2.2b: quadrilaterals m(g&m)–10–2 makes and defends conjectures, constructs geometric...

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2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving polygons GSE’s - Rectangle

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Page 1: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

2.2b: Quadrilaterals

M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving polygons

GSE’s

- Rectangle

Page 2: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

POLYGONS

Quadrilaterals

Parallelograms

RECTANGLES Has ALL the properties of the shapes above it

Page 3: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

In Addition to all those:

Rectangles also have:A

1) Congruent Diagonals

2) All 4 angles are right

What can we do to prove these on the coordinate plane?

BR

N

Use Distance Formula for AB and RN

Use slope to find if each consecutive side is opposite reciprocals

Page 4: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

Page 5: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

Solve for x and the m<ADB and m<BDC

Page 6: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

X, and the length of RB, VQ

Page 7: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

Page 8: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

Quadrilateral RSTU is a rectangle. If and find x.

The diagonals of a rectangle are congruent,

Definition of congruent segments

Substitution

Subtract 6x from each side.

Add 4 to each side.

Page 9: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

Quadrilateral LMNP is a rectangle. Find x and y.

908523 xx

90108 x808 x10x

322)10(3

588)10(5

NLPm

MLNm

5y

306y

3226y

AnglesInterior Alternate

PLNMNL

58 32

Page 10: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

Determine whether each statement is True or False. Explain

• If the quadrilateral is a rectangle, then it is a parallelogram

• If the quadrilateral is a parallelogram, then it is a rectangle.

• If a parallelogram has congruent diagonals, then it is a rectangle

• If a quadrilateral has congruent diagonals, then it is a rectangle

True, all rectangles are parallelograms

False, not all parallelograms are rectangles, they could be other shapes as well.

True, by definition, a parallelogram with congruent diagonals is a rectangle

False, a 4 sided figure (quadrilateral) could have congruent diagonals and not be a rectangle

Page 11: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

If you wanted to install a ceiling fan in your rectangular family room,Find where it would be in the room?

Page 12: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

The foundation for your house was poured, and you measured the outside walls to Be 40’ by 24’ . Does this make it rectangular? Why or why not?

Page 13: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

12

12

xx

yy

212

212 )()( yyxx

To determine if a parallelogram is a rectangle:

A) show the two diagonals are equal by the distance formula

Or B) show the four corners are perpendicular (90 degrees) by slopes

Page 14: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

Prove if it’s a rectangle.Easiest way = Find the length of the 2 diagonals (AC and BD)

212

212 )()( yyxx

Page 15: 2.2b: Quadrilaterals M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems

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