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Agenda

Agenda. Go over homework. Go over Exploration 8.13: more practice A few more details--they are easy. Lots more practice problems. Study hard! And bring a ruler and protractor. Homework 8.2.

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Agenda

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  1. Agenda • Go over homework. • Go over Exploration 8.13: more practice • A few more details--they are easy. • Lots more practice problems. • Study hard! And bring a ruler and protractor.

  2. Homework 8.2 • 1c. Hexagon, 6 sides, non-convex, no congruent sides, 2 acute angles, 3 obtuse angles, 1 reflex angle, no parallel sides, no right angles…

  3. Homework 8.2 • 4. Shape # diagonals • Quadrilateral 2 • Pentagon 5 • Hexagon 9 • Octagon 20 • N-gon • each vertex (n) can connect to all but 3 vertices (itself, left, and right). So, n(n-3). • But now diagonals have been counted twice. So n(n-3)/2

  4. Homework 8.2 • 11. Adjacent, congruent sides. Can be true for: • Trapezoid • Square • Rhombus • Non-convex kite • Convex kite.

  5. Homework 8.2 • 18a • Scalene obtuse

  6. Homework 8.2 • 18b • Equilateral Isosceles

  7. Homework 8.2 • 18c. • Parallelogram rectangle

  8. Homework 8.2 • 18b • rectangle rhombus

  9. Quadrilaterals • Look at Exploration 8.13. Do 2a, 3a - f. • Use these categories for 2a: • At least 1 right angle • 4 right angles • 1 pair parallel sides • 2 pair parallel sides • 1 pair congruent sides • 2 pair congruent sides • Non-convex

  10. Exploration 8.13 • Let’s do f together: • In the innermost region, all shapes have 4 equal sides. • In the middle region, all shapes have 2 pairs of equal sides. Note that if a figure has 4 equal sides, then it also has 2 pairs of equal sides. But the converse is not true. • In the outermost region, figures have a pair of equal sides. In the universe are the figures with no equal sides.

  11. 8.13 2a • At least 1 right angle: A, E, G, J, O, P • 4 right angles: J, O, P • At least 1 pair // lines: E, F, J - P • 2 pair // lines: J - P • At least 1 pair congruent sides: not A, B, C, E • 2 pair congruent sides: G - P • Non-convex: I

  12. 8.13 • 3a: at least 1 obtuse angle (or no right angle, 1 obtuse and 1 acute angle), 2 pair parallel sides (or 2 pair congruent sides) • 3b: at least 1 pair parallel sides,at least 1 pair congruent sides • 3c:at least1 pair sides congruent, at least 1 right angle

  13. 8.13 • 3d: kite, parallelogram • 3e:LEFT: exactly 1 pair congruent sides, RIGHT: 2 pair congruent sides, BOTTOM: at least 1 right angle • 3f: Outer circle: 1 pair congruent sides, Middle circle: 2 pair congruent sides, Inner circle: 4 congruent sides

  14. O P K, L, M, N E, G, Try these now • What are the attributes? parallelogram 1 right angle

  15. E G, J, O, P D, F Try these now • What are the attributes? At least 1 right angle Trapezoid

  16. J, O, P E, G Try this one • What are the attributes? 4 right angles At least 1 right angle

  17. Discuss answers to Explorations 8.11 and 8.13 • 8.11 • 1a - c • 3a: pair 1:same area,not congruent;pair 2: different area, not congruent; • Pair 3: congruent--entire figure is rotated 180˚.

  18. Warm Up • Use your geoboard to make: • 1. A hexagon with exactly 2 right angles • 2. A hexagon with exactly 4 right angles. • 3. A hexagon with exactly 5 right angles. • Can you make different hexagons for each case?

  19. Warm-up part 2 • 1. Can you make a non-convex quadrilateral? • 2. Can you make a non-simple closed curve? • 3. Can you make a non-convex pentagon with 3 collinear vertices?

  20. A F B G C D Warm-up Part 3 • Given the diagram at the right, name at least 6 different polygons using their vertices. E

  21. A visual representation of why a triangle has 180˚ • Use a ruler and create any triangle. • Use color--mark the angles with a number and color it in. • Tear off the 3 angles. • If the angles sum up to 180˚, what should I be able to do with the 3 angles?

  22. Diagonals, and interior angle sum (regular) • Triangle • Quadrilateral • Pentagon • Hexagon • Heptagon (Septagon) • Octagon • Nonagon (Ennagon) • Decagon • 11-gon • Dodecagon

  23. Congruence vs. Similarity Two figures are congruent if they are exactly the same size and shape. Think: If I can lay one on top of the other, and it fits perfectly, then they are congruent. Question: Are these two figures congruent? Similar: Same shape, butmaybe different size.

  24. Let’s review • Probability: • I throw a six-sided die once and then flip a coin twice. • Event? • Possible outcomes? • Total possible events? • P(2 heads) • P(odd, 2 heads) • Can you make a tree diagram? Can you use the Fundamental Counting Principle to find the number of outcomes?

  25. Probability: • I have a die: its faces are 1, 2, 7, 8, 9, 12. • P(2, 2)--is this with or without replacement? • P(even, even) = • P(odd, 7) = • Are the events odd and 7 disjoint? Are they complementary?

  26. Combinations and Permutations • These are special cases of probability! • I have a set of like objects, and I want to have a small group of these objects. • I have 12 different worksheets on probability. Each student gets one: • If I give one worksheet to each of 5 students, how many ways can I do this? • If I give one worksheet to each of the 12 students, how many ways can I do this?

  27. More on permutations and combinations • I have 15 french fries left. I like to dip them in ketchup, 3 at a time. How may ways can I do this? • I am making hamburgers: I can put 3 condiments: ketchup, mustard, and relish, I can put 4 veggies: lettuce, tomato, onion, pickle, and I can use use 2 types of buns: plain or sesame seed. How many different hamburgers can I make? • Why isn’t this an example of a permutation or combination?

  28. When dependence matters • If I have 14 chocolates in my box: 3 have fruit, 8 have caramel, 2 have nuts, one is just solid chocolate! • P(nut, nut) • P(caramel, chocolate) • P(caramel, nut) • If I plan to eat one each day, how many different ways can I do this?

  29. Geometry • Sketch a diagram with 4 concurrent lines. • Now sketch a line that is parallel to one of these lines. • Extend the concurrent lines so that the intersections are obvious. • Identify: two supplementary angles, two vertical angles, two adjacent angles. • Which of these are congruent?

  30. Geometry • Sketch 3 parallel lines segments. • Sketch a line that intersects all 3 of these line segments. • Now, sketch a ray that is perpendicular to one of the parallel line segments, but does not intersect the other two parallel line segments. • Identify corresponding angles, supplementary angles, complementary angles, vertical angles, adjacent angles.

  31. Name attributes • Kite and square • Rectangle and trapezoid • Equilateral triangle and equilateral quadrilateral • Equilateral quadrilateral and equiangular quadrilateral • Convex hexagon and non-convex hexagon.

  32. Consider these triangles acute scalene, right scalene, obtuse scalene, acute isosceles, right isosceles, obtuse isosceles, equilateral • Name all that have: • At least one right angle • At least two congruent angles • No congruent sides

  33. Consider these figures: Triangles: acute scalene, right scalene, obtuse scalene, acute isosceles, right isosceles, obtuse isosceles, equilateral Quadrilaterals: kite, trapezoid, parallelogram, rhombus, rectangle, square Name all that have: At least 1 right angle At least 2 congruent sides At least 1 pair parallel sides At least 1 obtuse angle and 2 congruent sides At least 1 right angle and 2 congruent sides

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