Topics
6. Quadrilaterals & Other Polygons
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United States of AmericaFL
Grade 912

6.02 Properties of parallelograms

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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

Outcomes

MA.912.GR.1.4

Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships and theorems of parallelograms.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

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