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

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

Who Am I? I have no right angles My total angle measure is not 360˚ I have fewer than 3 congruent sides.

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

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

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

Types of Curves closed curves: a closed curve is a curve with no endpoints and which completely encloses an area

Types of Curves convex curve: If a plane closed curve be such that a straight line can cut it in at most two points, it is called a convex curve. Convex Curves Not Convex Curves

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

Types of Triangles Classified by Angles Equiangular: all angles congruent Acute: all angles acute Obtuse: one obtuse angle Right: one right angle Classified by Sides Equilateral: all sides congruent Isosceles: at least two sides congruent Scalene: no sides congruent

Triangles Scalene (No sides equal) Isosceles (at least two sides equal) Equilateral (all sides equal)

What’s possible? Equilateral Isosceles Scalene Equiangular Acute Right Obtuse NO NO NO

Homework Textbook pages 444-446 #9-12, #23-26, #49-52

Pythagorean Theorem c2 a2 b2 a2 + b2 = c2

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

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

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

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

Testing for acute, obtuse, right a2 + b2 = c2 Pythagorean theorem says: What happens if or a2 + b2 > c2 a2 + b2 < c2

Testing for acute, obtuse, right Right triangle: Acute triangle: Obtuse triangle: a2 + b2 = c2 a2 + b2 > c2 a2 + b2 < c2

Types of Angles Website www.mrperezonlinemathtutor.com Complementary Supplementary Adjacent Vertical

Transversals

Let’s check the homework! Textbook pages 444-446 #9-12, #23-26, #49-52

What is the value of x? 2x + 5 3x + 10

Angles in pattern blocks

Diagonals Joining two nonadjacent vertices of a polygon

For which shapes will the diagonals always be perpendicular? Type of Quadrilateral Are diagonals perpendicular? Trapezoid Parallelogram Rhombus Rectangle Square Kite

For which shapes will the diagonals always be perpendicular? Type of Quadrilateral Are diagonals perpendicular? Trapezoid maybe Parallelogram Rhombus Rectangle Square Kite

For which shapes will the diagonals always be perpendicular? Type of Quadrilateral Are diagonals perpendicular? Trapezoid maybe Parallelogram Rhombus Rectangle Square Kite

For which shapes will the diagonals always be perpendicular? Type of Quadrilateral Are diagonals perpendicular? Trapezoid maybe Parallelogram Rhombus yes Rectangle Square Kite

For which shapes will the diagonals always be perpendicular? Type of Quadrilateral Are diagonals perpendicular? Trapezoid maybe Parallelogram Rhombus yes Rectangle Square Kite

For which shapes will the diagonals always be perpendicular? Type of Quadrilateral Are diagonals perpendicular? Trapezoid maybe Parallelogram Rhombus yes Rectangle Square Kite

For which shapes will the diagonals always be perpendicular? Type of Quadrilateral Are diagonals perpendicular? Trapezoid maybe Parallelogram Rhombus yes Rectangle Square Kite

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

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

Sum of the angles of a polygon Use 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 3 4 9 http://www.arcytech.org/java/patterns/patterns_j.shtml 8 2 5 7 1 3 6 4 7 5 6

Sum of the angles of a polygon # sides Total degrees Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Triskaidecagon 13 Nth N What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 1080 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 1080 Nonagon 9 1260 Decagon 10 Undecagon 11 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 1080 Nonagon 9 1260 Decagon 10 1440 Undecagon 11 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 1080 Nonagon 9 1260 Decagon 10 1440 Undecagon 11 1620 Dodecagon 12 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 1080 Nonagon 9 1260 Decagon 10 1440 Undecagon 11 1620 Dodecagon 12 1800 Triskaidecagon 13 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 1080 Nonagon 9 1260 Decagon 10 1440 Undecagon 11 1620 Dodecagon 12 1800 Triskaidecagon 13 1980 nth n What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 1080 Nonagon 9 1260 Decagon 10 1440 Undecagon 11 1620 Dodecagon 12 1800 Triskaidecagon 13 1980 nth n ? What patterns do you see?

Sum of the angles of a polygon # sides Total degrees Triangle 3 180 Quadrilateral 4 360 Pentagon 5 540 Hexagon 6 720 Heptagon 7 900 Octagon 8 1080 Nonagon 9 1260 Decagon 10 1440 Undecagon 11 1620 Dodecagon 12 1800 Triskaidecagon 13 1980 nth n 180(n-2) What patterns do you see?

Total degree of angles in polygon

Area Ideas Triangles Parallelograms Trapezoids Irregular figures

Area Formulas: Triangle http://illuminations.nctm.org/LessonDetail.aspx?ID=L577

Area Formulas: Triangle 1. Using a ruler, draw a diagonal (from one corner to the opposite corner) on shapes A, B, and C. 2. Along the top edge of shape D, mark a point that is not a vertex. Using a ruler, draw a line from each bottom corner to the point you marked. (Three triangles should be formed.) 3. Cut out the shapes. Then, divide A, B, and C into two parts by cutting along the diagonal, and divide D into three parts by cutting along the lines you drew. 4. How do the areas of the resulting shapes compare to the area of the original shape?

Area Formulas: Triangle

Area Formulas: Triangle

Area Formulas: Trapezoids http://illuminations.nctm.org/LessonDetail.aspx?ID=L580

Area 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: Trapezoids 18cm 15 cm 13 cm 11cm 24 cm

Connect Math Shapes Set http://phcatalog.pearson.com/component.cfm?site_id=6&discipline_id=806&subarea_id=1316&program_id=23245&product_id=3502 CMP Cuisenaire® Connected Math Shapes Set (1 set of 206) ISBN-10: 157232368X ISBN-13: 9781572323681 Price: $29.35

Area Formulas: Trapezoids A = ½h(b1 + b2) 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.

Area Formulas: Trapezoids 18cm 15 cm 13 cm 11cm 24 cm

Area Formulas: Trapezoids Websites: http://argyll.epsb.ca/jreed/math9/strand3/trapezoid_area_per.htm

Parallelograms

A = Length x width http://illuminations.nctm.org/LessonDetail.aspx?ID=L578

Area of Parallelogram Can you estimate the area of Tennessee?

Area of irregular figure?

Find the area of the irregular figure.

Area of irregular figure?

Area of irregular figure?

Fact: m<1 = 30˚ and m<7 = 100 ˚ Find: m<2 m<3 m<4 m<5 m<6 m<8 m<9 m<10 m<11 m<12 5 6 8 7 2 10 4 9 1 12 3 11

Fact: m<1 = 30˚ and m<7 = 100 ˚ 5 80 ˚ 6 80 ˚ 8 7 100 ˚ 130 ˚ 150 ˚ 50 ˚ 10 4 30˚ 9 2 1 11 12 3 30˚ 50 ˚ 130 ˚ 150 ˚ m<1 + m<5 + m<12 = _______ m<2 + m<8 + m<11 = _______

The sum of which 3 angles will equal 180˚? 2 3 1 4 8 9 10 5 11 12 7 6

The sum of which 3 angles will equal 360˚? 2 3 1 4 8 9 10 5 11 12 7 6

Pentominos How many ways can you arrange five tiles with at least one edge touching another edge? Use your tiles to determine arrangements and cut out each from graph paper.

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

Archimedes’ Puzzle 1 8 2 4 9 3 10 12 13 7 6 11 5 14

http://mabbott.org/CMPUnitOrganizers.htm