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8.01 Properties of parallelograms

Lesson

Concept summary

Quadrilateral

A polygon with exactly four sides and four vertices.

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

A quadrilateral containing two pairs of parallel sides.

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.
Diagonal of a polygon

A line segment that connects the nonconsecutive vertices of a polygon.

A quadrilateral showing two line segments each connecting a pair of non-adjacent vertices. The two line segments intersect each other inside the quadrilateral.

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

Parallelograms have special properties regarding side lengths, angles, and diagonals. We can use these properties to find unknown angles or sides of parallelograms. The following are theorems about parallelograms:

Parallelogram opposite sides theorem

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

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

If a quadrilateral is a parallelogram, then 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 labelled A, B, C and D consecutively.
Parallelogram diagonals theorem

If a quadrilateral is a parallelogram, then 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.

To prove that a quadrilateral is a parallelogram, we can make use of the following theorems:

Parallelogram opposite sides converse

If both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.

A quadrilateral with 2 pairs of congruent opposite sides.
Parallelogram opposite angles converse

If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram.

A quadrilateral with 2 pairs of congruent opposite sides and 2 pairs of parallel sides.
Parallelogram consecutive angles converse

If an angle is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

Quadrilateral A B C D with consecutive angles C, A, and B marked. Angles B and C are congruent.
Parallelogram diagonals converse

If a quadrilateral has diagonals that bisect each other, then the quadrilateral is a parallelogram.

Quadrilateral A B C D with diagonals A D and B C intersecting at point E. Segments A E and E D are congruent, as well as segments B E and E C.
Opposite sides parallel and congruent theorem

If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

A quadrilateral with one pair of congruent and parallel opposite sides.

Worked examples

Example 1

Given parallelogram PQRS, find RS.

Parallelogram P Q R S with side Q R measuring 5.08 and side Q P measuring 2.41.

Approach

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

Solution

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

RS = PQ = 2.41

Example 2

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.

Approach

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.

Solution

\displaystyle (5x)+(2x+5)\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(25)=125.

m\angle DGF = 125 \degree

Example 3

Consider the quadrilateral ABCD with \overline{AD} \parallel \overline{BC} and \angle A \cong \angle C.

Determine whether ABCD is a parallelgram or not. Justify your answer.

Approach

We have many theorems that we can use to justify that a quadrilateral is a parallelogram. We need to decide which we want to use.

In this case, we have been told information about two of the four angles. If we knew some information about one of the other angles, we could use the parallelogram consecutive angles converse to decide if it is indeed a parallelogram or not.

Solution

Since we are told that \overline{AD} \parallel \overline{BC} we can use the Consecutive interior angles theorem to determine that \angle D and \angle C are supplementary.

Since \angle A \cong \angle C, this means that \angle D and \angle A are also supplementary.

Since we know that \angle D is supplementary to both of its consecutive angles, we have that the quadrilateral ABCD is a parallelogram using the parallelogram consecutive angles converse.

Outcomes

M2.G.CO.C.9

Use definitions and theorems about parallelograms to solve problems and to justify relationships in geometric figures.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP2

Reason abstractly and quantitatively.

M2.MP3

Construct viable arguments and critique the reasoning of others.

M2.MP5

Use appropriate tools strategically.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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