Geometry C Chapter 2 Inductive Reasoning
Name: ___________________________ Period: _____
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Geometry C Lesson 2.1 Inductive Reasoning (Patterning)
Name ______________________________
Inductive Reasoning is ______________________________________________________________________ _________________________________________________________________________________________
When we use inductive reasoning to make a generalization, the generalizations is called a _________________ (this is very similar to a ________________ made in science class) Let’s try some patterning! 1. Consider the sequence: 7, 10, 13, 16, … Make a conjecture about the rule for generating the sequence: ___________________________ Use your conjecture to find the next 5 terms of the sequence: ___________________________ 2. Consider the sequence: 5, 0, -5, -10, … Make a conjecture about the rule for generating the sequence: ___________________________ Use your conjecture to find the next 5 terms of the sequence: ___________________________ 3. Consider the sequence: 2, 4, 7, 11, … Make a conjecture about the rule for generating the sequence: ___________________________ Use your conjecture to find the next 5 terms of the sequence: ___________________________ 4. Consider the sequence: 1, 4, 9, 16, … Make a conjecture about the rule for generating the sequence: ___________________________ Use your conjecture to find the next 5 terms of the sequence: ___________________________ 5. Consider the sequence: 2048, 1024, 512, 256, … Make a conjecture about the rule for generating the sequence: ___________________________ Use your conjecture to find the next 5 terms of the sequence: ___________________________
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Shape Shifters 1. Look at the sequence of shapes. Pay close attention to the patterns that occur in every shape!
What pattern do you notice in the 1st, 3rd, and 5th shapes?
What patterns do you notice in the 2nd, 4th, and 6th shapes?
Use the patterns you discovered to draw the 25th shape.
Draw the next two shapes in the sequence.
What would the 100th shape look like? Describe it (you don’t have to draw it).
2. Draw the next 2 shapes in this sequence.
3. Draw the next 2 shapes in this sequence. 1
3
6
10
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Geometry C 2.1 Practice
Name ______________________________
For each exercise, describe the pattern of the sequence and use patterning to find the next 3 terms of the sequence.
1. 4, 8, 12, 16, _____, _____, _____
2. 400, 200, 100, 50, 25, _____, _____, ______
Description:
Description:
3. 1, 3, 9, 27, 81, _____, _____, _____
4. 7, 3, -1, -5, _____, _____, _____
Description:
Description:
Draw the next figure in the pattern. 5.
Draw the next two figures in the pattern. 6.
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Geometry C
2.3 – Finding the nth Term Part I
Name
Situation 1: During the first snowfall of the winter, snow is falling at a rate of 2 inches per hour for n hours.
Graph 1:
Table 1: n (hours)
calculation
pattern
f(n) total snow depth
0 1 2 3 … n Equation 1 to describe situation, table, and graph:
Situation 2: Initially, there are 6 inches of snow on the ground. Snow is falling at a rate of 2 inches per hour for n hours. Table 2: n (hours)
Graph 2: calculation
pattern
f(n) total snow depth
0 1 2 3 … n Equation 2 to describe situation, table, and graph: 5
Situation 3: On a sunny day in March there are 10 inches of snow on the ground. The snow is melting at a rate of 2 inches per day. Table 1: n (hours)
Graph 1:
calculation
pattern
f(n) total snow depth
0 1 2 3 … n Equation 1 to describe situation, table, and graph:
Situation 4: Initially, there are 4 inches of snow on the ground. Snow is falling at a rate of 3 inches per 2 hours for n hours. Table 2: n (hours)
Graph 2: calculation
pattern
f(n) total snow depth
0 1 2 3 … n Equation 2 to describe situation, table, and graph:
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Geometry C 2.3 – Finding the nth Term Part 2
Name
Write the equation for the line through these points: 1.
x 0 1 2 3 4 5
y 4 7 10 13 16 19
2.
y=
x 0 1 2 3 4 5
y 11 7 3 -1 -5 -9
y=
Notice that the term numbers don’t go up by 1 anymore. Think how that affects the slope. 9.
x 0 2 4 6 8 10
y 5 11 17 23 29 35
y=
10.
x 0 2 4 6 8 10
y 13 14 15 16 17 18
y=
Notice that the 0TH term is not included in the table. Use the slope to think about what that value would be. 11.
y=
x 1 2 3 4 5 6
y 8 11 14 17 20 23
12.
x 2 3 4 5 6 7
y 10 15 20 25 30 35
y=
7
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Geometry C 2.5 – Vertical Angles and Linear Pairs
Name
Vertical Angles On a piece of patty paper, draw two intersecting lines. Label the angles in order 1,2, 3, and 4. 1. Which angles are vertical? _____ and _____ , _____ and _____ 2. Fold the patty paper so that the vertical angles lie over each other. What do you notice about their measurements? (tape your construction below) Vertical Angle Conjecture:
Linear Pair of Angles On a piece of patty paper, draw PQ and place a point R between P and Q. Choose another point S not on PQ and draw . You have just constructed a linear pair of angles. What is the sum of the measures of the linear pair of angles? (tape your construction below) Linear Pair Conjecture:
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Use the Vertical Angles Conjecture and the Linear Pair Conjecture to find each lettered angle measure. 1.
a = _____
2.
b = _____
3.
a = _____
a = _____
4.
b = _____
5.
a = ______
c = _____
a = _____
b = _____
d = _____
e = _____
a = _____
b = _____
d = _____
e = _____
c = _____
6.
b = _____
c = _____
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Geometry C 2.5 – Vertical Angles and Linear Pairs Day 1 Practice
Name ______________________________
Find the measure of each missing angle: 1.
