Course: Grade 9 Applied Mathematics (MFM1P)
Unit 2:
Plane Geometry
Unit 2
Plane Geometry
Section
2.1.1
2.1.2
2.1.4
2.1.P
2.5.1
2.5.2
2.6.1
2.7.1
2.7.2
2.7.J
2.7.P
2.S
2.R
2.RLS
Activity
I Remember
Plane Geometry Record Sheet
Parallel Lines Exploration – Optional
Practice
What’s So Special Guide Sheet
What’s So Special Record Sheet
Learn the Lingo
Exterior and Interior Angles of a Polygon
Interior Angle Sums
Journal Activity
Practice
Unit Summary Page
Reflecting on My Learning (3, 2, 1)
Reflecting on Learning Skills
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.1: I remember….
Work with a partner to complete the definitions below that you know. Leave the ones you are unsure of
and come back to them throughout this unit as you learn more about them.
Word/Term
Definition
Diagram
Supplementary Angles
Complimentary Angles
Opposite Angles
Corresponding Angles
Alternate Angles
Co-Interior Angles
Parallel Lines
Transversal
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.1: I remember….(continued)
Word/Term
Definition
Diagram
Triangle
Isosceles Triangle
Equilateral Triangle
Right Triangle
Acute Triangle
Obtuse Triangle
Scalene Triangle
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.1: I remember….(continued)
Word/Term
Definition
Diagram
Quadrilateral
Parallelogram
Rhombus
Trapezoid
Square
Rectangle
Hexagon
Polygon
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.2: Plane Geometry Record Sheet
Use this page to record your observations and conclusions from the Plane Geometry GSP®4
file. Determine the unknown angle in the right column. Give reasons for your answer.
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.2: Plane Geometry Record Sheet (continued)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.2: Plane Geometry Record Sheet (continued)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.4: Parallel Lines Exploration - Optional
Explore and Reflect
1. How did you know when Line 1 and Line 2 were
parallel?
Sketch
Angle Relationships
2. Find one pair of equal angles. Explain how you
know they are equal.
3. Find another pair of equal angles. Explain how you
know they are equal.
4. Find as many pairs of angles that are
supplementary (add to 180°) as you can. Explain
how you know.
Summary (to be completed as a whole class)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice
1.
Look at this diagram.
(a)
Name two parallel line segments.
(b)
Name two transversals.
(c)
Name two corresponding angles.
(d)
Name two alternate angles.
(e)
Find the measures of ∠ ECD, ∠ ACE, and ∠ BCA.
A
E
•
65º
50º
B
C
•D
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice (continued)
2.
(a)
Draw parallelogram ABCD with ∠ A = 51º.
(b)
How can you use what you know about parallel line segments and a transversal
to find the measures of the other 3 angles in the parallelogram? Explain your
work.
(c)
When is a quadrilateral a parallelogram? Explain.
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice Sheet (continued)
Define each principle and determine the unknown angles.
1.
xo =
Reason:
=
x° 85°
2.
70°
r°
ro =
=
Reason:
3.
mo =
=
Reason:
m°
30°
4.
q°
50°
55°
qo =
=
Reason:
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice Sheet (continued)
5.
bo =
60°
Reason:
=
b°
45°
6.
ao =
=
75°
Reason:
a°
7.
xo =
=
Reason:
x°
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice Sheet (continued)
8.
x°
xo =
Reason:
=
72°
68°
mo =
Reason:
wo =
Reason:
m°
83°
w°
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice (continued)
9.
Find the measure of a.
Give reasons.
5.
Find the value of x.
Give reasons.
68º
aº
xº
71º
10.
49º
46º
Find x. Give reasons.
7.
Find the values of the missing angles.
Give reasons.
yº
89º
44º
xº
42º
11.
42º
96º
xº
The diagram shows two parallel lines cut by a transversal. The measure of a + b is
_____. Give reasons.
aº
bº
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.1.P: Practice (continued)
12.
For the following diagram, list as many examples of each Angle Theorem as possible.
a° b°
c° d°
Z – pattern
C – pattern
F - pattern
e.g. ∠c = ∠ g
e° f°
g°
h°
13.
Solve for x and y
a)
b)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.5.1: “What’s So Special?” Guide Sheet
Explore!
Drag each vertex in the figure.
