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

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

Use the applet to complete the following sentences:

A quadrilateral is a parallelogram if and only if its opposite sides are â¬š.

A quadrilateral is a parallelogram if and only if its opposite angles are â¬š.

In a parallelogram, consecutive angles will be â¬š.

A quadrilateral is a parallelogram if and only if its diagonals â¬š each other.

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

The diagonal of a parallelogram is a transversal between a pair of parallel lines

The other diagonal of a parellelogram is also a transversal between a pair of parallel lines

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

Consider the quadrilateral shown:

a

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

Worked Solution

b

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

Worked Solution

c

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

Worked Solution

Consider the quadrilateral shown:

a

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

Worked Solution

b

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

Worked Solution

c

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

Worked Solution

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

Worked Solution

Idea summary

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

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

Find the missing parts of the parallelograms.

a

Given parallelogram PQRS, find RS.

Worked Solution

b

Given parallelogram DEFG, find m \angle DGF.

Worked Solution

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

a

Worked Solution

b

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c

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

Worked Solution

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