Glencoe Geometry - Concepts and Applications HQ.pdf

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Glencoe Geometry - Concepts and Applications HQ
1
Reasoning in
Geometry
What You’ll Learn
Key Ideas
• Identify patterns and use
inductive reasoning.
• Identify, draw models of,
and use postulates about
points, lines, and planes.
• Write statements in if-then
form and write their
converses. (Lesson 1–4)
• Use geometry tools.
• Use a four-step plan to
solve problems. (Lesson 1–6)
Key Vocabulary
line (p. 12)
line segment (p. 13)
plane (p. 14)
point (p. 12)
ray (p. 13)
Why It’s Important
Interior Design The goal of an interior designer is to make a
room beautiful and functional. Designers listen carefully to their
client's needs and preferences and put together a design plan
and budget. The plan includes coordinating colors and selecting
furniture, floor coverings, and window treatments.
Reasoning in geometry is used to solve real-life problems. You
will use the four-step plan for problem solving to find the
amount of wallpaper border an interior designer would need for
a room in Lesson 1-6.
2 Chapter 1 Reasoning in Geometry
CHAPTER
Reasoning in
CHAPTER
1
Geometry
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Study these lessons
to improve your skills.
✓
✓
Check Your Readiness
Algebra
Review, p. 718
Evaluate each expression.
1. 2
9
2
3
2. 2(4)
2(7)
3. 2(9)
2(12)
4. 2(14)
2(18)
5. 2
11
6. 9(10)
7. 9
3
8. 9
8
9. 12(7)
✓
Algebra
Review, p. 719
Solve each equation.
10. 10.1
0.2
x
11. y
2.6 – 1.4
12. n
4.7 – 3.1
13. j
100.4 – 94.9
14. 1.43
0.84
p
15. 4.6
2.9
n
16. 0.8
1.3
d
17. 11.1
0.2
0.2
t
18. h
7.4(4.1)
✓
Algebra
Review, p. 720
19. m
2.3(8.8)
20. (10.7)(15.5)
a
21. 0.6(143.5)
g
1 5 2 1 1 2
7
0
1
0
2
1
22. q
23. 1
– 1
t
3
6
w
25. y
1
1
1
–
1
3
26. b
5
1
4
5
2
c
2
28. v
5
1
3
29.
3
5
8 d
30. z
9
3
4
Make this Foldable to help you organize your Chapter 1 notes. Begin with
a sheet of 8
1
2 ” by 11” paper.
➊
Fold lengthwise in
fourths.
➋
Draw lines along the
folds and label each
column sequences,
patterns, conjectures,
and conclusions.
sequences patterns conjectures conclusions
Reading and Writing As you read and study the chapter, record different sequences and
describe their patterns. Also, record conjectures and state whether they are true or false; if
false, provide at least one counterexample.
Chapter 1 Reasoning In Geometry 3
24.
3
4
27.
1
4
5
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1–1
Patterns and
Inductive Reasoning
What You’ll Learn
You’ll learn to identify
patterns and use
inductive reasoning.
If you see dark, towering clouds
approaching, you might want
to take cover. Why? Even
though you haven’t heard a
weather forecast, your past
experience tells you that
a thunderstorm is likely
to happen. Every day
you make decisions
based on past
experiences or
patterns that
you observe.
Why It’s Important
Business Businesses
look for patterns in
data. See Example 5.
Rain clouds approaching
When you make a conclusion based on a pattern of examples
or past events, you are using inductive reasoning . Originally,
mathematicians used inductive reasoning to develop geometry
and other mathematical systems to solve problems in their
everyday lives.
You can use inductive reasoning to find the next terms in a sequence.
Example
1
Find the next three terms of the sequence 33, 39, 45,....
Study the pattern in the sequence.
33, 39, 45
6
6
Each term is 6 more than the term before it. Assume that this pattern
continues. Then, find the next three terms using the pattern of adding 6.
33, 39, 45, 51, 57, 63
6
6
6
6
6
The next three terms are 51, 57, and 63.
Your Turn
Find the next three terms of each sequence.
a. 1.25, 1.45, 1.65,...
b. 13, 8, 3,...
c. 1, 3, 9,...
d. 32, 16, 8,...
4 Chapter 1 Reasoning in Geometry
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Example
2
Find the next three terms of the sequence 1, 3, 7, 13, 21,....
1, 3, 7, 13, 21
2
4
6
8
Notice the pattern 2, 4, 6, 8, . . . . To find the next terms in the
sequence, add 10, 12, and 14.
1, 3, 7, 13, 21, 31, 43, 57
2
4
6
8
10
12
14
The next three terms are 31, 43, and 57.
Your Turn
Find the next three terms of each sequence.
e. 10, 12, 15, 19,...
f. 1, 2, 6, 24,...
Some patterns involve geometric figures.
Example
3
Draw the next figure in the pattern.
There are two patterns to study.
• First, the pattern with the squares (S) and triangles (T) is SSTTSS.
The next figure should be a triangle (T).
• Next, the pattern with the colors white (W) and blue (B) is
WBWBWB. The next figure should be white.
Therefore, the next figure should be a white triangle.
Your Turn
g.
Lesson 1–1 Patterns and Inductive Reasoning 5
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Throughout this text, you will study many patterns and make
conjectures. A conjecture is a conclusion that you reach based on
inductive reasoning. In the following activity, you will make a conjecture
about rectangles.
Materials:
grid paper
ruler
Step 1 Draw several rectangles on the
grid paper. Then draw the
diagonals by connecting each
corner with its opposite corner.
Step 2 Measure the diagonals of each
rectangle. Record your data in a
table.
diagonals
Try These
1. Make a conjecture about the diagonals of a rectangle.
2. Verify your conjecture by drawing another rectangle and measuring
its diagonals.
A conjecture is an educated guess. Sometimes it may be true, and other
times it may be false. How do you know whether a conjecture is true or
false? Try out different examples to test the conjecture. If you find one
example that does not follow the conjecture, then the conjecture is false.
Such a false example is called a counterexample .
Example
Number Theory Link
4
Akira studied the data in the table at the right and made the
following conjecture.
Factors Product
28 6
515 75
20 38 760
54 62 3348
The product of two positive numbers
is always greater than either factor.
Find a counterexample for his conjecture.
The numbers
1
2
and 10 are positive numbers.
1
2
However, the product of
and 10 is 5, which is less than 10.
Therefore, the conjecture is false.
Businesses often look for patterns in data to find trends.
6 Chapter 1 Reasoning in Geometry
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