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Verifying Conjecture: Law of Detachment
:
(WS#G10208)
There are two laws of deductive reasoning: the Law of Detachment and the Law
of Syllogism.
The Law of Detachment:
If ab is a true statement and a is true, then b is true.
Typically, this appears as a series of statements, including a conditional (the ab
portion of the law) and a separate “a” statement. See the example below:
Example:
ab:
If you are late to class three times, you must go
to detention.
a:
Sarah was late to class three times.
Conjecture: Sarah must go to detention.
Notice that the “a” of the conditional ab matches “a:” This is essential to
making the conjecture valid. Because Sarah was late to class three times matches
the hypothesis of the conditional, it is a valid statement.
Example:
ab:
If you want to go to the museum, you must have
asigned permission slip.
a:
Jonathan has a signed permission slip.
Conjecture: Jonathan wants to go to the museum.
Notice that the “a” of the conditional ab matches the conjecture in this second
example. Since the hypothesis, “a” of the conditional must match the proposed
hypothesis in order to be valid, this example is invalid.
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Practice. Use the Law of Detachment to verify whether the following conjectures
are valid. Support your conclusions.
1. Given: If you are from Texas, you are American. Janie is from Texas.
Conjecture: Janie is American.
2. Given: When babies are hungry, they cry. Baby Sarah is hungry.
Conjecture: Sarah is crying.
3. Given: If you plant marigold seeds, you will grow marigolds. Ginny planted
marigold seeds. Conjecture: Ginny will grow marigolds.
4. Given: If you are attentive in class, you will know what the assignment is. James
knows what the assignment is. Conjecture: James is attentive in class.
5. Given: If you slice onions, you might cry. Fargo sliced onions.
Conjecture: Forgo might cry.
6. Given: Drinking milk helps your body retain calcium. Heather drinks a lot of milk.
Conjecture: Her body will retain calcium.
7. Given: If you practice your free throw every day, you will be able to sink more
free throw shots. Billy practices his free throw every day.
Conjecture: Billy will is able to sink more free throw shots.
8. Given: If you brush your teeth after each meal, you can prevent cavities. Zoe
has no cavities. Conjecture: Zoe brushes her teeth after each meal.
9. Given: If you use your turn signal, other drivers will know that you are changing
lanes. Marquis uses his turn signal.
Conjecture: Other drivers know that he is changing lanes.
10. Given: If you poke a turtle in the mouth, he may bite you. Jomar got bit by a
turtle.
Conjecture: Jomar poked the turtle in the mouth.
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Proving Lines Parallel Postulate (WS#G10325)
DEFINITION
Alternate Interior Angles Theorem: alternate interior angles are equal when
two parallel lines are cut by a transversal. The converse view of this statement
is also true. The Converse of Alternate Interior Angles Theorem states that
lines that are cut by a transversal are parallel if the alternate interior angles
are equal.
EXAMPLE
P
R
5
1
6
2
7
Q
3
8
4
Given: m
2=m
7
Prove line P is parallel to Q
Statement
m
e=m
line A is parallel to line B
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Reason
Given
Converse of Corresponding
Angles Postulate
QUESTIONS
Write a column proof for each problem below. The proof should contain two
columns, a statement and a reason. Notice that each problem has two statements
that must be proved.
M
L
1.
O
A
F
E
Given: m
Prove m
B+m
B=m
B
G
C
H
D
C = 180˚
G and line M is parallel to line L.
Statements
Reasons
1.
2.
3.
4.
2.
A
B
C
Given: m
Prove m
3x
h = 3x
b=m
b
e
c
f
d
g
h
g and line A is parallel to line B.
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Statements
Reasons
1.
2.
3.
4.
5.
6.
3. H
I
J
Given: m
Prove m
w
110˚
y
x
x = 70˚
y=m
x and line H is parallel to line I.
Statements
Reasons
1.
2.
3.
4.
M
4.
O
2x
q
o
p
N
60˚
r
Given: x = 60
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Prove m
q=m
p and line M is parallel to line N.
Statements
Reasons
1.
2.
3.
4.
5.
6.
7.
Y
5.
