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Parallelograms

Parallelograms. Objectives. Recognize and apply properties of the sides and angles of parallelograms. Recognize and apply properties of the diagonals of parallelograms. Parallelograms. A quadrilateral with parallel opposite sides is called a parallelogram ( ABCD). A. B. D. C.

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Parallelograms

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  1. Parallelograms

  2. Objectives • Recognize and apply properties of the sides and angles of parallelograms. • Recognize and apply properties of the diagonals of parallelograms.

  3. Parallelograms • A quadrilateral with parallel opposite sides is called a parallelogram ( ABCD). A B D C

  4. Parallelograms Theorems • Theorem 6.3 – Opposite sides of are ≅. • Theorem 6.4 – Opposite s in are ≅. • Theorem 6.5 – Consecutive s in are supplementary. • Theorem 6.6 – If has 1 rt. , then it has 4 rt. s.

  5. Given: Prove: Example 1: Prove that if a parallelogram has two consecutive sides congruent, it has four sides congruent.

  6. Proof: Statements Reasons 1. 1. Given 2. 2. Given 3. 3. Opposite sides of a parallelogram are . 4. 4. Transitive Property Example 1:

  7. Prove that if and are the diagonals of , and Given: Prove: Your Turn:

  8. Proof: Statements Reasons 1. 1. Given 2. 2. Opposite sides of a parallelogram are congruent. 3. 3. If 2 lines are cut by a transversal, alternate interior s are . 4. 4. Angle-Side-Angle Your Turn:

  9. RSTU is a parallelogram. Find and y. If lines are cut by a transversal, alt. int. Example 2: Definition of congruent angles Substitution

  10. Example 2: Angle Addition Theorem Substitution Subtract 58 from each side.

  11. Answer: Example 2: Definition of congruent segments Substitution Divide each side by 3.

  12. ABCD is a parallelogram. Answer: Your Turn:

  13. Diagonals of Parallelograms • Theorem 6.7 – The diagonals of a bisect each other. • Theorem 6.8 – Each diagonal of a separates the into two ≅ ∆s.

  14. MULTIPLE-CHOICE TEST ITEM What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)? A B C D Read the Test ItemSince the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of Example 3:

  15. Find the midpoint of Midpoint Formula Example 3: Solve the Test Item The coordinates of the intersection of the diagonals of parallelogram MNPR are (1, 2). Answer: C

  16. MULTIPLE-CHOICE TEST ITEM What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with verticesL(0, –3), M(–2, 1), N(1, 5), O(3, 1)? A B C D Your Turn: Answer: B

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