11.02 Properties of parallelograms

Solving for missing sides or angles in parallelograms

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

A parallelogram - it has two pair of parallel sides

 

The following are five useful properties of a parallelogram:

We can use these properties to find unknown angles or sides of parallelograms.

 

Worked examples

Question 1

Consider the parallelogram below. Find the value of $x$x and $y$y.

Think: There are two properties we can use to find $x$x and $y$y. For a parallelogram, opposite angles are congruent, and consecutive angles are supplementary.

Do: The angle with marked with $y$y is opposite the angle marked $144$144 so they will be equal.

$y=144$y=144

Similarly the angle marked $x$x is consecutive to the angle marked $144$144.

$x+144$x+144	$=$=	$180$180
$x$x	$=$=	$36$36

Reflect: The remaining unlabeled angle is opposite the angle marked $x$x, so it will also measure $36^\circ$36°. We can then check the sum of the internal angle measures to be $144^\circ+36^\circ+144^\circ+36^\circ=360^\circ$144°+36°+144°+36°=360°, as expected for a quadrilateral.

Question 2

Consider the parallelogram below. Find the value of $x$x.

Think: For a parallelogram, opposite sides are congruent, so the lengths given will be equal.

Do: We can create the equation $x-5=10$x−5=10, and solve for $x$x.

$x-5$x−5	$=$=	$10$10
$x$x	$=$=	$15$15

Question 3

Consider the parallelogram below. Find the value of $x$x.

Think: For a parallelogram the diagonals bisect each other. This means they will create two segments equal to each other.

Do: Each half of the diagonal will be equal, so $2x-1=7$2x−1=7. We can then solve this equation for $x$x:

$2x-1$2x−1	$=$=	$7$7
$2x$2x	$=$=	$8$8
$x$x	$=$=	$4$4

 

Reflect: The length marked on the diagram is $2x-1$2x−1. When $x=4$x=4 this length will be equal to $2\times4-1=7$2×4−1=7, which is the same length as the other half of the diagonal.

 

Practice questions

Question 4

Question 5

Question 6

 

 

Proving properties of parallelograms

We define a parallelogram as any quadrilateral whose opposite side lengths are parallel.

Note that there are several definitions that achieve the same result. It is important to distinguish them, as we consider them to be properties or theorems of our definition.

Exploration

Consider the following theorem.

If a quadrilateral is a parallelogram, then its opposite sides are congruent.

To begin proving the above theorem, we can first label the parallelogram by its set of vertices, $ABCD$ABCD.

Parallelogram $ABCD$ABCD

 

From the definition of a parallelogram, we know that the pair of opposite sides are parallel. We write this as geometric statements $\overline{AB}\parallel\overline{DC}$AB∥DC and $\overline{AD}\parallel\overline{BC}$AD∥BC.

We can then create a pair of triangles by joining the vertices $A$A and $C$C. If we can show that the two triangles $\triangle ABC$△ABC and $\triangle CDA$△CDA are congruent, then we can show that the opposite sides of the parallelogram are congruent.

Parallelogram $ABCD$ABCD with segment $\overline{AC}$AC

 

Next we can observe that we have two pairs of alternate interior angles, where the segment $\overline{AC}$AC traverses both pairs of parallel lines. This means that $\angle BAC\cong\angle DCA$∠BAC≅∠DCA and $\angle BCA\cong\angle DAC$∠BCA≅∠DAC.

We also notice that both triangles have the common side, $\overline{AC}$AC.

Hence, $\triangle ABC$△ABC and $\triangle CDA$△CDA have angle-side-angle congruence.

Now since the two triangles are congruent, their corresponding parts must also be congruent.

If we rotate $\triangle ABC$△ABC we can see that the segment $\overline{AB}$AB is congruent to $\overline{CD}$CD and the segment $\overline{BC}$BC is congruent to $\overline{DA}$DA.

$\triangle ABC$△ABC is congruent to $\triangle CDA$△CDA

 

We can formalize the above steps into a two-column proof where each line contains a geometric statement in the left column and a corresponding reason in the right column.

 

Two-column proof

Given the parallelogram $ABCD$ABCD, show that the opposite sides are congruent.
Statements	Reasons
$ABCD$ABCD is a parallelogram.	Given
$\overline{AB}\parallel\overline{DC}$AB∥DC and $\overline{AD}\parallel\overline{BC}$AD∥BC	Definition of a parallelogram
$\angle BAC\cong\angle DCA$∠BAC≅∠DCA and $\angle BCA\cong\angle DAC$∠BCA≅∠DAC	Alternate interior angles theorem
$\overline{AC}\cong\overline{AC}$AC≅AC	Reflexive property of congruence
$\triangle ABC\cong\triangle CDA$△ABC≅△CDA	Angle-side-angle congruence
$\overline{AB}\cong\overline{CD}$AB≅CD and $\overline{BC}\cong\overline{DA}$BC≅DA	Corresponding parts of congruent triangles are congruent (CPCTC)

The final line contains the statement that the opposite sides of the parallelogram are congruent, which is what we wanted to show.

We may be able to change the order of some lines without changing the validity of the proof, while there are other lines that require certain preceding statements. For example, the following line

$\overline{AC}\cong\overline{AC}$AC≅AC

Reflexive property of congruence

can be introduced at any point before the statement $\triangle ABC\cong\triangle CDA$△ABC≅△CDA with no effect on the proof. In contrast, the statement that $\triangle ABC\cong\triangle CDA$△ABC≅△CDA must appear after this line, and not before.

We summarize the properties of parallelograms as theorems below.

Practice questions

question 7

question 8