geometry and trigonometry unit€¦ · geometry and trigonometry unit: # assignment completed?...

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Name:__________________________

Geometry and Trigonometry Unit:

# Assignment Completed? Comments

1. Review Lesson

2. Assignment #1

3. Sine Law

4. Assignment #2

5. Cosine Law

6. Assignment #3

7. Practice Solving with Trig

(no new lesson)

8. Assignment #4

9. How to Answer Exam Questions

10. Assignment #5

11. More Triangles

12. Assignment #6

13. Properties of Quadrilaterals

14. Assignment #7

15. Regular Polygons

16. Assignment #8

17. How to Answer Exam Questions

18. Assignment #9

Geo and Trig Test:_________________________

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Review Lesson #1

Labelling Triangles

Angles are labelled with upper case letters

Sides are labelled with lower case letters

The angles and opposite sides are labelled with

the same letters of the alphabet

Labelling Angles

Angles are labelled with upper case letters

An angle is named using three letters (the vertex, or the point where the

two arms of an angle meet, is the middle letter)

Labelling Triangles for SOH CAH TOA (trig)

The HYPOTENUSE is always the side across from the right angle

The OPPOSITE side is across from the angle you are working with

The ADJACENT side is next to the

angle you are working with

The opposite and adjacent sides are

labelled differently depending on which

angle you are using

*Remember, we don’t use the right angle

(90) when doing SOH CAH TOA

For example: Name this angle _____________

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5 Angle Rules

#1 – Complementary Angles

when two angles add up to 90

#2 – Supplementary Angles

when two angles add up to 180

#3 – Corresponding Angles

corresponding angles are equal

#4 – Vertically Opposite

Angles

vertically opposite angles are

equal

#5 – Alternate Angles

exterior and interior alternate

angles are equal

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Match the following terms with the appropriate examples:

#1 – Complementary Angles #2 – Supplementary Angles

#3 – Corresponding Angles #4 – Vertically Opposite Angles

#5 – Alternate Angles

#____

Angle X = ________

Angle Y = ________

#____ 45+135 = _______

#____

Angle 1 = Angle ___

Angle 2 = Angle ___

Angle 3 = Angle ___

Angle 4 = Angle ___

#____

25+65 = _______

#____

Angle X = ________

Angle Y = ________

Angle Z = ________

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SOH CAH TOA: This is a reminder of the following trig formulas

Sin θ = ________

Cos θ = ________

Tan θ = ________

Trigonometry can be used in everyday life to

solve for angles and missing sides for many

problems.

Example #1 – How to solve for a missing side:

Calculator steps:

Example #2 – How to solve for a missing angle:

Calculator steps:

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Assignment #1: Review

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Lesson #2 Sine Law

The Sine Law can be used with triangles that do not contain a 900 angle (called

oblique triangles) to solve for unknown sides and angles.

The Sine Law formula:

or

Example 1: Find the measures of side a, and side b, and angle C.

B

1100

59.6 m

23.10

A C

First:

Second:

Third:

When you are solving for an angle:

sin sinA B

a b

sin sinA C

a c

sin sinB C

b c

When you are solving for a side:

sin sin

a b

A B

sin sin

a c

A C

sin sin

b c

B C

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Example 2: Find the measures of angle D, angle F, and side f.

Example 3: Boats are anchored at positions J, K and M on a lake. Boats J and K are

80 m apart and J and M are 110 m apart. The angle between the lines of sight from

K to J, and K to M is 1200.

a) What is the angle between the lines of sight from J to K and J to M?

b) How far is it from K to M?

D

13 cm

F

45° 14 cm

E

First:

Second:

Third:

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Assignment #2: Sine Law

1. Given Δ JKL, if ∠J = 34°, ∠K = 72°, what is ∠L?

2. Find side b, given ∠B = 40°, ∠A = 70°, and a = 6 cm.

A

B C

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3. Determine the measure of ∠X.

X

11.7cm

5.6cm 108°

Z Y

NOTE: For the following questions, make sure to draw a diagram to help you

with the question.

4. There were two cabins on one side of the pond, Cabin A and Cabin B. The

distance to a boathouse on the opposite side of the pond, was found to be

2000 m from Cabin A and 1474 m from Cabin B.