X
Given: m
Prove m
e
a+m
b+m
a
f
b
g
c
h
Z
d
g = 180˚
g = 180˚ and line Y is parallel to line Z.
Statements
Reasons
1.
2.
3.
4.
5.
6.
7.
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Law of Cosines #1
:
(WS#G10828)
Law of Cosines:
c2 = a2 + b2 – (2ab)cosC
Where a, b, c are side lengths of an arbitrary
triangle, and A, B, C are the respective opposite
angles. Since a, b, and c are arbitrarily
defined. This means that a2 = c2 + b2 – (2cb)cosA
and b2 = a2 + c2 – (2ac)cosB are alternate ways to
write this law.
The law of cosines can be used to solve obtuse and acute triangles when two
sides and the angle enclosed by those sides are known, or when three sides are
known.
Notice what happens when C is a right angle:
c2 = a2 + b2 – (2ab)cos90˚ = c2 = a2 + b2 – (2ab)(0) = c2 = a2 + b2
The law of cosines reduces to the Pythagorean Theorem when the triangle is a
right triangle. That is because the law of cosines is a generalization of
Pythagoras’ Theorem.
Example: Solve the triangle, using the notation from the figure above (although
not to scale), given a = 21.0 b = 38.794 c = 46.003 A = ? B = ? C = ?
cosC = (a2 + b2 – c2)/(2ab)
C = arccos( (a2 + b2 – c2) / (2ab) ) =
arccos( (212 + 38.7942 – 46.0032) / (2(21)(38.794) ) = 96
C = 96˚
cosB = (a2 + c2 – b2)/(2ac)
B = arccos( (a2 + c2 – b2) / (2ac) ) =
arccos( (212 + 46.0032 - 38.7942) / (2(21)(46.003) ) = 57
B = 57˚
A = 180˚ - (B + C) = 180˚ - 153˚ = 27˚
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Law of Cosines #1 – Page 2
For each question, side a is opposite angle A, side b is opposite angle B, and side c
is opposite angle C. Use the triangle above as reference.
1. a = 25 b = ? c = 25.448 A = ? B = 5 C = ?
2. a = ? b = 30.248 c = 63.545 A = 9 B = ? C = ?
3. a = ? b = 29.445 c = 42.381 A = 47 B = ? C = ?
4. a = 10 b = 47.214 c = ? A = ? B = ? C = 89
5. a = ? b = 31.89 c = 22.39 A = 84 B = ? C = ?
6. a = 35 b = 140.967 c = ? A = ? B = ? C = 89
7. a = 24 b = 7.202 c = 24.571 A = ? B = ? C = ?
8. a = 7 b = 7.346 c = 5.268 A = ? B = ? C = ?
9. a = 44 b = ? c = 38.741 A = ? B = 72 C = ?
10. a = 11 b = ? c = 15.424 A = ? B = 64 C = ?
11. a = 15 b = 3.678 c = 16.621 A = ? B = ? C = ?
12. a = 17 b = 10.777 c = ? A = ? B = ? C = 76
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Geometric Probability
(WS#G10934)
Geometric probability is the relation of the number of outcomes in an event to
the number of outcomes in a sample space. Geometric probability is used in the
special case that there are infinite number of outcomes to an experiment. You
can use length to develop a probable ratio between geometric lengths or areas.
Example: A point is chosen randomly on segment AD.
The point is on AC
P = AC/AD
= 5+8/5+8+6
= 13/19
A
The point is not on BC
P(BC) = 8/19
19
/19 - 8/19 = 11/19
5
B
8
C
6
Subtract from one to find the probability
that the point is not on BC.
The point is on AB or CD.
P(AB or CD) = P(AB) + P(CD) = 5/19 + 6/19 = 11/19
Practice. Give the probability of the following events.
N
3
O
5
P
4
Q
1-4. A point is chosen randomly on NQ. Find the probability of each event.
1. The point is on PQ.
2. The point is on OP.
3. The point is not on NO.
4. The point is on NO or OP.
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D
5-6. A point is chosen randomly on AZ. Find the probability of each event.
5
A
4
P
5. A point on PT.
T
5
Z
6. A point not on TZ.
7-8. A point is chosen randomly on QR. Find the probability of each event.
