Math 11 Home
Book 6: Angles and Triangles
Name: _____________________________
Start Date: ______________
Completion Date: ________________
Book 6: Math 11 Home - Angles and Triangles
Edited April 2015
Year Overview:
Earning and
Spending Money
1. Earning Money
2. Pay Statements
and Deductions
3. Responsible
Spending Habits
Home
4. Data in Your Life
5. Measurement
6. Angles and
Triangles
Travel and
Transportation
7. Let’s Travel
Project
Recreation and
Wellness
8. Personal Health
and Wellness
9. Puzzles and
Games
Topic Overview
The intent of this theme is to develop a deeper understanding of the
applications of data collection and analysis, measurement, and geometry for
the purpose of designing, building, and maintaining a home and yard.
Outcomes
Overlapping Outcomes
M11.1 Extend understanding of arithmetic operations to rational numbers to
solve problems within the home, money, recreation, and travel themes.
Theme Specific Outcomes
M11.5 Demonstrate understanding of angles to solve problems within the home
theme.
M11.6 Demonstrate understanding of the Pythagorean Theorem to
solve problems within the home theme.
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Book 6: Math 11 Home - Angles and Triangles
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Contents
Topic Overview.................................................................................................................. 1
Outcomes ....................................................................................................................... 1
Overlapping Outcomes ........................................................................................... 1
Theme Specific Outcomes....................................................................................... 1
Contents ......................................................................................................................... 2
Glossary of Terms ........................................................................................................... 3
Angles and Triangles......................................................................................................... 5
Check What You Know ................................................................................................ 6
6.1 Angles ........................................................................................................................... 7
Discuss the Ideas ........................................................................................................... 7
6.1a Practice Your Skills: Measuring and Drawing Angles ...................................... 8
6.1bPractice Your Skills – Identify and Name Angles ............................................. 11
6.1c Practice Your Skills – Drawing Angles............................................................... 13
6.2 Angle Construction and Bisection ......................................................................... 14
6.2 Practice Your Skills – Constructing Angle Bisectors .......................................... 15
6.3 Complementary, Supplementary and Vertically Opposite Angles.................. 16
A. Complementary Angles ........................................................................................ 16
6.3A Practice Your Skills: Complementary Angles.................................................. 17
B. Supplementary Angles ........................................................................................... 18
6.3B Practice Your Skills: Supplementary Angles .................................................... 19
C. Vertically Opposite Angles ................................................................................... 20
6.3C Practice Your Skills – Vertically Opposite Angles ........................................... 21
6.3 Practice Your Skills - Go Fish or Memory Card Games .................................... 22
6.4 Square Roots and Irrational Numbers .................................................................... 24
6.4 Practice Your Skills – Determining Square Roots ............................................... 25
6.5 The Right Triangle and Pythagorean Theorem .................................................... 26
6.5a Practice Your Skills – Pythagorean Theorem .................................................. 29
6.5b Practice Your Skills – Missing Leg Length in a Right Triangle......................... 30
6.5c Practice Your Skills – Right Triangle Problem Solving ..................................... 31
Student Evaluation .......................................................................................................... 39
Learning Log .................................................................................................................... 41
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Book 6: Math 11 Home - Angles and Triangles
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Glossary of Terms
acute angle
an angle that is less than 90 degrees
angle bisector
a line passing through the vertex of an angle that cuts it into two equal smaller
angles
angles
created whenever two lines of straight surfaces meet
complementary angles
two angles that add up to 90 degrees
irrational numbers
a real number that cannot be written as a fraction
non-perfect squares
a square root that has a root that is not an integer
obtuse angle
an angle that is greater than 90 degrees but less than 180 degrees
perfect squares
a number that is the square of a whole number
protractor
an instrument used to measure angles
Pythagorean Theorem
a triangle in which the square of the hypotenuse is equal to the sum of the
squares of the other two sides
ray
a portion of a line which starts at a point and goes off in a particular direction
to infinity
reflex angle
an angle that is greater than 180 degrees but less than 360 degrees
right angle
an angle that is exactly 90 degrees
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Book 6: Math 11 Home - Angles and Triangles
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square roots
a number which, when multiplied by itself, results in a given number
straight angle
an angle that is exactly 180 degrees
supplementary angles
two angles that add up to 180 degrees
triangle
a three sided figure created when connecting any three points
vertex
the point at which two rays meet to form an angle
vertically opposite angles
the angles opposite each other when two lines cross
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Math 11 Home
Angles and Triangles
Angles are created
whenever two lines
or straight surfaces
meet. In our homes,
we need to know
the size of angles
when cutting
baseboards,
creating rafters,
and landscaping
our yards.
