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9. Polygons
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Geometry

9.02 Parallelograms

Lesson
Practice

Explore properties of parallelograms

Quadrilateral

A polygon with exactly four sides and four vertices

A set of different kinds of four-sided polygons.
Parallelogram

A quadrilateral with both pairs of opposite sides parallel.

A polygon showing one pair of opposite sides marked with single parallel markings, and the other pair of opposite sides marked with double parallel markings.

Consecutive angles are angles of a polygon that share a side.

Exploration

Drag the points to change the quadrilateral and use the checkboxes to explore the applet.

Loading interactive...

Use the applet to complete the following sentences:

  1. A quadrilateral is a parallelogram if and only if its opposite sides are ⬚.

  2. A quadrilateral is a parallelogram if and only if its opposite angles are ⬚.

  3. In a parallelogram, consecutive angles will be ⬚.

  4. A quadrilateral is a parallelogram if and only if its diagonals ⬚ each other.

We have many tools in our mathematical tool box to help with proofs now, for example:

A parallelogram with diagonals drawn. the top and bottom sides, together with the diagonal from upper right to bottom left are highlighted.
The diagonal of a parallelogram is a transversal between a pair of parallel lines
A parallelogram with diagonals drawn. the top and bottom sides, together with the diagonal from upper left to bottom right are highlighted.
The other diagonal of a parellelogram is also a transversal between a pair of parallel lines
Parallelogram ABCD. See your teacher for more info.

If we extend the sides of a parallelogram, one pair of sides can be seen as transversals of the other pair of parallel sides.

We can utilize congruent triangle theorems since the diagonals of a parallelogram break it into triangles:

  • Side-side-side, or SSS: The two triangles have three pairs of congruent sides

  • Side-angle-side, or SAS: The two triangles have two pairs of congruent sides, and the angles between these sides are also congruent

  • Angle-side-angle, or ASA: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent

  • Angle-Angle-Side, or AAS: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent

Examples

Example 1

Consider the quadrilateral shown:

Quadrilateral ABCD with a diagonal drawn from vertex A to C.
a

If ABCD is a parallelogram, prove the opposite sides are congruent.

Worked Solution
Create a strategy

Start by stating what is given and the definition of a parallelogram to build the proof.

Apply the idea
To prove: Opposite sides of a parallelogram are congruent
StatementsReasons
1.ABCD is a parallelogramGiven
2.\overline{AB}\parallel \overline{DC} \text{ and } \overline{AD}\parallel \overline{BC}Definition of a parallelogram
3.\angle{BAC}\cong\angle{DCA} \text{ and } \angle{ACB}\cong\angle{CAD}Alternate interior angles theorem
4.\overline{AC}\cong \overline{AC}Reflexive property of congruence
5.\triangle{ABC}\cong \triangle{CDA}ASA congruence theorem
6. \overline{AB}\cong\overline{CD} \text{ and } \overline{BC}\cong \overline{DA}CPCTC
b

Prove that if opposite sides of a quadrilateral are congruent, then it is a parallelogram.

Worked Solution
Create a strategy

Construct a diagonal from B to D. Include congruence markings based on what is given.

Quadrilateral ABCD with diagonals drawn and congruent side markings. Diagonal BD is highlighted.
Apply the idea
To prove: If opposite sides of a quadrilateral are congruent, then it is a parallelogram
StatementsReasons
1.\overline{AB}\cong\overline{CD} \text{ and } \overline{BC}\cong \overline{DA}Given
2.\overline{BD}\cong \overline{BD}Reflexive property of congruence
3.\triangle{ABD}\cong \triangle{CDB}SSS congruence theorem
4.\angle ABD \cong \angle CDB and \angle ADB \cong \angle CBDCPCTC
5.\overline{AB} \parallel \overline {CD}Since \angle ABD \cong \angle CDB by converse of alternate interior angles theorem
6.\overline{AD} \parallel \overline {BC}Since \angle ADB \cong \angle CBD by converse of alternate interior angles theorem
7. ABCD is a parallelogramDefinition of a parallelogram
c

Verify the opposite sides of a parallelogram are congruent using constructions.

