Geometric Conclusions Determine if each statement is a SOMETIMES, ALWAYS, or NEVER.

Geometric ConclusionsGeometric Conclusions

Determine if each statement is a Determine if each statement is a SOMETIMES, ALWAYS, or NEVERSOMETIMES, ALWAYS, or NEVER

Who Am I?Who Am I?

My total angle measure is 360˚.My total angle measure is 360˚. All of my sides are different lengths.All of my sides are different lengths. I have no right angles.I have no right angles.

Who Am I? Who Am I?

I have no right anglesMy total angle measure is not 360˚I have fewer than 3 congruent sides.

Who Am I?Who Am I?

My total angle measure is 360˚ or less.My total angle measure is 360˚ or less. I have at least one right angle.I have at least one right angle. I have more than one pair of congruent sides.I have more than one pair of congruent sides.

Who Am I?Who Am I?

I have at least one pair of parallel sides. I have at least one pair of parallel sides. My total angle measure is 360˚.My total angle measure is 360˚. No side is perpendicular to any other side. No side is perpendicular to any other side.

Types of curvesTypes of curves

simple curves:simple curves: A curve is simple if it does A curve is simple if it does not cross itself.not cross itself.

Types of CurvesTypes of Curves

closed curvesclosed curves: a closed curve is a curve : a closed curve is a curve with no endpoints and which completely with no endpoints and which completely encloses an areaencloses an area

Types of CurvesTypes of Curves

convex curveconvex curve: If a plane closed curve be : If a plane closed curve be such that a straight line can cut it in at such that a straight line can cut it in at most two points, it is called a convex most two points, it is called a convex curve.curve.

Convex Curves

Not Convex Curves

Triangle DiscoveriesTriangle Discoveries

Work with a part to see what discoveries can Work with a part to see what discoveries can you make about triangles.you make about triangles.

Types of TrianglesTypes of Triangles

Classified by AnglesClassified by Angles Equiangular: all angles congruentEquiangular: all angles congruent Acute: all angles acuteAcute: all angles acute Obtuse: one obtuse angleObtuse: one obtuse angle Right: one right angleRight: one right angle

Classified by SidesClassified by Sides Equilateral: all sides congruentEquilateral: all sides congruent Isosceles: at least two sides congruentIsosceles: at least two sides congruent Scalene: no sides congruentScalene: no sides congruent

TrianglesTriangles

Isosceles (at least two sides equal)

Scalene

(No sides equal)

Equilateral (all sides equal)

What’s possible?What’s possible?

EquilateralEquilateral IsoscelesIsosceles ScaleneScalene

EquiangularEquiangular

AcuteAcute

RightRight

ObtuseObtuse

NO

NO

NO

HomeworkHomework

Textbook pages 444-446 Textbook pages 444-446

#9-12, #23-26, #49-52#9-12, #23-26, #49-52

Pythagorean TheoremPythagorean Theorem

a2

b2

c2

a2 + b2 = c2


http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm







Testing for acute, obtuse, rightTesting for acute, obtuse, right

Pythagorean theorem says: Pythagorean theorem says:

What happens if What happens if

or or

a2 + b2 = c2

a2 + b2 > c2

a2 + b2 < c2

Testing for acute, obtuse, rightTesting for acute, obtuse, right

Right triangle: Right triangle:

Acute triangle:Acute triangle:

Obtuse triangle: Obtuse triangle:

a2 + b2 = c2

a2 + b2 > c2

a2 + b2 < c2

Types of AnglesTypes of Angles

Website Website

www.mrperezonlinemathtutor.comwww.mrperezonlinemathtutor.com Complementary Complementary SupplementarySupplementary AdjacentAdjacent VerticalVertical

TransversalsTransversals

Let’s check the homework!Let’s check the homework!

Textbook pages 444-446 Textbook pages 444-446

#9-12, #23-26, #49-52#9-12, #23-26, #49-52

What is the value of x?What is the value of x?

2x + 5

3x + 10

Angles in pattern blocksAngles in pattern blocks

DiagonalsDiagonals

Joining two nonadjacent vertices of a Joining two nonadjacent vertices of a polygonpolygon

For which shapes will the diagonals For which shapes will the diagonals always be perpendicular?always be perpendicular?

Type of Type of QuadrilateralQuadrilateral

Are diagonals Are diagonals perpendicular?perpendicular?

