Geometry (A): READ: Proving Quadrilaterals are Parallelograms Reading

Proving Quadrilaterals are Parallelograms

Learning Objectives

Prove a quadrilateral is a parallelogram given congruent opposite sides.
Prove a quadrilateral is a parallelogram given congruent opposite angles.
Prove a quadrilateral is a parallelogram given that the diagonals bisect each other.
Prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.

Introduction

You’ll remember from earlier in this course that you have studied converse statements. A converse statement reverses the order of the hypothesis and conclusion in an if-then statement, and is only sometimes true. For example, consider the statement: “If you study hard, then you will get good grades.” Hopefully this is true! However, the converse is “If you get good grades, then you study hard.” This may be true, but is it not necessarily true—maybe there are many other reasons why you get good grades—i.e., the class is really easy!

An example of a statement that is true and whose converse is also true is as follows: If I face east and then turn a quarter-turn to the right, I am facing south. Similarly, if I turn a quarter-turn to the right and I am facing south, then I was facing east to begin with.

Also all geometric definitions have true converses. For example, if a polygon is a quadrilateral then it has four sides and if a polygon has four sides then it is a quadrilateral.

Converse statements are important in geometry. It is crucial to know which theorems have true converses. In the case of parallelograms, almost all of the theorems you have studied this far have true converses. This lesson explores which characteristics of quadrilaterals ensure that they are parallelograms.

Proving a Quadrilateral is a Parallelogram Given Congruent Sides

In the last lesson, you learned that a parallelogram has congruent opposite sides. We proved this earlier and then looked at one example of this using the distance formula on a coordinate grid to verify that opposite sides of a parallelogram had identical lengths.

Here, we will show on the coordinate grid that the converse of this statement is also true: If a quadrilateral has two pairs of opposite sides that are congruent, then it is a parallelogram.

Example 1

Show that the figure on the grid below is a parallelogram.

We can see that the lengths of opposite sides in this quadrilateral are congruent. For example, to find the length of $\overline {EF}$ we can find the difference in the $x-$ coordinates $(6-1 = 5)$ because $\overline {EF}$ is horizontal (it’s generally very easy to find the length of horizontal and vertical segments). $EF = CD = 5$ and $CF = DE = 7$ . So, we have established that opposite sides of this quadrilateral are congruent.

But is it a parallelogram? Yes. One way to argue that $CDEF$ is a parallelogram is to note that $m \angle {CFE} = m \angle {FED} = 90^\circ$ . We can think of $\overline {FE}$ as a transversal that crosses $\overline {CF}$ and $\overline {DE}$ . Now, interior angles on the same side of the transversal are supplementary, so we can apply the postulate if interior angles on the same side of the transversal are supplementary then the lines crossed by the transversal are parallel.

Note: This example does not prove that if opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram. To do that you need to use any quadrilateral with congruent opposite sides, and then you use congruent triangles to help you. We will let you do that as an exercise, but here’s the basic picture. What triangle congruence postulate can you use to show $\triangle {GHI} \cong \triangle {IJG}?$

Proving a Quadrilateral is a Parallelogram Given Congruent Opposite Angles

Much like the converse statements you studied about opposite side lengths, if you can prove that opposite angles in a quadrilateral are congruent, the figure is a parallelogram.

Example 2

Complete the two-column proof below.

Given: Quadrilateral $DEFG$ with $\angle {D} \cong \angle {F}$ and $\angle {E} \cong \angle {G}$
Prove: $DEFG$ is a parallelogram

Statement	Reason
1. $DEFG$ is a quadrilateral with $\angle {D} \cong \angle {F}$ and $\angle {E} \cong \angle {G}$	1. Given
2. $m \angle {D} + m \angle {E} + \angle {F} + m \angle {G} = 360^\circ$	2. Sum of the angles in a quadrilateral is $360^\circ$
3. $m \angle {D} + m \angle {E} + \angle {D} + m \angle {E} = 360^\circ$	3. Substitution $(\angle {D} \cong \angle {F}$ and $\angle {E} \cong \angle {G})$
4. $2 ( m \angle {D}) + 2 (m \angle {E}) = 360^\circ$	4. Combine like terms
5. $2 ( m \angle {D} + m \angle {E}) = 360^\circ$	5. Factoring
6. $m \angle {D} + m \angle {E} = 180^\circ$	6. Division property of equality (divided both sides by 2)
7. $\overline {DG} \\| \overline {EF}$	7. If interior angles on the same side of a transversal are supplementary then the lines crossed by the transversal are parallel
8. $m \angle {D} + m \angle {G} = 180^\circ$	8. Substitution on line 6 $( \angle {E} \cong \angle {G})$
9. $\overline {DE} \\| \overline {FG}$	9. Same reason as step 7
10. $DEFG$ is a parallelogram	10. Definition of a parallelogram $\blacklozenge$

Proving a Quadrilateral is a Parallelogram Given Bisecting Diagonals

In the last lesson, you learned that in a parallelogram, the diagonals bisect each other. This can also be turned around into a converse statement. If you have a quadrilateral in which the diagonals bisect each other, then the figure is a parallelogram. See if you can follow the proof below which shows how this is explained.

