Geometry (A): READ: Parallelogram Reading

Using Parallelograms

Learning Objectives

Describe the relationships between opposite sides in a parallelogram.
Describe the relationship between opposite angles in a parallelogram.
Describe the relationship between consecutive angles in a parallelogram.
Describe the relationship between the two diagonals in a parallelogram.

Introduction

Now that you have studied the different types of quadrilaterals and their defining characteristics, you can examine each one of them in greater depth. The first shape you’ll look at more closely is the parallelogram. It is defined as a quadrilateral with two pairs of parallel sides, but there are many more characteristics that make a parallelogram unique.

Opposite Sides in a Parallelogram

By now, you recognize that there are many types of parallelograms. They can look like squares, rectangles, or diamonds. Either way, opposite sides are always parallel. One of the most important things to know, however, is that opposite sides in a parallelogram are also congruent.

To test this theory, you can use pieces of string on your desk. Place two pieces of string that are the same length down so that they are parallel. You’ll notice that the only way to connect the remaining vertices will be two parallel, congruent segments. There will be only one possible fit given two lengths.

Try this again with two pieces of string that are different lengths. Again, lay them down so that they are parallel on your desk. What you should notice is that if the two segments are different lengths, the missing segments (if they connect the vertices) will not be parallel. Therefore, it will not create a parallelogram. In fact, there is no way to construct a parallelogram if opposite sides aren’t congruent.

So, even though parallelograms are defined by their parallel opposite sides, one of their properties is that opposite sides be congruent.

Example 1

Parallelogram $FGHJ$ is shown on the following coordinate grid. Use the distance formula to show that opposite sides in the parallelogram are congruent.

You can use the distance formula to find the length of each segment. You are trying to prove that $FG$ is the same as $HJ$ , and that $GH$ is the same as $FJ.$ (Recall that $FG$ means the same as $m \overline {FG}$ , or the length of $\overline {FG}$ .)

Start with $FG$ . The coordinates of $F$ are $(-4,5)$ and the coordinates of $G$ are $(3,3)$ .

$FG &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(3 - (-4))^2 + (3-5)^2} \\ &= \sqrt{(3+4)^2 + (3-5)^2} \\ &= \sqrt{(7)^2 + (-2)^2} \\ &= \sqrt{49 + 4} \\ &= \sqrt{53}$

So $FG = \sqrt{53}$ .

Next find $GH$ . The coordinates of $G$ are $(3, 3)$ and the coordinates of $H$ are $(6,-4)$ .

$GH &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(6 - 3)^2 + (-4-3)^2} \\ &= \sqrt{(3)^2 + (-7)^2} \\ &= \sqrt{9 + 49} \\ &= \sqrt{58}$

So $GH = \sqrt{58}$ .

Next find $HJ$ . The coordinates of $H$ are $(6,-4)$ and the coordinates of $J$ are $(-1,-2)$ .

$HJ &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(-1 - 6)^2 + (-2-(-4))^2} \\ &= \sqrt{(-7)^2 + (2)^2} \\ &= \sqrt{49 + 4} \\ &= \sqrt{53}$

So $HJ = \sqrt{53}$ .

Finally, find the length of $FJ$ . The coordinates of $F$ are $(-4,5)$ and the coordinates of $J$ are $(-1,-2)$ .

$FJ &= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \\ &= \sqrt{(-1 - (-4))^2 + (-2-5)^2} \\ &= \sqrt{(3)^2 + (-7)^2} \\ &= \sqrt{9 + 49} \\ &= \sqrt{58}$

So $FJ = \sqrt{58}$ .

Thus, in parallelogram $FGHJ$ , $FG=HJ$ and $GH=FJ$ . The opposite sides are congruent.

This example shows that in this parallelogram, the opposite sides are congruent. In the last section we proved this fact is true for all parallelograms using congruent triangles. Here we have shown an example of this property in the coordinate plane.

Opposite Angles in a Parallelogram

Not only are opposite sides in a parallelogram congruent. Opposite angles are also congruent. You can prove this by drawing in a diagonal and showing ASA congruence between the two triangles created. Remember that when you have congruent triangles, all corresponding parts will be congruent.

