PROVING QUADRILATERALS ARE PARALLELOGRAMS
About "Proving Quadrilaterals are Parallelograms"
Proving Quadrilaterals are Parallelograms :
In this section, you will learn how to prove that a quadrilateral is a parallelogram.
We can use the following Theorems to prove the quadrilateral are parallelograms.
Theorems
Theorem 1 :
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
It has been illustrated in the diagram shown below.
In the diagram above,
AB ≅ DC
AD ≅ BC
Theorem 2 :
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
It has been illustrated in the diagram shown below.
In the diagram above,
m∠A ≅ m∠C
m∠B ≅ m∠D
Theorem 3 :
If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
It has been illustrated in the diagram shown below.
In the diagram above,
m∠A + m∠B = 180°
m∠B + m∠C = 180°
m∠C + m∠D = 180°
m∠A + m∠D = 180°
Theorem 4 :
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
It has been illustrated in the diagram shown below.
In the diagram above,
AM ≅ CM
BM ≅ DM
Theorem 5 :
If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.
It has been illustrated in the diagram shown below.
In the diagram above,
AD ≅ BC
AD || BC
Proving Quadrilaterals are Parallelograms - Practice Problems
Problem 1 :
In the diagram given below, if AB ≅ CD, AD ≅ CB, then prove that ABCD is a parallelogram.
|
Statements AB ≅ CD, AD ≅ CB aaaaaaa AC ≅ AC aaaaa aaaaaaaaaaaaaaaaaaaaaa ΔABC ≅ ΔCDA aa m∠BAC ≅ m∠DCA a aa m∠DAC ≅ m∠BCA aa aaaaaaaaaaaaaaaaaaaaaa aa AB || CD, AD || CB a aaaaaaaaaaaaaaaaaaaaa ABCD is a parallelogram |
Reasons Given Reflexive Property of Congruence SSS Congruence Postulate Corresponding parts of congruent triangles are congruent Alternate Interior Angles Converse Definition of parallelogram |
Problem 2 :
In the diagram given below, if BC || DA, BC ≅ DA, then prove that ABCD is a parallelogram.
|
Statements BC || DA aa m∠DAC ≅ m∠BCA aa aaaaaaaaaaaaaaaaaaaa aaaaaaa AC ≅ AC aaaaa aaaaaaaaaaaaaaaaaaaaaa ΔABC ≅ ΔCDA BC ≅ DA ΔABC ≅ ΔCDA aaaaaa BC ≅ DA aaaaa aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaa ABCD is a parallelogram |
Reasons Given Alternate Interior Angles Converse Reflexive Property of Congruence SSS Congruence Postulate Given SAS Congruence Postulate Corresponding parts of congruent triangles are congruent If opposite sides of a quadrilateral are congruent, then it is a parallelogram |
Problem 3 :
Show that A(2, - 1), B(1, 3), C(6, 5) and D(7, 1) are the vertices of a parallelogram.
Solution :
Let us plot the given points in a coordinate plane as shown below.
Solution :
There are many ways to prove that the given points are the vertices of a parallelogram.
Method 1 :
Show that opposite sides have the same slope, so they are parallel.
Using slope formula to find the slopes of AB, CD, BC and DA.
Slope of AB = [3 - (-1)] / [1 - 2] = - 4
Slope of CD = [1 - 5] / [7 - 6] = - 4
Slope of BC = [5 - 3] / [6 - 1] = 2 / 5
Slope of DA = [- 1 - 1] / [2 - 7] = 2 / 5
AB and CD have the same slope. So they are parallel.
Similarly, BC and DA are parallel.
Because opposite sides are parallel, ABCD is a parallelogram.
Method 2 :
Show that opposite sides have the same length.
Using distance formula to find the lengths of AB, CD, BC and DA.
AB = √[(1 - 2)2 + (3 + 1)2] = √17
CD = √[(7 - 6)2 + (1 - 5)2] = √17
BC = √[(6 - 1)2 + (5 - 3)2] = √29
DA = √[(2 - 7)2 + (- 1 - 1)2] = √29
From the lengths AB, CD, BC and DA, it is clear that
AB ≅ CD and BC ≅ DA
Because both pairs of opposite sides are congruent, ABCD is a parallelogram.
Method 3 :
Show that one pair of opposite sides is congruent and parallel.
Find the slopes and lengths of AB and CD as shown in Methods 1 and 2.
Slope of AB = Slope of CD = - 4
Because AB and CD have the same slope, they are parallel.
AB = CD = √17
Because AB and CD have the same length, they are congruent.
So, ABCD is a parallelogram.
After having gone through the stuff given above, we hope that the students would have understood, "Proving quadrilaterals are parallelograms".
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