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Understanding Properties of Parallelograms: Theorems, Proofs, and Examples

Explore the properties of parallelograms, including congruent sides and angles, parallel lines, and diagonal bisecting. Learn through theorems, proofs, and practical examples. Test your knowledge with quizzes and algebraic exercises.

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Understanding Properties of Parallelograms: Theorems, Proofs, and Examples

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  1. Chapter 6 lesson 2 Properties of Parallelograms

  2. Warm-up ASA HGE GHE HEG GH HE EG They are parallel.

  3. Theorem 6.1 • Opposite sides of a parallelogram are congruent.

  4. Consecutive Angles • Angles of a polygon that share a side are consecutive angles. • Because opposite sides of a parallelogram are parallel, consecutive angles are same-side interior angles • They are therefore SUPPLEMENTARY. • ∠a and ∠d are consecutive angles m∠a + m∠d = 180

  5. Theorem 6-2 • Opposite angles of a parallelogram are congruent Opposite angles are supplements of the same angle. Therefore, they are congruent.

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

  7. Proof of Theorem 6.3 • Given: Parallelogram ABCD Prove: AC and BD bisect each other at point O • If ABCD is a parallelogram, then AB and DC are parallel. • ∠1≅ ∠4 and ∠2≅ ∠3 because alternate Interior angles are congruent. • AB ≅ DC because opposite sides of a parallelogram are congruent. • ∆ADO ≅ ∆BCO by ASA • AE≅CE and BE≅DE by CPCTC 4 2 1 3

  8. 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.

  9. Example 1: Using Consecutive Angles What is the measure of ∠P? Consecutive angles are supplementary 64 + P = 180 P = 180 – 64 P = 116

  10. Your Turn! Consecutive angles are supplementary 86 + P = 180 P = 180 –86 P = 94

  11. Example 2: Proofs

  12. Proof #2

  13. Example 3: Algebra

  14. Your Turn! Algebra

  15. Example 4: Parallel Lines and Transversals

  16. Lesson Quiz

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