If ∠ A = 25° and ∠ B = 35°, find the distance between the two cabins.

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5. A children’s slide is 10 feet long and inclines at 43° with the ground. The

ladder is 8 feet in length. What angle does the ladder make with the slide?

Note: the ladder is at a bit of an angle (it is not straight up).

6. Mr. Krahn is trying to make a winter fishing shack with the following

dimensions.

a) How long is the other section of the roof? 4.5 m

48° 48°

b) How wide is the shed, to the nearest tenth of a meter?

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Lesson #3: Cosine Law

The Sine Law allows you to solve for triangles where 2 sides and a corresponding

angle are known, or 2 angles and a corresponding side.

The cosine law allows you to solve for:

the third side of the triangle if you know 2 sides of a triangle and the angle

that is formed between these two sides. (SAS or side – angle – side)

any angle if you know the three side lengths of the triangle.

The Cosine Law formula:

To Find A Side:

a2 = b2 + c2 – 2bc cos A

To Find An Angle:

cos A = b2 + c2 – a2

(2bc)

Now that you know another trig formula, remember…

When solving triangles:

Check for right angles (900). Use basic trigonometric ratio’s (SOH CAH TOA)

Check for Sine Law Ratios (a side and an opposite angle). Use Sine Law.

If none of the above possibilities exist: Use Cosine Law.

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Example 1: Write the cosine formula for the missing side R of the following

triangle PQR.

To Find A Side:

a2 = b2 + c2 – 2bc cos A

Example 2: Write the cosine formula for the missing angle Q of the following

triangle PQR.

To Find An Angle:

cos A = b2 + c2 – a2

(2bc)

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Example 3: Find the measure of side a, angle B, and angle C.

First check… A

1. Right angle triangle?

2. Sine-Law ratio’s? (opposites?)

3. If no to both, use Cosine Law 55° 14 cm 18 cm

C

B

To Find A Side:

a2 = b2 + c2 – 2bc cos A

First: Find side a

Second: Find angle B

Third: Find angle C

To Find An Angle:

cos A = b2 + c2 – a2

(2bc)

Option #2

Second: Find angle B

Third: Find angle C

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Assignment #3: Cosine Law

1. Given ∆ABC. Solve for side a.

A 68°

c = 350 b= 475

B

C

2. Given ∆ABC. Solve for ∠A.

A

c = 55 b = 75

B a = 70 C

3. From a lighthouse, a cruise ship can be seen 8.3 km away and a freighter can also be

seen 12.5 km away. How far away is the cruise ship from the freighter if the angle

between the lines of observation are 68°?

Cruise Ship

8.3

Lighthouse 68°

?

12.5

Freighter

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4. Solve for all the interior angles.

A

c = 18 cm b = 20 cm

B C

a = 19 cm

NOTE: For the following questions, make sure to draw a diagram to help you with

the question.

5. At a provincial park, there is a sign, a reception area, and a picnic area. The

reception area is 350 m away from the picnic area, the picnic area is 475 m away

from the sign. From the picnic area, the angle between the 2 lines of sight for the

reception area and the sign is 64°. How far apart is the sign from the reception

area?

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6. An Art Gallery is in the shape of a triangle. Two of the walls are 114 m and 61 m in

length. The angle between these 2 walls is 72°.

a. How long is the 3rd wall?

b. What are the angles of the other 2 corners of the triangle?

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7. Construction has been started on a building as shown by the diagram.

12 ft

Pier

10 ft

Braces

a. What is the length of each brace?

b. What is the angle between both braces?

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Assignment #4: Applications of Sine & Cosine Laws (no new lesson; use your knowledge from the previous lessons)

When to use Cosine vs Sine Law?? Complete the following chart.

Information Give Measurement to be

Determined

Sine Law or Cosine Law

2 sides and the angle opposite one side

X

Y

Angle

2 angles and a side

X

Side

2 sides and the

contained angle Y

X

Side

3 sides

X y

z

Angle

INSTRUCTIONS: Diagrams are not to scale. If no diagram is given, sketch one to

represent the situation before completing the exercise.

Express all lengths to the nearest tenth and all angles to the nearest degree.

1. For each triangle, determine the indicated measures.

Find side a.

