# PROVING QUADRILATERALS ARE PARALLELOGRAMS WORKSHEET

Problem 1 :

In the diagram given below, if AB  ≅  CD, AD  ≅  CB, then prove that ABCD is a parallelogram.

Problem 2 :

In the diagram given below, if BC || DA, BC  ≅  DA, then prove that ABCD 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.

## Solutions

Problem 1 :

In the diagram given below, if AB  ≅  CD, AD  ≅  CB, then prove that ABCD is a parallelogram.

 Statements AB  ≅  CD, AD  ≅  CBaaaaaaa AC  ≅  AC aaaaa aaaaaaaaaaaaaaaaaaaaaaΔABC  ≅  ΔCDAaa m∠BAC  ≅  m∠DCA  a aa m∠DAC  ≅  m∠BCA aa aaaaaaaaaaaaaaaaaaaaaaaa AB || CD, AD || CB a aaaaaaaaaaaaaaaaaaaaaABCD is a parallelogram ReasonsGivenReflexive Property of CongruenceSSS Congruence PostulateCorresponding parts of congruent triangles are congruentAlternate Interior Angles ConverseDefinition of parallelogram

Problem 2 :

In the diagram given below, if BC || DA, BC  ≅  DA, then prove that ABCD is a parallelogram.

 Statements BC || DAaa m∠DAC  ≅  m∠BCA aa aaaaaaaaaaaaaaaaaaaaaaaaaaa AC  ≅  AC aaaaa aaaaaaaaaaaaaaaaaaaaaaΔABC  ≅  ΔCDABC  ≅  DAΔABC  ≅  ΔCDAaaaaaa BC  ≅  DA aaaaa aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaABCD is a parallelogram ReasonsGivenAlternate Interior Angles ConverseReflexive Property of CongruenceSSS Congruence PostulateGivenSAS 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.

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