Example 2

Fill in the blanks in the two-column proof below.

Given: $LMNO$ is a parallelogram
Prove: $\angle {OLM} \cong \angle {MNO}$

Statement	Reason
1. $LMNO$ is a parallelogram	1. Given
2. $\overline {LM} \\| \overline {ON}$	2. Definition of a parallelogram
3. $\angle\ \underline{\;\;\;\;\;\;\;} \cong \angle\ \underline{\;\;\;\;\;\;\;}$	3. Alternate Interior Angles Theorem
4. $\underline{\;\;\;\;\;\;\;}\ \\|\ \underline{\;\;\;\;\;\;\;}$	4. Definition of a parallelogram
5. $\angle {2} \cong \angle {3}$	5. _______________________
6. $\overline {MO} \cong \overline{MO}$	6. Reflexive Property
7. $\triangle\ \underline{\;\;\;\;\;\;\;} \cong \triangle\ \underline{\;\;\;\;\;\;\;}$	7. ASA Triangle Congruence Postulate
8. $\angle {OLM} \cong \angle {MNO}$	8. Corresponding parts of congruent triangles are congruent

The missing statement in step 3 should be related to the information in step 2. $\overline {LM}$ and $\overline {ON}$ are parallel, and $\overline {MO}$ is a transversal. Look at the following figure (with the other segments removed) to see the angles formed by these segments:

Therefore the missing step is $\angle {1} \cong \angle {4}$ .

Work backwards to fill in step 4. Since step 5 is about $\angle {2} \cong \angle {3}$ , the sides we need parallel are $\overline {LO}$ and $\overline {MN}$ . So step 4 is $\overline {LO} \| \overline {MN}$ .

The missing reason on step 5 will be the same as the missing reason in step 3: alternate interior angles.

Finally, to fill in the triangle congruence statement, BE CAREFUL to make sure you match up corresponding angles. The correct form is $\triangle {LMO} \cong \triangle {NOM}$ . (Students commonly get this reversed, so don’t feel bad if you take a few times to get it correct!)

As you can imagine, the same process could be repeated with diagonal $\overline{LN}$ to show that $\angle {LON} \cong \angle {LMN}$ . Opposite angles in a parallelogram are congruent. Or, even better, we can use the fact that $\angle {1} \cong \angle {4}$ and $\angle {2} \cong \angle {3}$ together with the Angle Addition Postulate to show $\angle {LON} \cong \angle {LMN}$ . We leave the details of these operations for you to fill in.

Consecutive Angles in a Parallelogram

So at this point, you understand the relationships between opposite sides and opposite angles in parallelograms. Think about the relationship between consecutive angles in a parallelogram. You have studied this scenario before, but you can apply what you have learned to parallelograms. Examine the parallelogram below.

Imagine that you are trying to find the relationship between $\angle {SPQ}$ and $\angle {PSR}$ . To help you understand the relationship, extend all of the segments involved with these angles and remove $\overline {RQ}$ .

What you should notice is that $PQ$ and $SR$ are two parallel lines cut by transversal $PS$ . So, you can find the relationships between the angles as you learned in Chapter 1. Earlier in this course, you learned that in this scenario, two consecutive interior angles are supplementary; they sum to $180^\circ$ . The same is true within the parallelogram. Any two consecutive angles inside a parallelogram are supplementary.

Example 3

Fill in the remaining values for the angles in parallelogram $ABCD$ below.

You already know that $m \angle {DAB} = 30^\circ$ since it is given in the diagram. Since opposite angles are congruent, you can conclude that $m \angle {BCD} = 30^\circ$ .

Now that you know that consecutive angles are supplementary, you can find the measures of the remaining angles by subtracting $30^\circ$ from $180^\circ$ .

$m \angle {BAD} + m \angle {ADC} &= 180^\circ \\ 30^\circ +m \angle {ADC} &= 180^\circ \\ 30^\circ +m \angle {ADC} - 30^\circ &= 180^\circ - 30^\circ \\ m\angle {ADC} &= 150^\circ$

So, $m\angle {ADC} = 150^\circ$ . Since opposite angles are congruent, $\angle {ABC}$ will also measure $150^\circ$ .

