Download Answers 8.2 plus review - University of Arizona Math

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
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–
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–
–
# 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:
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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.
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–
–
–
–
–
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