Download Unit 1 Section 1 9-10

Unit
Proof, Parallel and 1
Perpendicular Lines
Unit Overview
In this unit you will begin the study of an axiomatic
system, Geometry. You will investigate the concept
of proof and discover the importance of proof in
mathematics. You will extend your knowledge of the
characteristics of angles and parallel and perpendicular
lines and explore practical applications involving
angles and lines.
Essential Questions
?
Why are properties,
postulates, and theorems
important in mathematics?
?
How are angles and parallel
and perpendicular lines used
in real-world settings?
© 2010 College Board. All rights reserved.
Academic Vocabulary
Add these words and others you encounter in this unit
to your Math Notebook.
angle bisector
midpoint of a segment
complementary angles
parallel
conditional statement
perpendicular
congruent
postulate
conjecture
proof
counterexample
supplementary angles
deductive reasoning
theorem
inductive reasoning
EMBEDDED
ASSESSMENTS
These assessments, following
Activities 1.4, 1.7, and 1.9, will give
you an opportunity to demonstrate
what you have learned about
reasoning, proof, and some basic
geometric figures.
Embedded Assessment 1
Conditional Statements and Logic
p. 37
Embedded Assessment 2
Angles and Parallel Lines
p. 65
Embedded Assessment 3
Slope, Distance, and Midpoint
p. 81
1
UNIT 1
Getting
Ready
1. Solve each equation.
a. 5x - 2 = 8
b. 4x - 3 = 2x + 9
c. 6x + 3 = 2x + 8
6. Give the characteristics of a rectangular prism.
7. Draw a right triangle and label the hypotenuse
and legs.
8. Use a protractor to find the measure of each
angle.
a.
2. Graph y = 3x - 3 and label the x- and
y-intercepts.
3. Tell the slope of a line that contains the points
(5, -3) and (7, 3).
1
4. Write the equation of a line that has slope __
3
and y-intercept 4.
5. Write the equation of the line graphed below.
10
b.
y
8
c.
6
4
2
–10 –8 –6 –4 –2
–2
2
4
6
8
10
x
–4
–6
–8
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–10
2
SpringBoard® Mathematics with Meaning™ Geometry
Geometric Figures
What’s My Name?
ACTIVITY
1.1
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Interactive Word Wall, Activating Prior Knowledge,
Group Presentation
Below are some types of figures you have seen in earlier mathematics
courses. Describe each figure. Using geometric terms and symbols, list as
many names as possible for each figure.
1. Q
2. F
3. X Y
5.
!
G
H
4. D
Z
6.
m
E
J
χ
K
© 2010 College Board. All rights reserved.
7.
P
T
N
A
D
B
C
TRY THESE A
Identify each geometric figure. Then use symbols to write two different
names for each.
a. Q
P
b.
N
C
Unit 1 • Proof, Parallel and Perpendicular Lines
3
ACTIVITY 1.1
continued
Geometric Figures
What’s My Name?
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share,
Interactive Word Wall, Activating Prior Knowledge,
Discussion Group, Self/Peer Revision
My Notes
8. Draw four angles with different characteristics. Describe each angle.
Name the angles using numbers and letters.
9. Compare and contrast each pair of angles.
When you compare and contrast
two figures, you describe how
they are alike or different.
4
a.
b.
1
SpringBoard® Mathematics with Meaning™ Geometry
A
2
D
B
C
© 2010 College Board. All rights reserved.
MATH TERMS
Geometric Figures
ACTIVITY 1.1
continued
What’s My Name?
SUGGESTED LEARNING STRATEGIES: Discussion Group,
Peer/Self Revision
c.
R
D
60°
30°
E
d.
F
S
T
D
150°
X
30°
E
My Notes
F
Y
Z
10. a. The figure below shows two intersecting lines. Name two angles
that are supplementary to ∠4.
b. Explain why the angles you named in part a must have the same
measure.
3
4
5
© 2010 College Board. All rights reserved.
6
Unit 1 • Proof, Parallel and Perpendicular Lines
5
ACTIVITY 1.1
continued
Geometric Figures
What’s My Name?
My Notes
SUGGESTED LEARNING STRATEGIES: Discussion Group,
Create Representations, Activating Prior Knowledge
TRY THESE B
Complete the chart by naming all the listed angle types in each figure.
C
8 7
10 9
A
D
B
E
F
Acute angles
Obtuse angles
Angles with the
same measure
Supplementary
angles
11. In the circle below, draw and label each geometric term.
a. Radius OA
b. Chord BA
c. Diameter CA
12. Refer to your drawings in circle above. What is the geometric term for
point O?
6
SpringBoard® Mathematics with Meaning™ Geometry
© 2010 College Board. All rights reserved.
Complementary
angles
Geometric Figures
ACTIVITY 1.1
continued
What’s My Name?
SUGGESTED LEARNING STRATEGIES: Discussion Group,
Create Representations, Activating Prior Knowledge
My Notes
13. In the space below, draw a circle with center P and radius
1 in. Locate a point B
PQ = 1 in. Locate a point A so that PA = 1__
2
3 in.
so that PB = __
4
14. Use your diagram to complete these statements.
a. A lies _________________ the circle
because ___________________________________.
b. B lies _________________ the circle
because ___________________________________.
© 2010 College Board. All rights reserved.
15. In your diagram above, draw circles with radii PA and PB.
These three circles are called _______________________.
Unit 1 • Proof, Parallel and Perpendicular Lines
7
Geometric Figures
ACTIVITY 1.1
continued
What’s My Name?
My Notes
TRY THESE C
Classify each segment in circle O. Use all terms that apply.
___
AF : ________________________
___
BO : ________________________
A
___
B
C
CO : ________________________
___
DO : ________________________
O
___
EO : ________________________
___
CE : ________________________
F
E
D
CHECK YOUR UNDERSTANDING
Writeyour
youranswers
answersononnotebook
notebook
paper.
Show
Write
paper.
Show
youryour work.
6. Compare and contrast the terms acute angle,
work.
obtuse angle, right angle, and straight angle.
2. Describe all acceptable ways to name a plane.
3. Compare and contrast collinear and coplanar
points.
4. What are some acceptable ways to name an
angle?
5. a. Why is there a problem with using ∠B to
name the angle below?
b. How many angles are in the figure?
A
C
D
B
8
SpringBoard® Mathematics with Meaning™ Geometry
7. The measure of ∠A is 42°.
a. What is the measure of an angle that is
complementary to ∠A?
b. What is the measure of an angle that is
supplementary to ∠A?
8. Draw a circle P.
a. Draw a segment that has one endpoint on
the circle but is not a chord.
b. Draw a segment that intersects the circle in
two points, contains the center, but is not a
radius, diameter, or chord.
9. MATHEMATICAL How can you use the
R E F L E C T I O N figures in this activity to
describe real-world objects and situations?
Give examples.
© 2010 College Board. All rights reserved.
1. Describe all acceptable ways to name a line.