Diagonals in a Parallelogram

There is one more relationship to examine within parallelograms. When you draw the two diagonals inside parallelograms, they bisect each other. This can be very useful information for examining larger shapes that may include parallelograms. The easiest way to demonstrate this property is through congruent triangles, similarly to how we proved opposite angles congruent earlier in the lesson.

Example 4

Use a two-column proof for the theorem below.

Given: $WXYZ$ is a parallelogram
Prove: $\overline {WC} \cong \overline {CY}$ and $\overline {XC} \cong \overline {ZC}$

Statement	Reason
1. $WXYZ$ is a parallelogram	1. Given.
2. $\overline {WX} \cong \overline {YZ}$	2. Opposite sides in a parallelogram are congruent.
3. $\angle {WCX} \cong \angle {ZCY}$	3. Vertical angles are congruent.
4. $\angle {XWC} \cong \angle {CYZ}$	4. Alternate interior angles are congruent.
5. $\triangle {WXC} \cong \triangle {YZC}$	5. AAS congruence theorem: If two angles and one side in a triangle are congruent, the triangles are congruent.
6. $\overline {WC} \cong \overline {CY}$ and $\overline {XC} \cong \overline {ZC}$	6. Corresponding parts of congruent triangles are congruent. $\blacklozenge$

Lesson Summary

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

How to describe and prove the distance relationships between opposite sides in a parallelogram.
How to describe and prove the relationship between opposite angles in a parallelogram.
How to describe and prove the relationship between consecutive angles in a parallelogram.
How to describe and prove the relationship between the two diagonals in a parallelogram.

It is helpful to be able to understand the unique properties of parallelograms. You will be able to use this information in many different ways.

Points to Consider

Now that you have learned the many relationships in parallelograms, it is time to learn how you can prove that shapes are parallelograms.

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

Review Questions

$DG =\underline{\;\;\;\;\;\;\;}, DF =\underline{\;\;\;\;\;\;\;}, AD =\underline{\;\;\;\;\;\;\;}$
$a =\underline{\;\;\;\;\;\;\;}, b =\underline{\;\;\;\;\;\;\;}$

Use the following figure for exercises 3-6.

Find the slopes of $\overline{AD}$ and $\overline{CB}$ .
Find the slopes of $\overline{DC}$ and $\overline{AB}$ .
What kind of quadrilateral is $ABCD$ ? Give an answer that is as detailed as possible.
If you add diagonals to $ABCD$ , where will they intersect?

Use the figure below for questions 7-11. Polygon $PQRSTUVW$ is a regular polygon. Find each indicated measurement.

$m\angle{RST} =$
$m\angle{VWX} =$
$m\angle{WXY} =$
What kind of triangle is $\triangle {WVX}$ ?
Copy polygon and add auxiliary lines to make each of the following:
1. a parallelogram
2. a trapezoid
3. an isosceles triangle

Review Answers

$DG = 5 \;\mathrm{cm},\ DF = 7.25 \;\mathrm{cm}\ ,AD = 11 \;\mathrm{cm}$
$a = 76^\circ,\ b = 104^\circ$

Slopes of $\overline{AD}$ and $\overline{CB}$ $\mathrm{both} = 1$
$\mathrm{Both} = -1$
This figure is a parallelogram since opposite sides have equal slopes (i.e., opposite sides are parallel). Additionally, it is a rectangle because each angle is a $90^\circ$ angle. We know this because the slopes of adjacent sides are opposite reciprocals
The diagonals would intersect at $(0,0)$ . One way to see this is to use the symmetry of the figure—each corner is a $90^\circ$ rotation around the origin from adjacent corners
$m \angle RST = 135^\circ$
$m \angle VWX = 45^\circ$
$m \angle WXY = 90^\circ$
$\triangle WVX$ is an isosceles right triangle
There are many possible answers. Here is one: Auxiliary lines are in red:
1. $SRWV$ is a parallelogram (in fact it is a rectangle).
2. $STUV$ is a trapezoid.
3. $QPG$ is an isosceles triangle

Last modified: Monday, June 28, 2010, 4:23 PM