A triangle is a three sided figure. You can create a triangle by connecting any
three points. There is only 1 situation where you cannot make a triangle from 3
points. Can you think of what three points cannot make a triangle?
Book 6: Math 11 Home - Angles and Triangles
Edited April 2015
Check What You Know
1. Circle the numbers that are perfect squares?
1
2
5
4
8
9
14
16
2. Find the square roots:
a) √25
b) √100
c) √36
Math skills are embedded
into real life situations. In this
unit, you will use the
following skills:
• Angle terms: ray,
vertex
• Angle types: acute,
obtuse, reflex, straight
and right
• Perfect squares
• Square roots
d) √1
3. Match the following terms with
their definitions.
a) an angle that is greater than 90°
but less than 180°.
_____ 1) Ray
b) an angle that is exactly 180°.
_____ 2) Vertex
c) an angle that is greater than 180°
but less than 360°.
_____ 3) Acute angle
d) an angle that is exactly 90°.
_____ 4) Obtuse angle
e) an angle that is less than 90°
_____ 5) Reflex angle
f) the point at which two rays meet
to form an angle
_____ 6) Straight angle
_____ 7) Right angle
g. a portion of a line which starts at a
point and goes off in a particular
direction to infinity.
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Book 6: Math 11 Home - Angles and Triangles
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6.1 Angles
To measure angles, we can use a protractor. There are different types of
protractors, including those that are a full circle and those that are a half circle.
Half circle protractors are very common in school.
Discuss the Ideas
Brainstorm: “What is an angle?” Jot down all of your ideas. Share your ideas.
Consider the following questions:
1. If an angle is a measure of rotation, how many degrees is one rotation?
2. How can you estimate the measure of an angle?
3. How are angles measured?
4. Can you measure a reflex angle?
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Book 6: Math 11 Home - Angles and Triangles
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6.1a Practice Your Skills: Measuring and Drawing Angles
Common Angles
1.
One of these angles is 30°.
Which one do you think it is?
2.
One of these angles is 300°.
Which one do you think it is?
3.
One of these angles is 130°.
Which one do you think it is?
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Book 6: Math 11 Home - Angles and Triangles
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4. Create referent angles of 30°, 45°, 60°, 90°, 180° using a clock face, folding
paper, etc. Sketch an angle of given measure (e.g. 38°) using the referents.
Referent Angle
Sketch of Given Measure
30°
38°
45°
51°
60°
66°
90°
95°
180°
192°
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Book 6: Math 11 Home - Angles and Triangles
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5. Use a circle to create and label acute, obtuse, right, straight and reflex
angles.
a)
b)
c)
d)
e)
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Book 6: Math 11 Home - Angles and Triangles
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6.1bPractice Your Skills – Identify and Name Angles
Part 1 - Identify Angles: Identify and name types of angles (e.g. acute, obtuse,
right, straight and reflex) seen in objects located in your classroom.
Angle type
Object displaying
Practice: Angle Worksheets www.math-aids.com/Geometry/Angles/
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Book 6: Math 11 Home - Angles and Triangles
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Part 2 - Name Angles: Classify each angle as acute, obtuse, right or straight.
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Book 6: Math 11 Home - Angles and Triangles
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6.1c Practice Your Skills – Drawing Angles
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Book 6: Math 11 Home - Angles and Triangles
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6.2 Angle Construction and Bisection
Materials: Mira, compass, protractor, straight edge
Constructions
Angle Bisectors
Paper Folding (Informal
Construction)
MIRA (Informal
Construction)
Compass and Straight
Edge
(Formal Construction)
Note: It is easier to draw
the intersecting arcs when
the radius of the compass
is greater than half of the
length of the line segment
shown.
Note: No measuring is
involved in constructions!
Fold paper at the vertex, making sure that one ray is
directly matched over the other.
Place MIRA on the vertex, through the middle of the
angle. Reflect one ray onto the other. Draw in the dotted
line.
Place compass on vertex. Draw an arc of the same
length on each ray. Place compass on points where arcs
intersect the rays and draw a new arc from each. Draw a
line connecting vertex and the new intersection point.
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Book 6: Math 11 Home - Angles and Triangles
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6.2 Practice Your Skills – Constructing Angle Bisectors
Use any strategy or tool you need to complete the following:
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Book 6: Math 11 Home - Angles and Triangles
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6.3 Complementary, Supplementary and Vertically
Opposite Angles
A. Complementary Angles
Two angles are complementary when they add up to
90 degrees (a right angle). They don’t have to be
next to each other, just so long as the total is 90
degrees. For example, 40° and 50° are
complementary angles.