As you drag vertices, look for some of the following:
• measurements that always seem to be equal to each other
• measurements that never seem to change
• measurements that might have a constant ratio (proportional)
• lines that always seem to be parallel or perpendicular
• line segments that always seem to be bisected
• figures that always seem to be congruent
• objects that don’t seem to be connected, yet they move together when something is
dragged
Make an Hypothesis
Decide which measurements you need to test your hypothesis.
Drag each vertex again while you pay close attention to the way the object moves and to the
way the measurements change.
Test Your Hypothesis
Collect and record evidence to test your hypothesis.
What can you measure?
• angles
• lengths
• areas
• perimeters
• slopes
•
•
What can you calculate?
• sums
• ratios
• formulas
•
•
•
•
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.5.2: What’s So Special? Record sheet
Investigation 1. Special Triangles
Explore: Move the vertices of the triangles around and examine how the angles and side
lengths change.
Hypothesis: Make a hypothesis about what type of triangle each figure is and record it in the
chart below.
Test your Hypothesis: Make any measurements that will help test your hypothesis.
Hypothesis:
Type of Triangle
Conclusion:
Type of Triangle
Evidence:
(what measurements support your
conclusion)
ΔABC
ΔDEF
ΔGHI
ΔKLM
Investigation 2. Parallel or Perpendicular?
Explore: Drag the endpoints of the line segments.
Hypothesis: Make a hypothesis about which lines are parallel, which are perpendicular, and
which are neither.
Test your Hypothesis: Make any measurements that will help test your hypothesis.
Conclusions: Make a statement about which lines are parallel and which are perpendicular
and provide evidence (which measurements support your claim)
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.5.2: What’s So Special? Record sheet
Investigation 5: Special Quadrilaterals?
Explore: Drag each vertex of each figure.
Hypothesis: Make a hypothesis about what type of quadrilateral each figure is and record
your hypothesis in the chart below.
Test your Hypothesis: Make any measurements that will help test your hypothesis.
Conclusions:
Evidence
Quadrilateral
Hypothesis
Conclusions
(What measurements prove your
conclusions?)
A
B
C
D
E
F
G
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.6.1: Learn the Lingo
1. Part a) shows an example of how to complete a word chart.
Complete the remaining word charts.
a)
Term:
Visual
Representation:
b)
Term:
Equilateral
Triangle
Definition:
An equilateral
triangle is a triangle
for which all sides
have the same
length.
Visual
Representation:
Triangle
Association:
A Yield sign
c)
Definition:
Association:
d)
Term:
Visual
Representation:
Exterior
Angle
Definition:
Term:
Visual
Representation:
Interior
Angle
Association:
Definition:
Association:
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.6.1: Learn the Lingo (continued)
e)
Term:
Visual
Representation:
Parallel Lines
Definition:
Association:
Definition:
Association:
h)
Visual
Representation:
Perpendicular
Bisector
Definition:
Visual
Representation:
Transversal
g)
Term:
f)
Term:
Term:
Visual
Representation:
Diagonal
Association:
Definition:
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.6.1: Learn the Lingo (continued)
2. Determine the unknown angle. Give reasons for your answer.
b)
a)
c)
T
D
A
U
°
108
B
E
G
C
F
A
∠DEG = ____
W
Z
AB = AC = BC
∠ACB = ________
X
V
Y
TZ UY
∠TWX = 75o
∠UVW = __
d)
e)
C
f)
M
F
N
C
O
O
D
D
E
S
∠COD = 64
∠FOE = ___
h)
W X
°
42
S
T
∠BOC = 43o
∠COE = ____
∠EOD = ____
∠NQR = 115 o
∠MRQ = ____
i)
W
X
Y
R
Z
O
MP NO
∠COF = ___
V
E
R Q
P
o
g)
B
Create your own question!
Z
A
U
WX = WY
∠VRW = 42
RT = RU
o
∠YWX = 118o
∠WXZ = _____
∠SRT = 19o
∠RSZ =
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.7.1: Exterior and Interior Angles of a Polygon
Part A – Exterior Angles of a Triangle
Using the GSP files: Angles Triangles. gsp
1. Click the “Show Measurements” Tab.
2. Drag vertices A, B, and C.
3. What do you notice?
4. Click the “Reset the triangle” Tab.
5. Click the “Make the triangle smaller” Tab.
6. If we decrease the size of the triangle, what
do you notice about the sum of the exterior
angles?
Part B – Exterior Angles of a Quadrilateral
Similarly with the quadrilateral,
1. Click the “Make the quadrilateral smaller”
Tab.
