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Understanding Special Quadrilaterals: Properties, Proofs, and Relationships

Explore the properties of special quadrilaterals such as rhombuses, rectangles, and squares in this lesson. Discover how their definitions define their characteristics, including congruent sides and angles. Dive into proof processes with given conditions, such as proving ABCD as a rhombus using congruence and perpendicularity of diagonals. Engage with theorems detailing relationships between different types of parallelograms. Enhance your understanding of quadrilateral properties and their applications in geometric problems.

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Understanding Special Quadrilaterals: Properties, Proofs, and Relationships

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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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