6-2 Properties of Parallelograms

Quadrilaterals

• In a quadrilateral, opposite sides do not share a vertex and opposite angles do not share a side.– In other words, they are ACROSS from each other.

• Angles of a polygon that share a side are consecutive angles.

Parallelograms• A parallelogram is a quadrilateral

with both pairs of opposite sides parallel.

Theorem 6-3: If a quadrilateral is a parallelogram, then its opposite sides are congruent.

Theorem 6-4: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.

Theorem 6-5: If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Theorem 6-6: If a quadrilateral is a parallelogram, then its diagonals bisect each other.

Using Consecutive Angles

• What is mP?

Suppose you adjust the lamp so that mS = 86. What is mR?

Using Algebra to Find Lengths

• Solve a system of linear equations to find the values of x and y. What are KM and LN?

Find the values of x and y. What are PR and SQ?

Parallel Lines and Transversals

Theorem 6-7: If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

Using Parallel Lines and Transversals

• In the figure, marked lines are parallel, AB = BC = CD = 2, and EF = 2.25. What is EH?

If EF = FG = GH = 6 and AD = 15, what is CD?

6-3 Proving That a Quadrilateral Is a Parallelogram

Proving a Quadrilateral is a Parallelogram

Theorem 6-8: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 6-9: If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

Theorem 6-10: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 6-11: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorem 6-12: If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

Finding Values for Parallelograms

• For what value of y must PQRS be a parallelogram?

For what values of x and y must EFGH be a parallelogram?

Deciding Whether a Quadrilateral Is a Parallelogram

Can you prove that the quadrilateral is a parallelogram based on the given information?