2.
3.
a = ____ b = ____ c = ____
a = ____ b = ____ c = ____
a = ____ b = ____ c = ____ d = _____
4.
a = _____
5.
b = _____ c = _____
d = ____ e = ____ f = ____ g = ____
a = ____ b = ____ c = ____
h = ____ i = ____
d = ____ e = ____
6.
a = _____
b = _____
c = _____
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Geometry C 2.5 – Vertical Angles and Linear Pairs with Algebra
Name ______________________________
For each figure, set up an equation to solve for x. What kind of angles are in problems 1 and 2? _____________________ What is true about these kind of angles? _________________________
1.
Equation: _____________________________
A
D 8x + 25 145
C
B
x = _______
2.
Equation: ________________________________
A
D
5x - 7
C
3x + 21
B
x = ________
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What kind of angles are in problems 3 and 4? _______________________ What is true about these kind of angles? ____________________________
3.
Equation: _________________________________
G E
F
127
4x + 5
H
x = _________
4.
Equation: _____________________________
H
8x - 28
E
G
2x + 8
F
x = _________
m HGF = ___________
13
L
5.
Equation: ________________________________
I
66 6x + 42
12x - 30
M
J
K
x = __________ m IMK __________
6.
Find all angle measures. A F 4x+25 B
2x+4 E
5x+13 C
D
mAGB _____
mBGC _____
mAGF _____
mCGD _____
mFGE _____
mDGE _____
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Geometry C 2.5 – Vertical Angles and Linear Pairs with Algebra Practice
Name
1. Find: m YRK. Label the measure on the diagram. O
Y 4x + 65 6x - 9
R
K
2. Find: m NRP and m NRQ. Label the measures on the diagram.
Q
N 8x - 19
5x + 20
R O
P
3. Find x and the m TUV.
S V
37
U
5x + 13
T 15
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Geometry C Name ___________________________ 2-6 Special Angles Created by Intersecting Lines and a Transversal In the diagrams below, the bold line is intersecting two other lines and is called a transversal. A transversal creates different types of angle pairs. Interior Angles
Same Side Interior Angles
Alternate Interior Angles
Exterior Angles
Same Side Exterior Angles
Alternate Exterior Angles
Corresponding Angles:
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Geometry C 2-6 Special Angles Created by Parallel Lines and a Transversal
E
B
On a piece of patty paper, copy the diagram below:
D
G
F
Name ___________________________
Note: EB is parallel to GA . CH is a transversal. Place your copied picture onto the figure below. Slide it up … What do you notice?
C
A
H
E
B
D
G
A
F H
Corresponding Angle Conjecture:
Now trace CDB. What angle is its alternate exterior pair? What do you notice?
C
Alternate Exterior Angle Conjecture:
E
D
G
F
B A
H 18
Now trace BDF. What angle is its alternate interior pair? What do you notice?
C
Alternate Interior Angle Conjecture:
E
D
G
F
B A
H Now use your conjectures to find the measures of missing angles. In each diagram l || m and t is a transversal. t 1. m 1 = ________ m 5 = ________ l 2
1
30° 3 4
m 2 = ________
m 6 = ________
m 3 = ________
m 7 = ________
m
5
7 6
m 4 = ________
m
2.
1 2
3 4
l
5
100°
6 7
m 1 = ________
m 5 = ________
m 2 = ________
m 6 = ________
m 3 = ________
m 7 = ________
t
m 4 = ________
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Geometry C 2.6 Day 1 Practice
Name ________________________________
For 1 – 5, use the drawing at the right to give an example of each term. 1. Corresponding Angles:
__________________
2. Alternate Interior angles: __________________ 3. Alternate Exterior Angles: __________________ 4. Vertical Angles:
__________________
5. Linear Pair of Angles:
__________________
For 6 – 8, find each angle measure 6
a = _____ b = _____ c = _____
7.
8.
a = _____ b = _____ c = _____ d = _____
a = _____ b = _____
For Exercises 9 and 10, use your conjectures to determine whether or not the two lines are parallel, and explain why. 9. 10.
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Geometry C Parallel Lines and Transversals with Algebra
Name ___________________________________
REVIEW: 1. Complete the statements by filling in the blanks a) Line ______ is the transversal
a b
b) c)
c
3 and ______ are corresponding angles. 4 and ______ are alternate interior angles.
d) 6 and ______ are same – side exterior angles e) 1 and ______are alternate exterior angles f) 3 and ______ are vertical angles g) 2 and ______ are supplementary
2. Name the angles that are: a) Congruent to 2:
___________________
b) Supplementary to 5: ___________________ c) Congruent to 5:
___________________
d) Supplementary to 2: ___________________
3. Find the measure of each angle 1 :
__________
2 : __________ 3 :
__________
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Let’s mix some Algebra in! 4. Find the value of x:
5. Find x and the measure of each angle. Label the angle measures on the diagram.
6. Find x and the value of each angle. Label the angle measures on the diagram.
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Geometry C Parallel Lines and Transversals with Algebra --- Practice
Name __________________________________
For each problem, set up an equation for the angle relationship, solve for x, and find the measure of each angle
1.
What kind of angles are marked? ______________ Are these angles congruent or supplementary? Write an equation and solve for x:
2.
What kind of angles are marked? ______________ Are these angles congruent or supplementary? Write an equation and solve for x:
For Extra Challenge and Profit: 3. a) if m || n, find x:
b) if j || k, find x:
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