W ARM U P 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.

D EFINITIONS Rhombus A parallelogram with four congruent sides. Rectangle A parallelogram with four right angles. Square A parallelogram that is a rectangle and a rhombus.

C OROLLARY ABOUT S PECIAL Q UADRILATERALS 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.

G IVEN :ABCD IS A RHOMBUS. P ROVE : X A B C D 1. ABCD is a rhombus 2. AB ≅ CB 3. AX ≅ CX and BX ≅ DX 4.∆AXB ≅ ∆CXB 5. ∠ AXB ≅ ∠ CXB 6.AC ⊥ BD 1. Given 2. Definition of rhombus 3.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

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. A B C D AB CD

Theorem 6.13 A parallelogram is a rectangle if and only if its diagonals are congruent. ABCD is a rectangle if and only if. AB CD

LET’S CLARIFY… parallelograms rhombuses squares rectangles

S OMETIMES, A LWAYS, N EVER 1. A square is ___________ a rectangle. 2. A rhombus is _________ a square. 3. A rhombus is _________ a parallelogram. 4. A rectangle is _________ a rhombus. always sometimes always sometimes

M ORE D EFINITIONS 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.

C OOL D OWN Show that ABCD is a rhombus given A(-3,1), B(1,3), C(5,1), D(1,-1)