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 Shape# diagonals –Quadrilateral2 –Pentagon5 –Hexagon9 –Octagon20 –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 Adjacent, congruent sides. Can be true for: Trapezoid Square Rhombus Non-convex kite Convex kite.

Homework a Scaleneobtuse

Homework b EquilateralIsosceles

Homework c. Parallelogramrectangle

Homework b rectanglerhombus

Quadrilaterals Look at Exploration 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? E, G, K, L, M, N OPOP 1 right angle parallelogram

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

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

Discuss answers to Explorations 8.11 and a - 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. E G F D C B A

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