The Quadrilateral Family Tree

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Presentation transcript:

The Quadrilateral Family Tree Wednesday 12/3/08 – Thursday 12/4/08

1. _____________________ ____________ _________ ________________________ 2. _____________________ 3. _____________________ 4. _____________________ _____________________ 3. _______________ __________ __________________

Quadrilateral 1. Four-sided polygon __________ 1. _____________________ 2. _____________________ 3. _____________________ 4. _____________________ _____________________ 1. Four-sided polygon 1. _____________________ 2. _____________________ 3. _______________ __________ 1. _____________________ _____________________ 1. _____________________ 2. _____________________ 3. _____________________ __________________

Parallelograms

Parallelogram A quadrilateral with two pairs of parallel sides

Parallelogram QUADRILATERAL 1. Four-sided polygon Kite Trapezoid 2. 3. 4. Parallelogram Square Kite Isosceles Trapezoid

Parallelogram

Parallelogram

Parallelogram 2 1

Parallelogram 2 1

Parallelogram 3 2 1 4

Parallelogram 3 2 1 4

Parallelogram 3 2 1 4

Parallelogram 3 2 1 4

Parallelogram The two triangles formed by a diagonal are congruent by ASA. 3 2 1 4

Parallelogram What does that mean about opposite sides of a parallelogram? 3 2 1 4

Parallelogram What does that mean about opposite sides of a parallelogram? 3 2 1 4

Quadrilateral Kite Isosceles Trapezoid Rhombus Rectangle Square 1. TRAPEZOID 1. Four-sided polygon Quadrilateral 2. 3. 4. PARALLELOGRAM Square Rectangle Rhombus Kite Isosceles Trapezoid

Parallelogram QUADRILATERAL 1. Four-sided polygon 1. 2. 3. 4. Parallelogram Square 1. 1. Opposite sides are congruent 2. 3.

Parallelograms What about opposite angles in a parallelogram?

Parallelogram QUADRILATERAL 1. Four-sided polygon 1. 2. Opposite angles are congruent 3. 4. Parallelogram 2. Square 1. 1. Opposite sides are congruent 2. 3.

Parallelograms Are these two triangles congruent? By which postulate? 2 1

Parallelogram Since the triangles are congruent, the diagonals of the parallelogram bisect each other.

Parallelogram QUADRILATERAL 1. Four-sided polygon 1. 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Parallelogram 2. Square 3. 1. 1. Opposite sides are congruent 2. 3.

Parallelogram What kind of angles are angles 5 & 6? 5 6

Parallelogram Consecutive angles in a parallelogram are supplementary. 5 6

Parallelogram QUADRILATERAL 1. Four-sided polygon 1. 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 2. Square 3. 4. 1. 1. Opposite sides are congruent 2. 3.

Special Parallelograms Rectangles, rhombuses, and squares

Rectangles

Rectangle A quadrilateral with four right angles

Rectangle Has all the properties of a parallelogram

Quadrilateral Parallelogram RECTANGLE 1. Four-sided polygon 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 2. Square 2. All properties above 3. 1. Four right angles RECTANGLE 4. 1. Opposite sides are congruent

Rectangle What should be true about the overlapping triangles formed by the diagonals?

Rectangle What should be true about the overlapping triangles formed by the diagonals?

Rectangle The triangles are congruent by SAS since opposite sides of a parallelogram are congruent

Rectangle So the diagonals in a rectangle must be congruent by CPCTC

Quadrilateral Parallelogram RECTANGLE 1. Four-sided polygon 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 2. Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 3. 4. 1. Opposite sides are congruent

Rhombuses

Rhombus A quadrilateral with four congruent sides

Rhombus Also has all the properties of a parallelogram

Quadrilateral Parallelogram RHOMBUS RECTANGLE 1. Four-sided polygon 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 2. Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. 4. RHOMBUS 1. Opposite sides are congruent

Rhombus What kind of triangles are formed by a diagonal?

Rhombus What kind of triangles are formed by a diagonal?

Rhombus What kind of triangles are formed by a diagonal?

Rhombus Each diagonal bisects a pair of opposite angles

Rhombus Each diagonal bisects a pair of opposite angles

Quadrilateral Parallelogram RHOMBUS RECTANGLE 1. Four-sided polygon 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 2. Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. RHOMBUS 3. 1. Opposite sides are congruent

Rhombus

Rhombus What should be true about angles 1 and 2? 2 1

Rhombus What would the angle measures have to be? 2 1

Rhombus What would the angle measures have to be?