Worked Solution
Create a strategy

Construct a copy of the opposite sides to two consecutive sides to show that the parallelogram's opposite sides are congruent.

Apply the idea
Quadrilateral ABCD with diagonal AC. Ask your teacher for more info.
Quadrilateral ABCD with diagonal AC. Ask your teacher for more info.

If the other end of the compass lines up with C, then \overline{AB} is congruent to \overline{CD}.

Now let's check the other pair of sides.

Quadrilateral ABCD with diagonal AC. Ask your teacher for more info.
Quadrilateral ABCD with diagonal AC. Ask your teacher for more info.

If the other end of the compass lines up with C, then \overline{AD} is congruent to \overline{BC}.

Example 2

Consider the quadrilateral shown:

Quadrilateral ABCD with diagonals drawn intersecting at P.
a

If ABCD is a parallelogram, prove that the diagonals bisect each other.

Worked Solution
Create a strategy

Use the alternate interior angles theorem, congruence theorems, and the definition of a bisector to help prove that the diagonals bisect each other.

Apply the idea
A diagram showing proof that quadrilateral ABCD is a parallelogram. See you teacher for more info.
b

Prove that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Worked Solution
Create a strategy

Draw congruence markings using the given diagram to support starting a proof.

Quadrilateral ABCD with diagonals drawn intersecting at P. congruent markings are drawn on each half of a diagonal.
Apply the idea
To prove: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram
StatementsReasons
1.\overline{AP} \cong \overline {CP} and \overline{BP} \cong \overline {DP}Given
2.\angle APB \cong \angle CPD and \angle APD \cong \angle BPCVertical angles theorem
3.\triangle {ABP} \cong \triangle{CDP} and \triangle {APD} \cong \triangle {CPB}SAS congruence theorem
4.\overline{AB} \cong \overline{CD} and \overline{AD} \cong \overline{BC}CPCTC
5.\angle ABP \cong \angle CDP and \angle ADP \cong \angle CBPCPCTC
6.\overline{AB} \parallel \overline{CD} and \overline{AD} \parallel \overline{BC}Since \angle ABP \cong \angle CDP and \angle ADP \cong \angle CBP by converse of alternate interior angles theorem
7.ABCD is a parallelogramDefinition of a parallelogram
c

Verify that both halves of each diagonal are congruent using constructions.

Worked Solution
Create a strategy

Construct a copy of one half of the diagonals for each diagonal to show that both halves of each diagonal in a parallelogram are congruent.

Apply the idea
Quadrilateral ABCD. Diagonals AC and BD are intersecting on P. Ask your teacher for more info.
Quadrilateral ABCD. Diagonals AC and BD are intersecting on P. Ask your teacher for more info.

If the other end of the compass lines up with P, then \overline{AP} is congruent to \overline{CP}.

Now let's check the other diagonal.

Quadrilateral ABCD. Diagonals AC and BD are intersecting on P. Ask your teacher for more info.
Quadrilateral ABCD. Diagonals AC and BD are intersecting on P. Ask your teacher for more info.

If the other end of the compass lines up with P, then \overline{BP} is congruent to \overline{DP}.

Example 3

Use geometric constructions to show that ABCD is a parallelogram if its opposite angles are congruent.

Parallelogram A B C D.
Worked Solution
Create a strategy

Construct a copy of the opposite angles to two consecutive angles to show that if the quadrilateral's opposite angles are congruent, it must be a parallelogram.

Apply the idea
Quadrilateral ABCD.
Quadrilateral ABCD. See your teacher for more info.
Quadrilateral ABCD. See your teacher for more info.
Quadrilateral ABCD. See your teacher for more info.