TrapezoidTrapezoid

ParallelogramParallelogram

RhombusRhombus

RectangleRectangle

SquareSquare

KiteKite




TrapezoidTrapezoid maybemaybe

ParallelogramParallelogram

RhombusRhombus

RectangleRectangle

SquareSquare

KiteKite





ParallelogramParallelogram maybemaybe

RhombusRhombus

RectangleRectangle

SquareSquare

KiteKite






RhombusRhombus yesyes

RectangleRectangle

SquareSquare

KiteKite







RectangleRectangle maybemaybe

SquareSquare

KiteKite








SquareSquare yesyes

KiteKite








SquareSquare yesyes

KiteKite yesyes

If m<A = 140If m<A = 140°, what is the m<B, m<C and °, what is the m<B, m<C and m<D?m<D?

A

D C

B

If m<D = 75If m<D = 75°, what is the m<B, m<C and °, what is the m<B, m<C and m<A?m<A?

A

DC

B

Sum of the angles of a polygonSum of the angles of a polygonUse a minimum of five polygon pieces to create a 5-sided, 6-sided, 7 sided, 8-sided, 9-sided, 10-sided, 11-sided, or 12-sided figure. Trace on triangle grid paper, cut out, mark and measure the total angles in the figure.

2

1 34

5

7

6

9

5

8

1

2

34

6

7

http://www.arcytech.org/java/patterns/patterns_j.shtml

Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #

sidessidesTotal degreesTotal degrees

TriangleTriangle 33

QuadrilateralQuadrilateral 44

PentagonPentagon 55

HexagonHexagon 66

HeptagonHeptagon 77

OctagonOctagon 88

NonagonNonagon 99

DecagonDecagon 1010

UndecagonUndecagon 1111

DodecagonDodecagon 1212

Triskaidecagon Triskaidecagon 1313

NthNth NN

What patterns do you see?



TriangleTriangle 33 180180

QuadrilateralQuadrilateral 44

PentagonPentagon 55

HexagonHexagon 66

HeptagonHeptagon 77

OctagonOctagon 88

NonagonNonagon 99

DecagonDecagon 1010




nnthth nn





QuadrilateralQuadrilateral 44 360360

PentagonPentagon 55

HexagonHexagon 66

HeptagonHeptagon 77

OctagonOctagon 88

NonagonNonagon 99

DecagonDecagon 1010




nnthth nn






PentagonPentagon 55 540540

HexagonHexagon 66

HeptagonHeptagon 77

OctagonOctagon 88

NonagonNonagon 99

DecagonDecagon 1010




nnthth nn







HexagonHexagon 66 720720

HeptagonHeptagon 77

OctagonOctagon 88

NonagonNonagon 99

DecagonDecagon 1010




nnthth nn








HeptagonHeptagon 77 900900

OctagonOctagon 88

NonagonNonagon 99

DecagonDecagon 1010




nnthth nn









OctagonOctagon 88 10801080

NonagonNonagon 99

DecagonDecagon 1010




nnthth nn










NonagonNonagon 99 12601260

DecagonDecagon 1010




nnthth nn











DecagonDecagon 1010 14401440




nnthth nn












UndecagonUndecagon 1111 16201620



nnthth nn













DodecagonDodecagon 1212 18001800


nnthth nn














Triskaidecagon Triskaidecagon 1313 19801980

nnthth nn















nnthth nn ??















nnthth nn 180(n-2)180(n-2)


Total degree of angles in polygonTotal degree of angles in polygon

Area IdeasArea Ideas TrianglesTriangles ParallelogramsParallelograms TrapezoidsTrapezoids Irregular figuresIrregular figures

Area Formulas: TriangleArea Formulas: Triangle

http://illuminations.nctm.org/LessonDetail.aspx?ID=L577http://illuminations.nctm.org/LessonDetail.aspx?ID=L577

Area Formulas: TriangleArea Formulas: Triangle1. Using a ruler, draw a 1. Using a ruler, draw a

diagonal (from one corner diagonal (from one corner to the opposite corner) on to the opposite corner) on shapes A, B, and C.shapes A, B, and C.

2. Along the top edge of shape 2. Along the top edge of shape D, mark a point that is not D, mark a point that is not a vertex. Using a ruler, a vertex. Using a ruler, draw a line from each draw a line from each bottom corner to the point bottom corner to the point you marked. (Three you marked. (Three triangles should be triangles should be formed.)formed.)

3. Cut out the shapes. Then, 3. Cut out the shapes. Then, divide A, B, and C into two divide A, B, and C into two parts by cutting along the parts by cutting along the diagonal, and divide D into diagonal, and divide D into three parts by cutting three parts by cutting along the lines you drew.along the lines you drew.

