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Warm Up

Warm Up. In ABCD, m∠A = (3x+15) 0 and m∠C=(5x-17) 0 . What is the value of x? The vertices of PQRS are P(-1,-3), Q(2,-4),R(4,-1), and S(2,0). Is PQRS a parallelogram?. CH. 6.4 RHOMBUSES, RECTANGLES, AND SQUARES. Students will use properties of special quadrilaterals. Definitions.

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Warm Up

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  1. Warm Up • In ABCD, m∠A = (3x+15)0 and m∠C=(5x-17)0. What is the value of x? • The vertices of PQRS are P(-1,-3), Q(2,-4),R(4,-1), and S(2,0). Is PQRS a parallelogram?

  2. CH. 6.4 RHOMBUSES, RECTANGLES, AND SQUARES Students will use properties of special quadrilaterals.

  3. Definitions • Rhombus • A parallelogram with four congruent sides. • Rectangle • A parallelogram with four right angles. • Square • A parallelogram that is a rectangle and a rhombus.

  4. Corollary about Special Quadrilaterals • Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides. • Rectangle Corollary A quadrilateral is a rectangle if and only if it has four right angles. • Square Corollary A quadrilateral is a square if and only if it is a rhombus and a rectangle.

  5. Given:ABCD is a rhombus.Prove: A B X D C 1. Given • 1. ABCD is a rhombus • 2. AB≅CB • 3. AX≅CX and BX≅DX • ∆AXB ≅ ∆CXB • ∠AXB ≅ ∠CXB • AC ⊥ BD 2. Definition of rhombus • Since ABCD is a parallelogram, • its diagonals bisect each other 4. SSS 5. CPCTC 6. Th. 3.1 if two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular

  6. A B Theorem 6.11 • A parallelogram is a rhombus if and only if its diagonals are perpendicular. ABCD is a rhombus if and only if AC  BD. Theorem 6.12 • A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. • ABCD is a rhombus if and only if AC bisects DAB and BCD and BD bisects ADC and CBA. D C A B D C

  7. Theorem 6.13 • A parallelogram is a rectangle if and only if its diagonals are congruent. ABCD is a rectangle if and only if . A B D C

  8. LET’S CLARIFY… parallelograms rhombuses rectangles squares

  9. Sometimes, Always, Never • A square is ___________ a rectangle. • A rhombus is _________ a square. • A rhombus is _________ a parallelogram. • A rectangle is _________ a rhombus. always sometimes always sometimes

  10. More Definitions • trapezoid: a quadrilateral with exactly one pair of parallel sides • kite: a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

  11. Cool Down Show that ABCD is a rhombus given A(-3,1), B(1,3), C(5,1), D(1,-1)

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