is a Parallelogram
Now let’s practice using these five ways and prove some quadrilaterals are parallelograms.
Example 1
Given these diagrams, let’s determine if we can conclude whether each diagram is a parallelogram or not. If not, explain why not. If so, state the reason why.
Example 1
Given these diagrams, let’s determine if we can conclude whether each diagram is a parallelogram or not. If not, explain why not. If so, state the reason why.
Solutions:
a) Yes, one pair of sides is both parallel and congruent.
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b) No, having alternate interior angles that are congruent leads to parallel sides. The parallel sides are not congruent, so the quadrilateral is not a parallelogram.
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Example 2:
Find the value of the following variables: \(x\) and \(y\)) that would make \(NCHS\) a parallelogram. Solutions: \(x\): For \(NCHS\) to be a parallelogram, the opposite sides must be congruent: \(NS = CH\) \(4x - 3 = 2x + 7\) \(2x = 10\) \(x = 5\) \(y\): For \(NCHS\) to be a parallelogram, the diagonals bisect each other: \(NR = RH\) \(5y = 37\) \(y = \Large\frac{37}{5} = \normalsize 7.4\) |
Proving Parallelograms on the Coordinate Plane
Let’s review some formulas that we may need to use on the coordinate plane to prove the quadrilaterals are parallelograms.
Slope formula: \(\text{slope}=\Large\frac{y_2-y_1}{x_2-x_1}\)
Distance formula: \(\text{distance}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Midpoint formula: \(\text{midpoint}=\left(\Large\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\)
Example 3:
Show the following set of points are the vertices of a parallelogram:
\(A\;(2, -1)\), \(B\;(1, 3)\), \(C\;(6, 5)\), \(D\;(7, 1)\)
Let’s review some formulas that we may need to use on the coordinate plane to prove the quadrilaterals are parallelograms.
Slope formula: \(\text{slope}=\Large\frac{y_2-y_1}{x_2-x_1}\)
Distance formula: \(\text{distance}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Midpoint formula: \(\text{midpoint}=\left(\Large\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\)
Example 3:
Show the following set of points are the vertices of a parallelogram:
\(A\;(2, -1)\), \(B\;(1, 3)\), \(C\;(6, 5)\), \(D\;(7, 1)\)
Solution:
Watch the video for the solution.
Watch the video for the solution.
Quick Check
1) Given these diagrams, let’s determine if we can conclude whether each diagram is a parallelogram or not. If not, explain why not. If so, state the reason why.
1) Given these diagrams, let’s determine if we can conclude whether each diagram is a parallelogram or not. If not, explain why not. If so, state the reason why.
3) Show the following set of points are the vertices of a parallelogram: \(Q\;(-2, -2), U\;(4, 0), A\;(3, 3) D\;(-3, 1)\)