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Book 6: Math 11 Home - Angles and Triangles
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6.3A Practice Your Skills: Complementary Angles
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Book 6: Math 11 Home - Angles and Triangles
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B. Supplementary Angles
Two angles are
supplementary when they
add up to 180 degrees. They
don’t have to be next to
each other, just so long as
they add up to 180 degrees.
For example, 40° and 140°
are supplementary angles.
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Book 6: Math 11 Home - Angles and Triangles
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6.3B Practice Your Skills: Supplementary Angles
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Book 6: Math 11 Home - Angles and Triangles
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C. Vertically Opposite Angles
Vertically opposite angles are the angles
opposite each other when two lines cross.
They are always equal. In this example a°
and b° are vertically opposite.
Notice in the diagram below, the 4 angles are actually two oairs of vertical
angles.
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Book 6: Math 11 Home - Angles and Triangles
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6.3C Practice Your Skills – Vertically Opposite Angles
Find the missing vertical angles.
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Book 6: Math 11 Home - Angles and Triangles
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6.3 Practice Your Skills - Go Fish or Memory Card Games
Resource: Grade 6-9 Math Workshop: Symmetry, Lines, and Angles, pp.11 - 12
Materials: Cards numbered 0° to 180° plus any extras. These are in Appendix 2.
Instructions:
• Each group is provided with a set of cards that are to be shuffled. The game
can be played in a variety of ways:
1. Each player is dealt 6 cards. Taking turns, each player asks someone else
if he/she has a particular card. The card that is requested must be one
that is a supplement to a card in the player’s hand. If the card is a
supplement, he/she must give it to the requester. If not, the player
requesting draws one card. If the card is a supplement to a card in the
player’s hand, he/she may lay the two cards down. If not, the player
keeps the card in his/her hand and play moves to the next player. The
game is completed when one player has laid down all of his/her cards as
supplementary angles.
2. Each player is dealt 6 cards. The goal of this version is also to collect pairs
of angle cards that are supplements, but in this version a player does not
request cards from other players. Rather he/she draws a single card at
the start of each turn or takes the last previously discarded card. The
player ends his/her turn by discarding one card. When a player has all of
the angle cards in his/her hand matched as supplements, the game is
completed.
3. Spread the cards out face down. Players take turns flipping over two
cards. If the cards are supplementary angles, the player removes the two
cards and draws again. If they are not supplementary, the cards are
flipped back over and the next player starts his/her turn. The game ends
after a set period of time or once there are no remaining pairs. The
player who has collected the most pairs is the winner.
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Book 6: Math 11 Home - Angles and Triangles
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Adaptations:
• Have students write the rules for their favorite version of the game.
• Play these games using complementary angles. Remove all cards which
have obtuse and straight angle measures.
• The “go fish” version may also be played where rather than asking for a
particular value (40) the person asks for the complement to 50 degrees.
• Increase the difficulty level by playing both supplementary and
complementary at the same time.
• Once students are familiar with integers, negative angles could be
introduced.
• Cards including angles with decimals may also be added to the game (e.g.,
37.5°, 78.6°).
Extension: In any of the versions of the game, students with remaining angle
cards could be asked to either construct the angle and its complement and/or
supplement or to sketch them.
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Book 6: Math 11 Home - Angles and Triangles
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6.4 Square Roots and Irrational Numbers
Check your understanding of square roots, perfect squares, non-perfect squares
and irrational numbers using benchmarks.
Knowing perfect squares are helpful when estimating the square roots of nonperfect squares.
Number
(Square Root)
1
2
3
4
5
6
7
8
9
10
Perfect Square
(Squared
Number)
1
4
9
16
25
36
49
64
81
100
Number
(Square Root)
11
12
13
14
15
16
17
18
19
20
Perfect Square
(Squared
Number)
121
144
169
196
225
256
289
324
361
400
Example:
 To estimate√7.5, visualize the number line and the closest perfect
square on each side of 7.5.
√4 = 2
√9 = 3
 √7.5 is closer to 3 than to 2. From the diagram, an approximate
value for √7.5 is 2.7. We write √7.5 ≈ 2.7.
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Book 6: Math 11 Home - Angles and Triangles
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6.4 Practice Your Skills – Determining Square Roots
Using the benchmark method, determine the approximate value for the
following. Between which two consecutive whole numbers is each square root?