If the constructed copy of \angle{A} lines up with \angle{C}, then \angle{A} is congruent to \angle{C}.

Quadrilateral ABCD.
Quadrilateral ABCD. See your teacher for more info.
Quadrilateral ABCD. See your teacher for more info.
Quadrilateral ABCD. See your teacher for more info.

If the constructed copy of \angle{B} lines up with \angle{D}, then \angle{B} is congruent to \angle{D}.

Idea summary

We can use the definition of a parallelogram, theorems about congruency, and transversals to prove properties of parallelograms.

Apply properties of parallelograms

Parallelograms have special properties regarding side lengths, angles, and diagonals. We can use these properties to find unknown angles or sides of parallelograms, or to prove that a quadrilateral is a parallelogram.

Parallelogram opposite sides theorem

A quadrilateral is a parallelogram if and only if its opposite sides are congruent

A parallelogram with opposite sides marked congruent with each other.
Parallelogram opposite angles theorem

A quadrilateral is a parallelogram if and only if its opposite angles are congruent

A parallelogram with opposite angles marked as congruent.
Parallelogram consecutive angles theorem

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

Example: \angle ADC and \angle DAB are supplementary

Parallelogram A B C D with vertices labeled A, B, C and D consecutively.
Parallelogram diagonals theorem

A quadrilateral is a parallelogram if and only if its diagonals bisect each other

Parallelogram A B C D with diagonal A C and B D bisecting each other and intersecting each other at point E. A E and E C are marked congruent as well as B E and D E.

We may use these properties to solve problems when we are told that a diagram is a parallelogram.

Examples

Example 4

Find the missing parts of the parallelograms.

a

Given parallelogram PQRS, find RS.

Parallelogram P Q R S with side Q R measuring 5.08 and side Q P measuring 2.41.
Worked Solution
Create a strategy

Since we know PQRS is a parallelogram, we want to use the theorems about parallelograms to determine RS.

Apply the idea

Opposite sides of a parallelogram are congruent so \overline{RS} \cong \overline{PQ}.

RS = PQ = 2.41

b

Given parallelogram DEFG, find m \angle DGF.

Parallelogram D E F G with angle D E F measuring quantity 5x degrees and angle E D C measuring quantity 2 x plus 5 degrees.
Worked Solution
Create a strategy

The two labeled angles, \angle DEF and \angle EDG, are consecutive angles. Since DEFG is a parallelogram, the consecutive angles are supplementary.

We want to write an equation relating the two labeled angles and then solve for x.

Once we solve for x, we then want to use the theorem that states that opposite angles of a parallelogram are congruent. Using this theorem, we know that \angle DEF \cong \angle DGF.

We want to substitute the value we solved for x and into 5x and evaluate m \angle DEF as this will be the same as m \angle DGF.

Apply the idea
\displaystyle \left(5x\right)+\left(2x+5\right)\displaystyle =\displaystyle 180Consecutive angles are supplementary
\displaystyle 7x+5\displaystyle =\displaystyle 180Combine like terms
\displaystyle 7x\displaystyle =\displaystyle 175Subtract 5 from both sides of equation
\displaystyle x\displaystyle =\displaystyle 25Divide both sides of equation by 7

Since we know that \angle DEF \cong \angle DGF, we know that \angle DGF = 5x

Substituting 25 for x and evaluating, we get 5\left(25\right)=125.

m\angle DGF = 125 \degree

Example 5

Determine whether or not each of the given quadrilaterals is a parallelogram.

a
A parallelogram with angle labels of 101 degrees, 79 degrees, and 101 degrees. One angle measurement is not indicated.
Worked Solution
Create a strategy

We know that if a quadrilateral is a parallelogram, its opposite angles are congruent and its consecutive angles are supplementary.

Use the polygon angle sum theorem to find the missing angle and determine if the quadrilateral satisfies the conditions of a parallelogram.

