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

Rhombus A parallelogram with four congruent sides.

Rectangle A parallelogram with four right angles.

Square A parallelogram that is a rectangle and a

rhombus.

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.

GIVEN:ABCD IS A RHOMBUS.PROVE:

€

AC⊥BD

X

A B

CD

1. ABCD is a rhombus2. AB CB≅3. AX CX and BX DX≅ ≅ 4.∆AXB ∆CXB≅5.∠AXB CXB≅ ∠6.AC BD⊥

1. Given2. Definition of rhombus

3. Since ABCD is a parallelogram,its diagonals bisect each other

4. SSS5. 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

CD

A B

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 .

€

AC ≅ BD

A B

CD

LET’S CLARIFY…

parallelograms

rhombuses

squares

rectangles

SOMETIMES, ALWAYS, NEVER

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

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.

COOL DOWN

Show that ABCD is a rhombus givenA(-3,1), B(1,3), C(5,1), D(1,-1)