1. √12
2. √20
3. √54
4. √95
5. √135
6. Select 2 of your favorite numbers (For example, hockey shirt, house address,
birth date, birth year, number of Facebook friends, etc.) With these numbers,
use the benchmark method, and determine the approximate value of the
square root of each.
a.
b.
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Book 6: Math 11 Home - Angles and Triangles
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6.5 The Right Triangle and Pythagorean Theorem
What is a right angled triangle?
A right angled triangle is a triangle that has a right angle (90°) in it.
The little square in the corner tells us that it is a right angled triangle.
In a right-angled triangle, we use certain words to describe its parts.
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Book 6: Math 11 Home - Angles and Triangles
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Pythagorean Theorem
Over 2000 years ago a Greek Mathematician noticed a special relationship
when a triangle has a right angle (90°). When squares are made on each side of
the triangle, the biggest square has the exact same area as the other two
squares put together.
Interactive: To learn about right triangles, try this site and select right triangles:
Triangles http://www.mathsisfun.com/geometry/triangles-interactive.html
Interactive: To learn about Pythagoras` Theorem, try this site:
Pythagoras http://www.mathsisfun.com/pythagoras.html
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Book 6: Math 11 Home - Angles and Triangles
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Pythagorean Theorem is used to find the hypotenuse in the right angles below.
a2 + b2 = c2
1. a = 15, b = 20
2. a = 48, b = 55
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Book 6: Math 11 Home - Angles and Triangles
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6.5a Practice Your Skills – Pythagorean Theorem
Find the hypotenuse in the right angles below.
1.
2.
3.
http://www.math-aids.com
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Book 6: Math 11 Home - Angles and Triangles
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6.5b Practice Your Skills – Missing Leg Length in a Right Triangle
Find the missing side of each triangle. Round your answer to the nearest tenth if
necessary.
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Book 6: Math 11 Home - Angles and Triangles
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6.5c Practice Your Skills – Right Triangle Problem Solving
Unknown Side Lengths in Right Triangles
Complete the problems. Make sure to draw pictures to help you solve the
problems.
1. Tom has a bird named Jerry. Jerry was sitting 5 metres away from the base
of a tree. He sees another bird on a branch of a tree 12 metres away. Find
the exact distance between Jerry and the other bird.
2. Jack had to paint the wall of his bedroom so he used a ladder. He placed
the ladder 10 feet from a wall. The ladder reaches 13 feet up on the wall.
Find out how long the ladder is.
3. Fred bought an LCD TV 12 inches long. The diagonal of the LCD TV
measures 20 inches. Find the width of the TV.
4. In triangle OPQ, OQ is 17 meters, PQ is 15 meters, find OP.
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Book 6: Math 11 Home - Angles and Triangles
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6.6 Right Triangles in Buildings and in Construction
Often, when builders want to make sure that they are building something that is
square, they use the Pythagorean Theorem. To do this, they check the side
lengths and diagonal length of a triangle to see if it satisfies the Pythagorean
Theorem. They Pythagorean Theorem is true for right triangles only, so if it works,
it means that the angle created is a right angle (90°)
Whether it is new construction of a house or a stairway that has seen some time,
right triangles and the Pythagorean Theorem also define relationships of
distances. Sometimes, distances can’t be determined with a carpenter’s tape,
either, because obstacles are in the way or the distances are too large. In these
cases, the Pythagorean Theorem has often been of assistance.
Example 1:
Determine whether the triangle below is a right triangle.
If a2 + b2 = c2, then it is a right triangle.
a2 + b2 ≟ c2
82 + 242 ≟ 252
64+ 576 ≟ 625
640 ≠ 625
This is not a right triangle.
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Book 6: Math 11 Home - Angles and Triangles
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Example 2:
Adam is building a rectangular patio with dimensions of 18 ft x 24 ft. To check if
his corners are square, he measures the diagonal which is 30 ft. Are his corners
square? (If his corners are square, it means that the angle between the two sides
a right angle). Justify your answer.
a2 + b2 ≟ c2
182 + 242 ≟ 302
324 + 576 ≟ 900
900 = 900
Yes, his corners are square because his two sides and the diagonal make a right
triangle.
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Book 6: Math 11 Home - Angles and Triangles
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6.6 Practice your Skills – Right Triangles in Building and Construction
1. Determine whether the triangle below is a right triangle.
2. Each set of measurements below represents the side lengths of a
triangle.
Identify which triangles are right triangles.
