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The Quadrilateral Family Tree

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The Quadrilateral Family Tree

Wednesday 1/5/11 –

Thursday 1/6/11

SPECIAL PARALLELOGRAMS

Rectangles, rhombuses, and squares

- A quadrilateral with four right angles

- 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.

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

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

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

- 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

- A quadrilateral with four congruent sides

- 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.

- What kind of triangles are formed by a diagonal?

- What kind of triangles are formed by a diagonal?

- What kind of triangles are formed by a diagonal?

- Each diagonal bisects a pair of opposite angles

- 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.

- What should be true about angles 1 and 2?

2

1

- What would the angle measures have to be?

2

1

- 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 mVTZ.

mVZT =90°

Rhombus diag.

14a + 20=90°

a=5

mVTZ =mZTX

mVTZ =(5a – 5)°

mVTZ =[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 mGCH.

mGCD = (b + 3)°

and mCDF = (6b – 40)°

Def. of rhombus

mGCD + mCDF = 180°

b + 3 + 6b –40 = 180°

7b = 217°

b = 31°

Check It Out! Example 2b Continued

CDFG is a rhombus.

Find mGCH.

mGCD = (b + 3)°

and mCDF = (6b – 40)°

mGCH + mHCD = mGCD

Rhombus each diag. bisects opp. s

2mGCH = mGCD

Substitute.

2mGCH = (b + 3)

Substitute.

2mGCH = (31 + 3)

Simplify and divide both sides by 2.

mGCH = 17°

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

- 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