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 m  P?  Suppose you adjust the lamp so that m  S = 86 . What is m  R?

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 = 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?