geometric conclusions determine if each statement is a sometimes, always, or never
TRANSCRIPT
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
Pythagorean TheoremPythagorean Theorem
http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm
Pythagorean TheoremPythagorean Theorem
http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm
Pythagorean TheoremPythagorean Theorem
http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm
Pythagorean TheoremPythagorean Theorem
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
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 maybemaybe
ParallelogramParallelogram
RhombusRhombus
RectangleRectangle
SquareSquare
KiteKite
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 maybemaybe
ParallelogramParallelogram maybemaybe
RhombusRhombus
RectangleRectangle
SquareSquare
KiteKite
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 maybemaybe
ParallelogramParallelogram maybemaybe
RhombusRhombus yesyes
RectangleRectangle
SquareSquare
KiteKite
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 maybemaybe
ParallelogramParallelogram maybemaybe
RhombusRhombus yesyes
RectangleRectangle maybemaybe
SquareSquare
KiteKite
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 maybemaybe
ParallelogramParallelogram maybemaybe
RhombusRhombus yesyes
RectangleRectangle maybemaybe
SquareSquare yesyes
KiteKite
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 maybemaybe
ParallelogramParallelogram maybemaybe
RhombusRhombus yesyes
RectangleRectangle maybemaybe
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?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44
PentagonPentagon 55
HexagonHexagon 66
HeptagonHeptagon 77
OctagonOctagon 88
NonagonNonagon 99
DecagonDecagon 1010
UndecagonUndecagon 1111
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55
HexagonHexagon 66
HeptagonHeptagon 77
OctagonOctagon 88
NonagonNonagon 99
DecagonDecagon 1010
UndecagonUndecagon 1111
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66
HeptagonHeptagon 77
OctagonOctagon 88
NonagonNonagon 99
DecagonDecagon 1010
UndecagonUndecagon 1111
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77
OctagonOctagon 88
NonagonNonagon 99
DecagonDecagon 1010
UndecagonUndecagon 1111
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88
NonagonNonagon 99
DecagonDecagon 1010
UndecagonUndecagon 1111
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88 10801080
NonagonNonagon 99
DecagonDecagon 1010
UndecagonUndecagon 1111
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88 10801080
NonagonNonagon 99 12601260
DecagonDecagon 1010
UndecagonUndecagon 1111
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88 10801080
NonagonNonagon 99 12601260
DecagonDecagon 1010 14401440
UndecagonUndecagon 1111
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88 10801080
NonagonNonagon 99 12601260
DecagonDecagon 1010 14401440
UndecagonUndecagon 1111 16201620
DodecagonDodecagon 1212
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88 10801080
NonagonNonagon 99 12601260
DecagonDecagon 1010 14401440
UndecagonUndecagon 1111 16201620
DodecagonDodecagon 1212 18001800
Triskaidecagon Triskaidecagon 1313
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88 10801080
NonagonNonagon 99 12601260
DecagonDecagon 1010 14401440
UndecagonUndecagon 1111 16201620
DodecagonDodecagon 1212 18001800
Triskaidecagon Triskaidecagon 1313 19801980
nnthth nn
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88 10801080
NonagonNonagon 99 12601260
DecagonDecagon 1010 14401440
UndecagonUndecagon 1111 16201620
DodecagonDodecagon 1212 18001800
Triskaidecagon Triskaidecagon 1313 19801980
nnthth nn ??
What patterns do you see?
Sum of the angles of a polygonSum of the angles of a polygonPolygonPolygon # #
sidessidesTotal degreesTotal degrees
TriangleTriangle 33 180180
QuadrilateralQuadrilateral 44 360360
PentagonPentagon 55 540540
HexagonHexagon 66 720720
HeptagonHeptagon 77 900900
OctagonOctagon 88 10801080
NonagonNonagon 99 12601260
DecagonDecagon 1010 14401440
UndecagonUndecagon 1111 16201620
DodecagonDodecagon 1212 18001800
Triskaidecagon Triskaidecagon 1313 19801980
nnthth nn 180(n-2)180(n-2)
What patterns do you see?
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: TriangleArea Formulas: Triangle
Area Formulas: TriangleArea Formulas: Triangle
Area Formulas: TrapezoidsArea Formulas: Trapezoids
http://illuminations.nctm.org/LessonDetail.aspx?ID=L580http://illuminations.nctm.org/LessonDetail.aspx?ID=L580
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?
Area Formulas: TrapezoidsArea Formulas: Trapezoids
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
Area Formulas: TrapezoidsArea Formulas: Trapezoids
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)
Area Formulas: TrapezoidsArea Formulas: Trapezoids
24 cm
18cm
15 cm13 cm 11cm
Area Formulas: TrapezoidsArea Formulas: Trapezoids
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
http://illuminations.nctm.org/LessonDetail.aspx?ID=L578http://illuminations.nctm.org/LessonDetail.aspx?ID=L578
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.
Area of irregular figure?Area of irregular figure?
Area of irregular figure?Area of 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