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The Quadrilateral Family Tree. Wednesday 1/5/11 – Thursday 1/6/11. SPECIAL PARALLELOGRAMS. Special Parallelograms. Rectangles, rhombuses, and squares. Rectangles. Rectangle. A quadrilateral with four right angles. Rectangle. Has all the properties of a parallelogram. Quadrilateral.

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The quadrilateral family tree

The Quadrilateral Family Tree

Wednesday 1/5/11 –

Thursday 1/6/11

SPECIAL PARALLELOGRAMS


Special parallelograms
Special Parallelograms

Rectangles, rhombuses, and squares



Rectangle
Rectangle

  • A quadrilateral with four right angles


Rectangle1
Rectangle

  • Has all the properties of a parallelogram


Quadrilateral

1. Four-sided polygon

Parallelogram

1.

2. Opposite angles are congruent

3. Diagonals bisect each other

4. Consecutive angles are supp.

RECTANGLE

1.

1.

1. Four right angles

2.

2.

3.

2. All properties above

3.

4.

3.

Square

1.

2.

1.

2.

1. Opposite sides are congruent

3.


Rectangle2
Rectangle

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


Rectangle3
Rectangle

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


Rectangle4
Rectangle

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


Rectangle5
Rectangle

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


Quadrilateral

1. Four-sided polygon

Parallelogram

1.

2. Opposite angles are congruent

3. Diagonals bisect each other

4. Consecutive angles are supp.

RECTANGLE

1.

1.

1. Four right angles

2.

2.

3.

2. All properties above

3.

4.

3. Diagonals are congruent

Square

1.

2.

1.

2.

1. Opposite sides are congruent

3.


diags. bisect each other

Example 1: Craft Application

A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.

Rect.  diags. 

KM = JL = 86


Check It Out! Example 1a

Carpentry The rectangular gate has diagonal braces.

Find HJ.

Rect.  diags. 

HJ = GK = 48


Check It Out! Example 1b

Carpentry The rectangular gate has diagonal braces.

Find HK.

Rect.  diags. 

Rect.  diagonals bisect each other

JL = LG

JG = 2JL = 2(30.8) = 61.6



Rhombus
Rhombus

  • A quadrilateral with four congruent sides


Rhombus1
Rhombus

  • Also has all the properties of a parallelogram


Quadrilateral

1. Four-sided polygon

Parallelogram

1.

2. Opposite angles are congruent

3. Diagonals bisect each other

4. Consecutive angles are supp.

RHOMBUS

RECTANGLE

1.

1. Four congruent sides

1. Four right angles

2. All properties above

2.

3.

2. All properties above

3.

4.

3. Diagonals are congruent

Square

1.

2.

1.

2.

1. Opposite sides are congruent

3.


Rhombus2
Rhombus

  • What kind of triangles are formed by a diagonal?


Rhombus3
Rhombus

  • What kind of triangles are formed by a diagonal?


Rhombus4
Rhombus

  • What kind of triangles are formed by a diagonal?


Rhombus5
Rhombus

  • Each diagonal bisects a pair of opposite angles


Rhombus6
Rhombus

  • Each diagonal bisects a pair of opposite angles


Quadrilateral

1. Four-sided polygon

Parallelogram

1.

2. Opposite angles are congruent

3. Diagonals bisect each other

4. Consecutive angles are supp.

RHOMBUS

1. Four congruent sides

RECTANGLE

1.

2. All properties above

1. Four right angles

2.

3. Diagonals bisect opposite angles

2. All properties above

3.

3. Diagonals are congruent

4.

Square

1.

2.

1.

2.

1. Opposite sides are congruent

3.



Rhombus8
Rhombus

  • What should be true about angles 1 and 2?

2

1


Rhombus9
Rhombus

  • What would the angle measures have to be?

2

1


Rhombus10
Rhombus

  • What would the angle measures have to be?


Quadrilateral

1. Four-sided polygon

Parallelogram

1.

2. Opposite angles are congruent

3. Diagonals bisect each other

4. Consecutive angles are supp.

RHOMBUS

1. Four congruent sides

RECTANGLE

1.

2. All properties above

1. Four right angles

2.

3. Diagonals bisect opposite angles

2. All properties above

3.

3. Diagonals are congruent

4. Diagonals are

Square

1.

2.

1.

2.

1. Opposite sides are congruent

3.


Example 2A: Using Properties of Rhombuses to Find Measures

TVWX is a rhombus. Find TV.

WV = XT

13b – 9=3b + 4

10b =13

b =1.3

TV = XT

TV =3b + 4

TV =3(1.3)+ 4 = 7.9


Example 2B: Using Properties of Rhombuses to Find Measures

TVWX is a rhombus. Find mVTZ.

mVZT =90°

Rhombus  diag. 

14a + 20=90°

a=5

mVTZ =mZTX

mVTZ =(5a – 5)°

mVTZ =[5(5) – 5)]°

= 20°


Check It Out! Example 2a

CDFG is a rhombus. Find CD.

CG = GF

Def. of rhombus

5a =3a + 17

a =8.5

GF = 3a + 17=42.5

CD = GF

CD = 42.5


Check It Out! Example 2b

CDFG is a rhombus.

Find mGCH.

mGCD = (b + 3)°

and mCDF = (6b – 40)°

Def. of rhombus

mGCD + mCDF = 180°

b + 3 + 6b –40 = 180°

7b = 217°

b = 31°


Check It Out! Example 2b Continued

CDFG is a rhombus.

Find mGCH.

mGCD = (b + 3)°

and mCDF = (6b – 40)°

mGCH + mHCD = mGCD

Rhombus  each diag. bisects opp. s

2mGCH = mGCD

Substitute.

2mGCH = (b + 3)

Substitute.

2mGCH = (31 + 3)

Simplify and divide both sides by 2.

mGCH = 17°



Square
Square

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


Square1
Square

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


Quadrilateral

1. Four-sided polygon

Parallelogram

1.

2. Opposite angles are congruent

3. Diagonals bisect each other

4. Consecutive angles are supp.

Rhombus

1. Four congruent sides

Rectangle

1.

2. All properties above

1. Four right angles

2.

3. Diagonals bisect opposite angles

2. All properties above

3.

SQUARE

3. Diagonals are congruent

4. Diagonals are

1. All properties of a rectangle

1.

2.

1. Opposite sides are congruent

3.

2. All properties of a rhombus


Parallelograms

Rectangles

Rhombuses

Squares


Quadrilateral

1. Four-sided polygon

Kite

TRAPEZOID

1.

PARALLELOGRAM

1. Opposite sides are congruent

1.

2.

2. Opposite angles are congruent

3.

3. Diagonals bisect each other

4. Consecutive angles are supplementary

Isosceles Trapezoid

Rhombus

1.

Rectangle

1. Properties of a parallelogram

1. All properties of a parallelogram

2. All sides are congruent

2.

3. Diagonals are perpendicular

2. Four right angles

3.

4. Diagonals bisect opposite angles

3. Diagonals are congruent

Square

1. Properties of a rectangle

2. Properties of a rhombus


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