Apply the idea

By the polygon angle sum theorem, we know that the sum of the angles of an quadrilateral must be \left(4-2\right) \cdot 180 \degree = 2 \cdot 180 \degree = 360 \degree. Let the missing angle be x.

For the given quadrilateral, we have

\displaystyle 79 + 101 + 101\ + \ x\displaystyle =\displaystyle 360Polygon angle sum theorem
\displaystyle 281 \ + \ x\displaystyle =\displaystyle 360Combine like terms
\displaystyle x\displaystyle =\displaystyle 79Subtract 281 from both sides

Since the unknown angle in the quadrilateral is 79 \degree, we know that the opposite angles are congruent and therefore the quadrilateral is a parallelogram.

Reflect and check

We can also use that consecutive angles are supplementary in a parallelogram, so {101 \degree + \ x = 180 \degree} and the unknown angle must be 79 \degree, so we have that opposite angles are congruent and therefore the quadrilateral is a parallelogram.

b
A polygon with measurements of 42m and 30m for the diagonal from top left to bottom right, and 30m and 42m for the diagonal from top right to bottom left.
Worked Solution
Apply the idea

Since 42 \text{ m} \neq 30 \text{ m}, the diagonals of the quadrilateral do not bisect each other and the quadrilateral is not a parallelogram.

Reflect and check

If the segments were instead the same length along each diagonal, we could use the fact that if a quadrilateral is a parallelogram, then its diagonals bisect each other.

c
A polygon with all sides measuring 15ft, no angle is equal to 90 degrees
Worked Solution
Create a strategy

We aren't given information about the angles or diagonals of the quadrilateral, so we rely on determining if the quadrilateral meets the criteria: If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Apply the idea

Since the opposite sides of the quadrilateral are congruent, the quadrilateral is a parallelogram.

Example 6

Solve for the unknown variables in the diagram that make the quadrilateral a parallelogram.

A quadrilateral with with diagonals drawn. See your teacher for more info.
Worked Solution
Create a strategy

We can use the fact that the diagonals of the parallelogram form transversals, so we can use the alternate interior angles theorem to state that 32 = 4z.

We know that diagonals of a parallelogram bisect each other, so 4y + 6 = 5y - 5.

Since opposite sides of a parallelogram are congruent, 4x = 16.

Apply the idea

We have

\displaystyle 32\displaystyle =\displaystyle 4zAlternate interior angles theorem
\displaystyle 8\displaystyle =\displaystyle zDivide both sides by 4
\displaystyle 4y+6\displaystyle =\displaystyle 5y-5Diagonals of a parallelogram bisect each other
\displaystyle 4y+11\displaystyle =\displaystyle 5yAdd 5 to both sides
\displaystyle 11\displaystyle =\displaystyle ySubtract 4y from both sides
\displaystyle 4x\displaystyle =\displaystyle 16Opposite sides of a parallelogram are congruent
\displaystyle x\displaystyle =\displaystyle 4Divide both sides by 4
Idea summary

Use the following about quadrilaterals to solve problems involving parallelograms:

  • A quadrilateral is a parallelogram if and only if its opposite sides are congruent

  • A quadrilateral is a parallelogram if and only if its opposite angles are congruent

  • In a parallelogram, consecutive angles will be supplementary

  • A quadrilateral is a parallelogram if and only if its diagonals bisect each other

Outcomes

G.PC.1

The student will prove and justify theorems and properties of quadrilaterals, and verify and use properties of quadrilaterals to solve problems, including the relationships between the sides, angles, and diagonals.

G.PC.1a

Solve problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids.

G.PC.1b

Prove and justify that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the slope formula, the distance formula, and the midpoint formula.

G.PC.1c

Prove and justify theorems and properties of quadrilaterals using deductive reasoning.

G.PC.1d

Use congruent segment, congruent angle, angle bisector, perpendicular line, and/or parallel line constructions to verify properties of quadrilaterals.

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