Example 3

Complete the two-column proof below.

Given: $\overline {QV} \cong \overline {VS}$ , and $\overline {TV} \cong \overline {VR}$
Prove: $QRST$ is a parallelogram

Statement	Reason
1. $\overline {QV} \cong \overline {VS}$	1. Given
2. $\overline {TV} \cong \overline {VR}$	2. Given
3. $\angle {QVT} \cong \angle {RVS}$	3. Vertical angles are congruent
4. $\triangle {QVT} \cong \triangle {SVR}$	4. $SAS \cong SAS$ If two sides and the angle between them are congruent, the two triangles are congruent
5. $\overline {QT} \cong \overline {RS}$	5. Corresponding parts of congruent triangles are congruent
6. $\angle {TVS} \cong \angle {RVQ}$	6. Vertical angles are congruent
7. $\triangle {TVS} \cong \triangle {RVQ}$	7. $SAS \cong SAS$ If two sides and the angle between them are congruent, then the two triangles are congruent
8. $\overline {TS} \cong \overline {RQ}$	8. Corresponding parts of congruent triangle are congruent
9. $QRST$ is a parallelogram	9. If two pairs of opposite sides of a quadrilateral are congruent, the figure is a parallelogram $\blacklozenge$

So, given only the information that the diagonals bisect each other, you can prove that the shape is a parallelogram.

Proving a Quadrilateral is a Parallelogram Given One Pair of Congruent and Parallel Sides

The last way you can prove a shape is a parallelogram involves only one pair of sides.

The proof is very similar to the previous proofs you have done in this section so we will leave it as an exercise for you to fill in. To set up the proof (which often IS the most difficult step), draw the following:

Given: Quadrilateral $ABCD$ with $\overline {DA} \| \overline {CB}$ and $\overline {DA} \cong \overline {CB}$
Prove: $ABCD$ is a parallelogram

Example 4

Examine the quadrilateral on the coordinate grid below. Can you show that it is a parallelogram?

To show that this shape is a parallelogram, you could find all of the lengths and compare opposite sides. However, you can also study one pair of sides. If they are both congruent and parallel, then the shape is a parallelogram.

Begin by showing two sides are congruent. You can use the distance formula to do this.

Find the length of $\overline{FG}$ . Use $(-1,5)$ for $F$ and $(3,3)$ for $G$ .

$FG &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(3 - (-1))^2 + (3 - 5)^2} \\ &= \sqrt{(4)^2 + (-2)^2} \\ &= \sqrt{16 + 4} \\ &= \sqrt{20}$

Next, find the length of the opposite side, $\overline{JH}$ . Use $(2,-2)$ for $J$ and $(6, -4)$ for $H$ .

$JH &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(6 - 2)^2 + ((-4)-(-2))^2} \\ &= \sqrt{(4)^2 + (-2)^2} \\ &= \sqrt{16 + 4} \\ &= \sqrt{20}$

So, $FG = JH =\sqrt{20}$ ; they have equal lengths. Now you need to show that $\overline{FG}$ and $\overline{JH}$ are parallel. You can do this by finding their slopes. Recall that if two lines have the same slope, they are parallel.

$\text{Slope of }\overline{FG} &= \frac{y_2-y_1} {x_2-x_1}\\ &=\frac{3-5} {3-(-1)} \\ &=\frac{-2} {4} \\ &=-\frac{1}{2}$

So, the slope of $\overline{FG}= -\frac{1}{2}$ . Now, check the slope of $\overline{JH}$ .

$\text{Slope of }\overline{JH} &= \frac{y_2-y_1} {x_2-x_1}\\ &=\frac{(-4)-(-2)} {6-2} \\ &=\frac{-2} {4} \\ &=-\frac{1}{2}$

So, the slope of $\overline{JH}=-\frac{1}{2}$ . Since the slopes of $\overline{FG}$ and $\overline{JH}$ are the same, the two segments are parallel. Now that have shown that the opposite segments are both parallel and congruent, you can identify that the shape is a parallelogram.

Lesson Summary

In this lesson, we explored parallelograms. Specifically, we have learned:

How to prove a quadrilateral is a parallelogram given congruent opposite sides.
How to prove a quadrilateral is a parallelogram given congruent opposite angles.
How to prove a quadrilateral is a parallelogram given that the diagonals bisect each other.
How to prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.

It is helpful to be able to prove that certain quadrilaterals are parallelograms. You will be able to use this information in many different ways.

