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Objectives: Use relationships among sides and angles of parallelogram

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:

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Objectives: Use relationships among sides and angles of parallelogram

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  1. Section 6-2 Properties of Parallelograms SPI 32A: identify properties of plane figures from information in a diagramSPI 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. 3 4 1 2

  2. Parallelogram Theorem Thm 6-1: Opposite sides of a parallelogram are congruent. Given: ABCD Prove: C B 3 4 1 2 D A || || If lines are  then alt int s are  If lines are  then alt int s are  Reflexive Prop of  ASA  CPCTC

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

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

  5. Theorem 6-2Opposite angles of a parallelogram are congruent. P N 3 4 1 2 Q M Transitive Property

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

  7. Theorem 6-3Diagonals of a parallelogram bisect each other. P N 2 4 R 1 3 Q M Proof of Theorem 6-3 1 4 Alt Int Angles 2  3 Alt Int Angles MN PQ Def of Parallelogram ∆MNR  ∆PQR ASA  MR  PR and NR  QR CPCTC

  8. Theorem 6-4If 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.

  9. x = 7y – 16 The diagonals of a parallelogram bisect each other. 2x + 5 = 5y 2(7y – 16) + 5 = 5y Substitute 7y – 16 for x in the second equation to solve for y. 14y – 32 + 5 = 5y Distribute. 14y – 27 = 5y Simplify. –27 = –9y Subtract 14y from each side. 3 = y Divide each side by –9. x = 7(3) – 16 Substitute 3 for y in the first equation to solve for x. x = 5 Simplify. Using Algebra to Find Measurements Find the values of x and y in KLMN. So x = 5 and y = 3.

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