Q
3
8
S
7. A point on QS or TR.
T
3
R
8. A point not on QS.
9-10. A point is chosen randomly on PV.
P
3
Q
3
9. A point on PR.
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R 4
T 2 V
10. A point not on RT.
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3D Formulas: Polyhedrons
:
(WS#G11014)
A polyhedron is formed by four or more polygons that intersect only at their
edges. As a result, cylinders, spheres, and cones could not be classified as
polyhedrons.
Euler’s Formula
When you are attempting to verify the number of vertices, edges, and faces for
a polyhedron, you can use Euler’s formula: V – E + F = 2, where V = Vertices,
E=Edges, and F = Faces.
Example: A quick count tells us that there are 7 vertices, 15 edges, and 10
faces.
Euler’s Formula: V-E+F=2
7 – 15 + 10 = ?
7 – 15 + 10 = 2
Euler’s formula verifies that our count was correct.
Practice. Identify the number of vertices, edges, and faces for each of the
following shapes. Verify using Euler’s formula.
1. V:____ E:____ F:____
2. V:____ E:____ F:____
3. V:____ E:____ F:____
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4. V:____ E:____ F:____
5. V:____ E:____ F:____
6. V:____ E:____ F:____
7. V:____ E:____ F:____
8. V:____ E:____ F:____
9. V:____ E:____ F:____
10. V:____ E:____ F:____
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Geometry (Grade 9/Grade 10 High School Math) Curriculum Overview
1. High School
Geometry Basics
2. Geometric
Reasoning
Points, lines and
planes;
Collinear, coplanar
points;
Construct bisector of
an angle;
Transformations in a
co-ordinate plane;
Interior/exterior
angles;
Inductive reasoning to
make a conjecture;
Conditional
statements;
Deductive reasoning;
Bi-conditional
statements;
Geometric proof;
Flowchart proofs,
Paragraph Proofs;
5. Properties and
Attributes of
Triangles
6. Polygon and
Quadrilaterals
Perpendicular and
angle bisectors;
Median of a triangle;
Centroid of a triangle;
Triangle midsegment;
Indirect proof and
inequalities;
Pythagorean theorem;
Special right triangle;
Properties and
attributes of
polygons;
Properties of
parallelograms,
special
parallelograms, kites,
trapezoids;
Identifying conditions
for a polygon to be a
parallelogram;
9. Perimeter,
Circumference and
Area
10. Spatial
Reasoning and
Solid Geometry
Perimeter,
Circumference and
Area;
Areas of composite
figures;
Effects of changing
dimensions;
Geometric probability;
Surface areas and
volumes in solid
geometry - prism,
cylinder, pyramid,
cone;
3-d formulas for
polyhedrons;
Pythagorean Theorem
in 3-d;
3-d distance/mid-pt
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3. Parallel and
Perpendicular
Lines Theorems
Corresponding angles,
alternate interior
angles, alternate
exterior angles etc.;
Proving lines parallel;
slopes of lines
problems;
Point slope form,
slope intercept form
of line;
7. Similarity
Ratios in similar
polygons problems;
Similarity problems
using AA, SSS, SAS
similarity;
Triangle
proportionality
theorem;
Two-transversal
proportionality;
Dilations and
similarity
11. Circles
Chord, secant,
tangent, point of
tangency;
Congruent,
concentric, tangent
circles;
Inscribed angles;
Problems on tangent,
secants and chords;
Circles in a coordinate
plane;
Equation of a circle;
4. Triangle
Congruence
Triangle congruence
by angle measure;
Triangle congruence
by side lengths;
Triangle congruence:
using SSS, SAS;
Triangle congruence:
using ASA, AAS, HL;
Triangle congruence
using CPCTC;
Coordinate proof;
8. Right Triangles
and Trigonometry
The altitude to the
hypotenuse of a right
triangle;
Geometric means
corollary in right
triangles;
Trigonometric ratios:
sine, cosine and
tangent;
Problems on Vectors;
12. Advanced
Transformational
Geometry
Reflection,
translations;
rotations;
Line symmetry;
Tessellations –
translation, glide,
frieze pattern;
Regular/ semiregular
tessellations;
Dilations in a
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