Quadrilateral Parallelogram RHOMBUS RECTANGLE 1. Four-sided polygon 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 2. Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS 3. 1. Opposite sides are congruent

Squares

Square A quadrilateral with four right angles (like a rectangle) and four congruent sides (like a rhombus)

Square So a square has all the properties of both a rectangle and a rhombus

Quadrilateral Parallelogram Rhombus Rectangle SQUARE 1. 1. Four-sided polygon Quadrilateral 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 1. All properties of a rectangle SQUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles Rectangle 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are Rhombus 2. 3. 1. Opposite sides are congruent 2. All properties of a rhombus

Parallelograms Rectangles Rhombuses Squares

Quadrilateral Kite Isosceles Trapezoid Rhombus Rectangle Square 1. TRAPEZOID 1. Four-sided polygon Quadrilateral 1. Opposite sides are congruent 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supplementary PARALLELOGRAM 2. Properties of a rhombus 1. Properties of a rectangle Square 2. Four right angles 3. Diagonals are congruent 1. All properties of a parallelogram Rectangle 1. Properties of a parallelogram 2. All sides are congruent 3. Diagonals are perpendicular 4. Diagonals bisect opposite angles Rhombus 2. 3. Kite Isosceles Trapezoid

Trapezoids

Trapezoid A quadrilateral with exactly one pair of parallel sides

Parallelogram Square TRAPEZOID RHOMBUS RECTANGLE QUADRILATERAL 1. One pair of parallel sides TRAPEZOID 1. Four-sided polygon QUADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 1. All properties of a rectangle Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS 1. 2. 3. 1. Opposite sides are congruent 2. All properties of a rhombus

Trapezoid If the legs are congruent, the trapezoid is an isosceles trapezoid

Quadrilateral Parallelogram Rhombus TRAPEZOID ISOSCELES TRAPEZOID 1. One pair of parallel sides TRAPEZOID 1. Four-sided polygon Quadrilateral 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 1. All properties of a rectangle SQUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are Rhombus 1. Legs are congruent 2. 3. ISOSCELES TRAPEZOID 1. Opposite sides are congruent 1. 2. All properties of a rhombus

Isosceles Trapezoid

Isosceles Trapezoid In an isosceles trapezoid, the base angles are congruent

Quadrilateral Parallelogram Rhombus TRAPEZOID ISOSCELES TRAPEZOID 1. One pair of parallel sides TRAPEZOID 1. Four-sided polygon Quadrilateral 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 1. All properties of a rectangle SQUARE 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are Rhombus 1. Legs are congruent 2. Base <s are congruent 3. ISOSCELES TRAPEZOID 1. Opposite sides are congruent 2. 1. 2. All properties of a rhombus

Isosceles Trapezoid What should be true about the diagonals of an isosceles trapezoid?

Isosceles Trapezoid What should be true about the diagonals of an isosceles trapezoid?

Quadrilateral Parallelogram Rhombus Square TRAPEZOID 1. One pair of parallel sides TRAPEZOID 1. Four-sided polygon Quadrilateral 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 1. All properties of a rectangle Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are Rhombus 1. Legs are congruent 2. Base <s are congruent ISOSCELES TRAPEZOID 1. Opposite sides are congruent 2. 3. 1. 2. All properties of a rhombus

Trapezoid Midsegment The segment formed by the midpoints of the legs

Trapezoid Midsegment Its length is the average of the two bases.

Trapezoid Midsegment Its length is the average of the two bases.

Kites

Kite A quadrilateral with exactly two pairs of congruent consecutive sides

1. 2 pairs of congruent consecutive sides 1. One pair of parallel sides Trapezoid 1. Four-sided polygon QUADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 1. All properties of a rectangle Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS Kite 1. Legs are congruent 2. Base <s are congruent Isosceles Trapezoid 1. Opposite sides are congruent 2. 3. 1. 2 pairs of congruent consecutive sides 2. All properties of a rhombus

Kite Diagonals are perpendicular

1. 2 pairs of congruent consecutive sides 1. One pair of parallel sides Trapezoid 1. Four-sided polygon QUADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 1. All properties of a rectangle Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS Kite 1. Legs are congruent 2. Base <s are congruent Isosceles Trapezoid 1. Opposite sides are congruent 2. Diagonals are _ 3. 1. 2 pairs of congruent consecutive sides 2. All properties of a rhombus

Kite Exactly one pair of opposite angles are congruent

3. One pair of opposite <s congruent 1. One pair of parallel sides Trapezoid 1. Four-sided polygon QUADRILATERAL 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supp. Parallelogram 1. All properties of a rectangle Square 2. All properties above 3. Diagonals are congruent 1. Four right angles RECTANGLE 1. Four congruent sides 3. Diagonals bisect opposite angles 4. Diagonals are RHOMBUS Kite 1. Legs are congruent 2. Base <s are congruent Isosceles Trapezoid 1. Opposite sides are congruent 2. Diagonals are _ 3. One pair of opposite <s congruent 1. 2 pairs of congruent consecutive sides 2. All properties of a rhombus

Quadrilateral Kite Isosceles Trapezoid Rhombus Rectangle Square 1. Exactly one pair of parallel sides TRAPEZOID 1. Four-sided polygon Quadrilateral 1. Opposite sides are congruent 2. Opposite angles are congruent 3. Diagonals bisect each other 4. Consecutive angles are supplementary PARALLELOGRAM 2. Properties of a rhombus 1. Properties of a rectangle Square 2. Four right angles 3. Diagonals are congruent 1. All properties of a parallelogram Rectangle 1. Properties of a parallelogram 2. All sides are congruent 3. Diagonals are perpendicular 4. Diagonals bisect opposite angles Rhombus 1. Exactly two pairs of congruent sides 2. Exactly one pair of congruent opposite angles 3. Diagonals are perpendicular Kite 1. Legs are congruent 2. Diagonals are congruent 3. Two pairs of congruent base angles Isosceles Trapezoid