4. How do the areas of the 4. How do the areas of the resulting shapes compare resulting shapes compare to the area of the original to the area of the original shape?shape?



Area Formulas: TrapezoidsArea Formulas: Trapezoids



Do you have suggestions for finding area? What other shapes could you use to help you? Are there any other shapes for which you already know how to find the area?


24 cm

18cm

15 cm13 cm 11cm

Connect Math Shapes SetConnect Math Shapes Set

http://phcatalog.pearson.com/component.cfm?http://phcatalog.pearson.com/component.cfm?site_id=6&discipline_id=806&subarea_id=1316&program_id=23245&prsite_id=6&discipline_id=806&subarea_id=1316&program_id=23245&product_id=3502oduct_id=3502

CMP Cuisenaire® Connected Math CMP Cuisenaire® Connected Math Shapes Set (1 set of 206)Shapes Set (1 set of 206)ISBN-10:ISBN-10: 157232368X 157232368XISBN-13:ISBN-13: 9781572323681 9781572323681Price:Price: $29.35 $29.35


When triangles are removed from each corner and rotated, a rectangle will be formed. It’s important for kids to see that the midline is equal to the average of the bases. This is the basis for the proof—the midline is equal to the base of the newly formed rectangle, and the midline can be expressed as ½(b1 + b2), so the proof falls immediately into place. To be sure that students see this relationship,

ask, "How is the midline related to the two bases?" Students might suggest that the length of the midline is "exactly between" the lengths of the two bases; more precisely, some students may indicate that it is equal to the average of the two bases, giving the necessary expression. Remind students that the area of a rectangle is base × height; for the rectangle formed from the original trapezoid, the base is ½(b1 + b2) and the height is h, so the area of the rectangle (and,

consequently, of the trapezoid) is A = ½h(b1 + b2). This is the traditional formula for finding the area of

the trapezoid.

A = ½h(b1 + b2)


24 cm

18cm

15 cm13 cm 11cm


Websites:Websites:

http://argyll.epsb.ca/jreed/math9/strand3/http://argyll.epsb.ca/jreed/math9/strand3/trapezoid_area_per.htmtrapezoid_area_per.htm

ParallelogramsParallelograms

dDwxNTM

A = Length x widthA = Length x width


dDwxNTM

Area of ParallelogramArea of Parallelogram

Can you estimate the area of Tennessee?Can you estimate the area of Tennessee?

Area of irregular figure?Area of irregular figure?

Find the area of the irregular figure.Find the area of the irregular figure.



Fact: Fact: m<1 = 30˚m<1 = 30˚ and m<7 = 100 ˚and m<7 = 100 ˚

3

2

5

4

76 8

1 11

10

129

Find:

m<2

m<3

m<4

m<5

m<6

m<8

m<9

m<10

m<11

m<12

Fact: Fact: m<1 = 30˚m<1 = 30˚ and m<7 = 100 ˚and m<7 = 100 ˚

150 ˚̊

100 ˚100 ˚

80 ˚̊

30˚30˚

130 ˚̊50 ˚̊

150 ˚̊

30˚30˚

100 ˚100 ˚

80 ˚̊

50 ˚̊130 ˚̊

21211

10

8

9

76

5

4

31

m<1 + m<5 + m<12 = _______

m<2 + m<8 + m<11 = _______

The sum of which 3 angles will The sum of which 3 angles will equal 180˚?equal 180˚?

32

5

4

7 6

8

1

11

10

12

9

The sum of which 3 angles will The sum of which 3 angles will equal 360˚?equal 360˚?

32

5

4

7 6

8

1

11

10

12

9

PentominosPentominos

How many ways can you arrange five tiles How many ways can you arrange five tiles with at least one edge touching another with at least one edge touching another edge? edge?

Use your tiles to determine arrangements Use your tiles to determine arrangements and cut out each from graph paper. and cut out each from graph paper.

PentominosPentominos

http://www.ericharshbarger.org/pentominoes/http://www.ericharshbarger.org/pentominoes/

Archimedes’ PuzzleArchimedes’ Puzzle

1

4

11

3

2

9

8

6 7

5 14

10

12 13

http://mabbott.org/CMPUnitOrganizers.htmhttp://mabbott.org/CMPUnitOrganizers.htm

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Geometric Conclusions Determine if each statement is a SOMETIMES, ALWAYS, or NEVER.

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