New York Geometry - 2020 Edition
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9.02 Properties of parallelograms
Lesson

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:

Properties of a parallelogram

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

If a quadrilaterals is a parallelogram, then its opposite angles are congruent.
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary ($x+y=180$x+y=180).
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
If a quadrilateral is a parallelogram, then its diagonals form two congruent triangles.

 

 

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 $x-5=10$x5=10, and solve for $x$x.

$x-5$x5 $=$= $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$2x1=7. We can then solve this equation for $x$x:

$2x-1$2x1 $=$= $7$7
$2x$2x $=$= $8$8
$x$x $=$= $4$4

 

Reflect: The length marked on the diagram is $2x-1$2x1. When $x=4$x=4 this length will be equal to $2\times4-1=7$2×41=7, which is the same length as the other half of the diagonal.

 

Practice questions

Question 4

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

  1. Enter each line of work as an equation.

Question 5

Consider the diagram below.

  1. Find the value of $a$a.

    Enter each line of work as an equation.

  2. Find the value of $b$b.

    Enter each line of work as an equation.

Question 6

The perimeter of the parallelogram given is $48$48 m.

  1. Find the value of $x$x in meters.

 

 

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 $\overline{AB}\parallel\overline{DC}$ABDC and $\overline{AD}\parallel\overline{BC}$ADBC.

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 $\angle BAC\cong\angle DCA$BACDCA and $\angle BCA\cong\angle DAC$BCADAC.

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

$\overline{AB}\parallel\overline{DC}$ABDC and $\overline{AD}\parallel\overline{BC}$ADBC

Definition of a parallelogram

$\angle BAC\cong\angle DCA$BACDCA and $\angle BCA\cong\angle DAC$BCADAC

Alternate interior angles theorem

$\overline{AC}\cong\overline{AC}$ACAC Reflexive property of congruence
$\triangle ABC\cong\triangle CDA$ABCCDA Angle-side-angle congruence
$\overline{AB}\cong\overline{CD}$ABCD and $\overline{BC}\cong\overline{DA}$BCDA

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

$\overline{AC}\cong\overline{AC}$ACAC Reflexive property of congruence

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

We summarize the properties of parallelograms as theorems below.

Theorems about parallelograms

If a quadrilateral is a parallelogram, then:

  • its opposite sides are congruent.
  • its opposite angles are congruent.
  • its consecutive angles are supplementary.
  • its diagonals bisect each other.
  • any diagonal forms two congruent triangles.

Practice questions

question 7

Given the proof below, select the correct statement and reason.

  1. Given the parallelogram $PQRS$PQRS, show that the opposite angles are congruent.

    Statements Reasons
    $PQRS$PQRS is a parallelogram. Given

    $\overline{PQ}\parallel\overline{SR}$PQSR and $\overline{PS}\parallel\overline{QR}$PSQR

    Definition of a parallelogram

    $\angle P$P and $\angle S$S, $\angle S$S and $\angle R$R and

    $\angle R$R and $\angle Q$Q are supplementary.

    Consecutive interior angles postulate
    $\left[\text{_____}\right]$[_____] $\left[\text{_____}\right]$[_____]
    $\angle P\cong\angle R$PR and $\angle Q\cong\angle S$QS

    Polygon interior angles sum theorem

    A
    $\angle P\cong\angle Q$PQ and $\angle R\cong\angle S$RS Congruent complement theorem
    B
    $\angle P\cong\angle R$PR and $\angle Q\cong\angle S$QS Congruent supplements theorem
    C
    $\angle P\cong\angle Q$PQ and $\angle R\cong\angle S$RS Congruent supplements theorem
    D
    $\angle P\cong\angle R$PR and $\angle Q\cong\angle S$QS

    Polygon interior angles sum theorem

    A
    $\angle P\cong\angle Q$PQ and $\angle R\cong\angle S$RS Congruent complement theorem
    B
    $\angle P\cong\angle R$PR and $\angle Q\cong\angle S$QS Congruent supplements theorem
    C
    $\angle P\cong\angle Q$PQ and $\angle R\cong\angle S$RS Congruent supplements theorem
    D

question 8

Select the error in the following proof.

  1.  

    Given the parallelogram $PQRS$PQRS, show that the opposite angles are congruent.

      Statements Reasons
    1. $PQRS$PQRS is a parallelogram. Given
    2.

    $\angle P$P and $\angle S$S$\angle S$S and $\angle R$R and 

    $\angle R$R and $\angle Q$Q are supplementary.

    Consecutive interior angles postulate
    3. $\overline{PQ}\parallel\overline{SR}$PQSR and $\overline{PS}\parallel\overline{QR}$PSQR Definition of a parallelogram
    4. $\angle P\cong\angle R$PR and $\angle Q\cong\angle S$QS Congruent supplements theorem

    The reason in line 3 uses the result of the proof, so it cannot be used.

    A

    Line 3 should follow after line 1.

    B

    Line 3 cannot come before line 4.

    C

    There are no errors.

    D

    Line 4 uses an incorrect reason.

    E

    The reason in line 3 uses the result of the proof, so it cannot be used.

    A

    Line 3 should follow after line 1.

    B

    Line 3 cannot come before line 4.

    C

    There are no errors.

    D

    Line 4 uses an incorrect reason.

    E

Outcomes

GEO-G.CO.11

Prove and apply theorems about parallelograms.

GEO-G.SRT.5a

Use congruence and similarity criteria for triangles to solve problems algebraically and geometrically.

GEO-G.SRT.5b

Use congruence and similarity criteria for triangles to prove relationships in geometric figures.

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