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6.4 Special Parallelograms

6.4 Special Parallelograms. Standard: 7.0 & 12.0. Properties of Special Parallelograms. T ypes of parallelograms: rhombuses, rectangles and squares. . A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four congruent sides.

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6.4 Special Parallelograms

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  1. 6.4 Special Parallelograms Standard: 7.0 & 12.0

  2. Properties of Special Parallelograms • Types of parallelograms: rhombuses, rectangles and squares. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four congruent sides A square is a parallelogram with four congruent sides and four right angles.

  3. Ex. 1: Using properties of special parallelogramsRhombus • Theorem 6-9 Each diagonal bisect two angles of the rhombus. • Theorem 6-10 The diagonals of a rhombus are perpendicular.

  4. Theorem 6-11 Diagonals of a rectangle are congruent. Ex. 2: Using properties of special parallelogramsRectangle

  5. Ex. 3: Using properties of a Rhombus • In the diagram at the right, PQRS is a rhombus. What is the value of y? All four sides of a rhombus are ≅, so RS = PS. 5y – 6 = 2y + 3 5y = 2y + 9 Add 6 to each side. 3y = 9 Subtract 2y from each side. y = 3 Divide each side by 3.

  6. Statements: ABCD is a rhombus AB ≅ CB AX ≅ CX BX ≅ DX ∆AXB ≅ ∆CXB AXB ≅ CXB AC  BD Reasons: Given Ex. 4: Proving Theorem 6.10Given: ABCD is a rhombusProve: AC  BD

  7. Statements: ABCD is a rhombus AB ≅ CB AX ≅ CX BX ≅ DX ∆AXB ≅ ∆CXB AXB ≅ CXB AC  BD Reasons: Given Given Ex. 4: Proving Theorem 6.10Given: ABCD is a rhombusProve: AC  BD

  8. Statements: ABCD is a rhombus AB ≅ CB AX ≅ CX BX ≅ DX ∆AXB ≅ ∆CXB AXB ≅ CXB AC  BD Reasons: Given Given Diagonals bisect each other. Ex. 4: Proving Theorem 6.10Given: ABCD is a rhombusProve: AC  BD

  9. Statements: ABCD is a rhombus AB ≅ CB AX ≅ CX BX ≅ DX ∆AXB ≅ ∆CXB AXB ≅ CXB AC  BD Reasons: Given Given Diagonals bisect each other. Diagonals bisect each other. Ex. 4: Proving Theorem 6.10Given: ABCD is a rhombusProve: AC  BD

  10. Statements: ABCD is a rhombus AB ≅ CB AX ≅ CX BX ≅ DX ∆AXB ≅ ∆CXB AXB ≅ CXB AC  BD Reasons: Given Given Diagonals bisect each other. Diagonals bisect each other. SSS congruence post. Ex. 4: Proving Theorem 6.10Given: ABCD is a rhombusProve: AC  BD

  11. Statements: ABCD is a rhombus AB ≅ CB AX ≅ CX BX ≅ DX ∆AXB ≅ ∆CXB AXB ≅ CXB AC  BD Reasons: Given Given Diagonals bisect each other. Diagonals bisect each other. SSS CPCTC Ex. 4: Proving Theorem 6.10Given: ABCD is a rhombusProve: AC  BD

  12. Statements: ABCD is a rhombus AB ≅ CB AX ≅ CX BX ≅ DX ∆AXB ≅ ∆CXB AXB ≅ CXB AC  BD Reasons: Given Given Diagonals bisect each other. Diagonals bisect each other. SSS CPCTC Congruent Adjacent s Ex. 4: Proving Theorem 6.10Given: ABCD is a rhombusProve: AC  BD

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