The following questions are for your own review. The answers are listed below for you to check your work and understanding.

Review Questions

Use the following diagram for exercises 1-3.

Find each angle:
1. $m \angle FBC =$
2. $m \angle FBA =$
3. $m \angle ADC =$
4. $m \angle BCD =$
If and , find each length:
1. $AD =$
2. $DC =$
If and , find each length:
1. $AF =$
2. $BD =$

Use the following figure for exercises 4-7.

Suppose that $A$ $(1, 6)$ , $B$ $(6, 6)$ , and $C$ $(3, 2)$ are three of four vertices (corners) of a parallelogram. Give two possible locations for the fourth vertex, $D$ , if you know that the $y-$ coordinate of $D$ is $2$ .
Depending on where you choose to put point $D$ in $4$ , the name of the parallelogram you draw will change. Sketch a picture to show why.
If you know the parallelogram is named $ABDC$ , what is the slope of the side parallel to $AC$ ?
Again, assuming the parallelogram is named $ABDC$ , what is the length of $\overline{BD}$ ?
Prove: If opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Given: $ABCD$ with $\overline{AB} \cong \overline{DC}$ and $\overline{AD} \cong \overline{BC}$

Prove: $\overline{AB} \| \overline{DC}$ and $\overline{AD} \| \overline{BC}$ (i.e., $ABCD$ is a parallelogram).

Prove: If a quadrilateral has one pair of congruent parallel sides, then it is a parallelogram.
Note in 9 that the parallel sides must also be the congruent sides for that theorem to work. Sketch a counterexample to show that if a quadrilateral has one pair of parallel sides and one pair of congruent sides (which are not the parallel sides) then the resulting figure it is not necessarily a parallelogram. What kind of quadrilaterals can you make with this arrangement?

Review Answers

1. $m \angle FBC = 20^\circ$
2. $m \angle {FBA} = 46^\circ$
3. $m \angle ADC = 66^\circ$
4. $m \angle BCD = 114^\circ$ (note: you need to find almost all angle measures in the diagram to answer this question)
1. $AD = 9.5\;\mathrm{m}, DC = 4.5 \;\mathrm{m}$
1. $AF = 4.35 \;\mathrm{m}$ ,
2. $BD = 12 \;\mathrm{m}$

$D$ can be at either $(-2,2)$ or $(8,2)$
If $D$ is at $(-2,2)$ the parallelogram would be named $ABCD$ (in red in the following illustration). If $D$ is at $(8,2)$ then the parallelogram will take the name $ABDC$ .
$BD$ would have a slope of $-2$ .
$BD = \sqrt{20}$

Given: $ABCD$ with $\overline{AB} \cong \overline{DC}$ and $\overline{AD} \cong \overline{BC}$

Prove: $\overline{AB} \| \overline{DC}$ and $\overline{AD} \| \overline{BC}$ (i.e., $ABCD$ is a parallelogram)

Statement	Reason
1. $\overline{AB} \cong \overline{DC}$	1. Given
2. $\overline{AD} \cong \overline{BC}$	2. Given
3. Add auxiliary line $\overline{AC}$	3. Line Postulate
4. $\overline{AC} \cong \overline{AC}$	4. Reflexive Property
5. $\triangle ACD \cong \triangle CAB$	5. SSS Congruence Postulate
6. $\angle 2 \cong \angle 3$	6. Definition of congruent triangles
7. $\overline{AD} \\| \overline{BC}$	7. Converse of Alternate Interior Angles Postulate
8. $\angle 4 \cong \angle 1$	8. Definition of congruent triangles
9. $\overline{AB} \\| \overline{DC}$	9. Converse of Alternate Interior Angles Postulate

First, translate the theorem into given and prove statements:

Given: $ABCD$ with $\overline{AB} \| \overline{CD}$ and $\overline{AB} \cong \overline{CD}$

Prove: $\overline{BC} \| \overline{AD}$

Statement	Reason
1. $\overline{AB} \cong \overline{CD}$	1. Given
2. $\overline{AB} \\| \overline{CD}$	2. Given
3. $\angle{4} \cong \angle{1}$	3. Alternate Interior Angles Theorem
4. Add auxiliary line $\overline{AC}$	4. Line Postulate
5. $\overline{AC} \cong \overline{AC}$	5. Reflexive Property
6. $\triangle ABC \cong \triangle CDA$	6. SAS Triangle Congruence Postulate
7. $\angle BCA \cong \angle DAC$	7. Definition of congruent triangles
8. $\overline{BC} \\| \overline{AD}$	8. Converse of Alternate Interior Angles Theorem

If the congruent sides are not the parallel sides, then you can make either a parallelogram (in black) or an isosceles trapezoid (in red):

Last modified: Monday, June 28, 2010, 4:32 PM