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Understanding Parallelograms: Properties of Diagonals, Angles, and Sides

This lesson covers the fundamental properties of parallelograms, focusing on the relationships among diagonals, angles, and sides. Key theorems such as the congruence of opposite sides and angles are introduced. Theorem 6-1 explains that opposite sides are congruent while consecutive angles are supplementary. Theorem 6-2 states that opposite angles are congruent, and Theorem 6-3 illustrates that diagonals bisect each other. Multiple examples incorporate algebraic methods to solve for unknown variables within parallelograms, enhancing understanding of geometric principles.

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Understanding Parallelograms: Properties of Diagonals, Angles, and Sides

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  1. Chapter 6 Lesson 2 Objective: To use relationships among diagonals, angles and sides of parallelograms.

  2. Properties of Parallelograms Theorem 6-1 Opposite sides of a parallelogram are congruent. Angles of a polygon that share a side are consecutive angles. A parallelogram has opposite sides parallel. Its consecutive angles are same-side interior angles so they are supplementary. In    ABCD, consecutive angles B and C are supplementary, as are consecutive angles C and D.

  3. Example 1: Using Consecutive Angles Find mS in     RSTW .                 R and   S are consecutive angles of a parallelogram. They are supplementary.

  4. Example 2: Using Consecutive Angles Find mO in     KMOQ .                   Q and   O are consecutive angles of a parallelogram. They are supplementary. K M 35° Q O

  5. Theorem 6-2 Opposite angles of a parallelogram are congruent.

  6. Example 3: Using Algebra Find the value of x in      PQRS. Then find QR and PS. 

  7. Example 4: Using Algebra Find the value of y in parallelogram EFGH.

  8. Theorem 6-3 The diagonals of a parallelogram bisect each other.

  9. Example 5: Using Algebra Solve a system of linear equations to find the values of x and y in       ABCD. Then find AE, EC, BE, and ED. Step 1: Write equations. Diagonals bisect each other. Step 2:Solve for a variable and Substitute Step 3: Solve for variables

  10. Example 6: Using Algebra Find the values of a and b. Step 1: Write equations. Diagonals bisect each other. Step 2:Solve for a variable and Substitute Step 3: Solve for variables

  11. Theorem 6-4 If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

  12. Assignment

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