Section 6-2 Properties of Parallelograms SPI 32A: identify properties of plane figures from information in a diagram SPI 32 H: apply properties of quadrilaterals to solve a real-world problem Objectives: Use relationships among sides and angles of parallelogram Use relationships involving diagonals or transversals of parallelograms Definition: Parallelogram is a quadrilateral with both pairs of opposite sides parallel.

Parallelogram Theorem Given: ABCD Prove: ABCD is a parallelogram Given Def of Parallelogram  1   4  3   2 ∆ ABC  ∆ CDA If lines are  then alt int  s are  Reflexive Prop of  ASA  CPCTC || Thm 6-1: Opposite sides of a parallelogram are congruent.

Consecutive Angles Consecutive Angles: angles of polygons that share a side are same side interior angles are supplementary Supplementary Angles:  A and  B  B and  C  C and  D  D and  A Based on the definition of Consecutive Angles, which angles are supplementary?

Use KMOQ to find m O. Q and O are consecutive angles of KMOQ, so they are supplementary. Definition of supplementary angles m O + m Q = 180 Substitute 35 for m Q.m O + 35 = 180 Subtract 35 from each side.m O = 145 Using Consecutive Angles

Theorem 6-2 Opposite angles of a parallelogram are congruent Transitive Property

Find the value of x in ABCD. Then find m A. 2x + 15 = 135Add x to each side. 2x = 120Subtract 15 from each side. x = 60Divide each side by 2. x + 15 = 135 – xOpposite angles of a are congruent. Substitute 60 for x. m B = = 75 Consecutive angles of a parallelogram are supplementary. m A + m B = 180 Subtract 75 from each side.m A = 105 m A + 75 = 180Substitute 75 for m B. Using Algebra to find Angle Measures

Theorem 6-3 Diagonals of a parallelogram bisect each other  1  4 Alt Int Angles  2   3 Alt Int Angles MN  PQDef of Parallelogram ∆ MNR  ∆ PQRASA  MR  PR and NR  QR CPCTC Proof of Theorem 6-3

Theorem 6-4 If 3 or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal Diagram shows AC  CE. Since line BF is also a transversal, then BD  DF.

Find the values of x and y in KLMN. x = 7y – 16The diagonals of a parallelogram bisect each other. 2x + 5 = 5y 2(7y – 16) + 5 = 5ySubstitute 7y – 16 for x in the second equation to solve for y. 14y – = 5yDistribute. 14y – 27 = 5ySimplify. –27 = –9ySubtract 14y from each side. 3 = yDivide each side by –9. x = 7(3) – 16Substitute 3 for y in the first equation to solve for x. x = 5Simplify. So x = 5 and y = 3. Using Algebra to Find Measurements