agenda go over homework. go over exploration 8.13: more practice a few more details--they are easy....

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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.

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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?

E, G, K, L, M, N

O

P

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, GJ, O, P

At least 1 right angle 4 right angles

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.

E

G

F

DC

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. SimilarityTwo 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 trianglesacute 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 angleAt least 2 congruent sidesAt least 1 pair parallel sidesAt least 1 obtuse angle and 2 congruent sidesAt least 1 right angle and 2 congruent sides