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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 • 1c. Hexagon, 6 sides, non-convex, no congruent sides, 2 acute angles, 3 obtuse angles, 1 reflex angle, no parallel sides, no right angles… 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 Homework 8.2 • 11. Adjacent, congruent sides. Can be true for: • Trapezoid • Square • Rhombus • Non-convex kite • Convex kite. Homework 8.2 • 18a • Scalene obtuse Homework 8.2 • 18b • Equilateral Isosceles Homework 8.2 • 18c. • Parallelogram rectangle Homework 8.2 • 18b • rectangle rhombus 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 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. 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 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 least 1 pair sides congruent, at least 1 right angle 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 Try these now • What are the attributes? parallelogram 1 right angle E, G, O P K, L, M, N Try these now • What are the attributes? At least 1 right angle Trapezoid D, F E G, J, O, P Try this one • What are the attributes? At least 1 right angle E, G 4 right angles J, O, P 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˚. 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? 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? Warm-up Part 3 • Given the diagram at the right, name at least 6 different polygons using their vertices. A F B G C D E 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? Diagonals, and interior angle sum (regular) • • • • • • • • • • Triangle Quadrilateral Pentagon Hexagon Heptagon (Septagon) Octagon Nonagon (Ennagon) Decagon 11-gon Dodecagon 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, but maybe different size. 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? • • • • • • 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? 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? 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? 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? 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? 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. Name attributes • Kite and square • Rectangle and trapezoid • Equilateral triangle and equilateral quadrilateral • Equilateral quadrilateral and equiangular quadrilateral • Convex hexagon and non-convex hexagon. 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 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