2. Describe what just happened.
3. If we decrease the size of the
quadrilateral, what do you notice about the
sum of the exterior angles?
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.7.1: Exterior and Interior Angles of a Polygon (continued)
Part C – Exterior Angles of Any Polygon
Similarly with any polygon,
1. Click on the tab “Show Sum”.
2. Drag vertices A, B, and C.
3. What do you notice?
4. Click the “Reset” Tab.
5. Click the “Shrink polygon” Tab.
6. If we decrease the size of the polygon, what
do you notice about the sum of the exterior
angles?
7. Click the “One less side” Tab. What shape do
you have now? What is the sum of the exterior
angles?
8. Click the “Another side less” Tab. What shape
do you have now? What is the sum of the
exterior angles?
Compare the conclusions you reached in Part A and Part B.
Write your final conclusion about the sum of the exterior angles of any polygon.
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.7.2: Interior Angle Sums
1. Complete the chart.
Diagram
Number
of sides
Sum of
interior angles
Understanding
The sum of the angles in any triangle is
180o.
3
180°
4
5
n
2. a) Determine the sum of the interior angles in a polygon with 15 sides. Show your work.
b) Determine the number of sides in a polygon if the sum of the interior angles is 5400°.
Show your work.
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.7.2: Interior Angle Sums (continued)
3. Derek is building a deck for his summer job in the shape of a regular octagon.
a) Define: regular octagon
?
b) Determine the measure of the interior angles of the deck.
Show your work.
4. A Canadian $1 coin, known as a loonie, is a regular polygon with 11 sides, called an
undecagon.
a) Define a regular polygon with 11 sides.
b) Determine the sum of the interior angles of the loonie.
c) What is the size of one of the interior angles?
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.7.J: Journal Activity
Write a letter to Abe, who missed Math class, explaining how he can determine the sum of the
interior and exterior angles in a decagon (10-sided polygon).
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.7.P: Practice
1. Determine the measure of each indicated angle and state reasons.
a)
b)
c)
107º
41º
100º
104º
xº
xº
2.
70º
49º
Determine the values of x, y, and z. Give reasons.
a)
b)
xº
zº
3.
yº
xº
xº
108º
yº
c)
84º
64º
96º
47º
yº
132º
zº
zº
Determine the measures of a and b. Give reasons.
a)
b)
55º
105º
bº
25º
aº
115º
aº
83º
bº
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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57º
yº
2.7.P: Practice (continued)
4.
Find the measure of x in the following pentagon. Give reasons.
100º
xº
100º
5.
100º
100º
Find the measures of a, b, and c. Give reasons.
135º
bº
aº
cº
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.S: Unit Summary Page
Complete the following concept map to relate all the terms from this unit.
PLANE GEOMETRY
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.R: Reflecting on My Learning (3, 2, 1)
3 Things I know well from this unit
2 Things I need explained more
1 Question I still have
MFM 1P - Grade 9 Applied Mathematics – Unit 2: Plane Geometry (DPCDSB July 2008)
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2.RLS: Reflecting on Learning Skills
Students should be aware of the importance that these skills have on your performance. After
receiving your marked assessment, answer the following questions. Be honest with yourself.
Good Learning Skills will help you now, in other courses and in the future.
•
•
•
•
E – Always
G – Sometimes
S – Need Improvement
N – Never
Organization
• E G S N
• E G S N
• E G S N
I came prepared for class with all materials
My work is submitted on time
I keep my notebook organized.
Work Habits
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
• E G S N
I attempt all of my homework
I use my class time efficiently
I limit my talking to the math topic on hand
I am on time
If I am away, I ask someone what I missed,
I complete the work from the day that I missed.
Team Work
• E G S N
• E G S N
• E G S N
I am an active participant in pairs/group work
I co-operate with others within my group
I respect the opinions of others
Initiative
• E G S
• E G S
• E G S
• E G S
I participate in class discussion/lessons
When I have difficulty I seek extra help
After I resolve my difficulties, I reattempt the problem
I review the daily lesson/ideas/concepts
N
N
N
N
Works Independently
I attempt the work on my own
• E G S N
• E G S N
I try before seeking help
• E G S N
If I have difficulties I ask others but I stay on task
• E G S N
I am committed to tasks at hand
Yes No
I know all the different ways available in my school, where I can seek extra help.
Yes No
I tried my best.
What will I do differently in the next unit to improve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
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