How do you know?
a) 3 cm, 4 cm, 6 cm
b) 7 m, 24 m, 25 m
c) 6 cm, 8 cm, 10 cm
d) 1 m, 2 m,
5m
e) 2 m, 3 m,
12 m
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Book 6: Math 11 Home - Angles and Triangles
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3. Joe is building a frame for his window.
The frame is 88 cm wide and 105 cm tall.
He measures the diagonal of his frame and finds that it is 137 cm.
Is the frame a rectangle? Justify your answer.
4. Danielle is laying a foundation for a farm shed with dimensions 10 m by
6 m. To check that the foundation is square, Danielle measures a
diagonal.
How long should the diagonal be?
Give your answer to one decimal place.
5. Juan is building a doorframe with dimensions of 1 m by 2 m. He
measures the diagonal to be 2.7 m. Is the angle between the two
sides a right angle? Justify your answer.
6. Ben is laying a plywood floor in his cabin. The floor is rectangular with
side lengths 9 m and 12 m. He measures the diagonal of the floor as 15
m. Is the angle between the two sides a right angle? Justify your
answer.
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Book 6: Math 11 Home - Angles and Triangles
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7. The photograph shows stairs
going up a rock wall in a
diagonal pattern. If the set of
stairs is 9.4 m long and the
horizontal distance that a
person travels going up the
stairs is 7.5 m, what is the height
of the stairs? Give your answer
in metres to one decimal
place.
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Book 6: Math 11 Home - Angles and Triangles
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8. In the photograph of the house
under construction, a long 2 by 4
diagonal brace is shown near the
centre of the picture. If the framers
of the house know that the height
of the wall is going to be 2.44 m
and they want to brace the wall at
right angles to the floor, how far
from the wall should they place the
bottom of the 3.66 m diagonal
brace? Give your answer in metres
to two decimal places.
In the same picture above, you can see a gable end of the roof. If the width of
the roof is 4.28 m and the height of the gable end is 1.22 m, what is going to be
the length of each side of the roof truss? Give your answer in metres to two
decimal places.
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Book 6: Math 11 Home - Angles and Triangles
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Insufficient
Evidence (IE)
Student has not
demonstrated the
criteria below.
Developing (D)
Growing (G)
Proficient (P)
Exceptional (E)
Student has rarely
demonstrated the
criteria below.
Student has
inconsistently
demonstrated the
criteria below.
Student has
consistently
demonstrated the
criteria below.
Student has consistently
demonstrated the criteria
below. In addition they
have shown their
understanding in novel
situations or at a higher level
of thinking than what is
expected by the criteria.
Proficient Level Criteria
IE
D
G
P
M11.1 Extend understanding of arithmetic operations to rational
numbers to solve problems within the home, money, recreation, and
travel themes.
d. I can round decimals.
f. I can determine if my answer is reasonable.
M11.5 [WA 10.9] Demonstrate understanding of angles to solve
problems within the home theme.
a. I can use appropriate referents for a variety of angles to estimate
angle measurements (e.g., a corner of a sheet of paper is 90° so ½
of a corner is 45°).
b. I can measure angles in different orientations
c. I can be replicate and explain how angles can be drawn using a
variety of tools. (e.g., Mira, protractor)
d. I can identify, classify, and sketch angles of various measures,
including acute, right, straight, obtuse, and reflex angles.
e./f. I can bisect angles and explain what bisect means.
g. I can identify adjacent angles that are complementary,
supplementary, or neither, and explain why.
h. I can solve problems involving complementary and
supplementary angles.
i. I can identify vertically opposite angles and solve situational
problems.
38
E
Book 6: Math 11 Home - Angles and Triangles
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Student Evaluation
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Book 6: Math 11 Home - Angles and Triangles
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Proficient Level Criteria
IE
D
G
P
M11.6 [WA 10.6] Demonstrate understanding of the Pythagorean
Theorem to solve problems within the home theme.
a. I can explain the meaning, role, and use of the Pythagorean
Theorem, with examples and non-examples.
b. I can apply the Pythagorean 3:4:5 ratios to determine if angles
are square (right angles) in home construction contexts.
c. Apply the Pythagorean Theorem to solve for a missing side.
d. Estimate the values of irrational numbers using a table of perfect
squares, multiplication chart, or a number line and show appropriate
rounding of irrational numbers.
e. I can observe and analyze the use of Pythagorean lengths of
diagonals of various building structures (eg. Trusses, frames, door
jambs, window casings).
40
E
Book 6: Math 11 Home - Angles and Triangles
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Learning Log
Date
Starting Point
Ending Point
41