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Find angle X.

Find angle A and angle C.

Find angle X

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Note: Some questions describe an angle using the phrase

“angle of elevation” or “angle of depression”.

Use the following two diagram examples to help you review these

concepts.

2. From a certain point, the angle of elevation to the top of a church steeple is 9°. At

a point 100 m closer to the steeple, the angle of elevation is 15°. Calculate the

height of the steeple.

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3. A tower is supported by two guy wires attached to the top of the tower and fixed

to the ground on opposite sides of the tower 27 m apart. One wire is 19.3 m long

and meets the ground at an angle of 53°.

NOTE: Guy wires act like the cable supports on a camping tent.

a) What is the height of the tower?

b) What is the length of the second wire?

c) What angle does the second wire make with the ground?

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4. A triangular park has sides of length 200 m, 155 m and 172 m.

a) Determine Angle A.

b) Determine h.

c) Calculate the area of the park.

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5. To determine the height of a cliff, a surveyor measured the angle of elevation of

the top of the cliff from a point away from the base to be 45°. He then moved

20 m further away from the base of the cliff and found the angle of elevation to

the top to be 37°. Determine the height of the cliff.

6. The end of a lean-to for cattle is in the shape of an obtuse triangle as shown below.

a) Determine the length of the roof.

b) Determine the angle that the roof of the shed makes with the ground.

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7. In the design of a ski chalet, the slant of the roof must be steep enough for the

snow to slide off. An architect originally designed the roof to span 45 feet with

slanted sides of 36 ft and 30 ft.

a) Looking at the diagram below, which angle is the smallest? Why?

b) Calculate the smallest angle measure.

The architect decided it would be better to modify the roof by increasing the

measure of the smaller angle by 10°.

c) Calculate the new angle measure.

d) Predict what happens to the side opposite the new angle. Why?

e) Calculate the new length of this side.

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Lesson #5: EXAM QUESTIONS:

Applications of Sine & Cosine Laws

Standard SINE LAW Exam Question: The Sine Law is often used in construction, commercial, industrial, or artistic applications.

A) Demonstrate one use of the Sine Law in the real world by performing the following two steps: (2 marks) • State a specific example where Sine Law is used.

• Support your example with a written explanation of how Sine Law is used.

B) Sketch a reasonably neat picture or diagram (not necessarily to scale) that supports your example in Part A. (1 mark)

Exemplars of Full Mark Student Answers:

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Standard COSINE LAW Exam Question: The Cosine Law is often used in construction, commercial, industrial, or artistic applications.

A) Demonstrate one use of the Cosine Law in the real world by performing the following two steps: (2 marks) • State a specific example where Cosine Law is used.

• Support your example with a written explanation of how Cosine Law is used.


Exemplar of Full Mark Student Answers:

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Assignment #5: EXAM QUESTIONS:

Applications of Sine & Cosine Laws

1. Create one response to the following exam question that is your own: The Sine or Cosine Law is often used in construction, commercial, industrial, or artistic applications.

A) Demonstrate one use of the Sine or Cosine Law in the real world by performing the following two steps: (2 marks) • State a specific example where Sine or Cosine Law is used.

• Support your example with a written explanation of how Sine or Cosine Law

is used.


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Lesson #6: More Triangles

Triangles are often used in construction because they are naturally rigid and do not

easily collapse or change shape. Roof trusses and shelf brackets are common

structures created using triangles.

Quadrilaterals are not rigid structures. A rectangle can be pushed into a

parallelogram (in other words, the side lengths stay the same, only the angles

change).

Similarly, any polygon with four or more sides is not a rigid structure. However, if

you push a triangle, the shape cannot change without changing the side lengths.

Triangles classified by the angle measurement:

1. Oblique Triangle – a triangle with no angle equal to 90°. Oblique triangles can be

classified as acute or obtuse.

Acute angle - Obtuse angle –

push

push

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2. Right Angle Triangle – a triangle with 1 angle = 90°.

What do we know about interior angles on a triangle?

What about exterior angles in triangles?

What rules can we make about exterior and interior angles of triangles?

Triangles classified by side length:

Equilateral triangle:

Isosceles triangle:

Scalene triangle:

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Example #1: Solve for ∠A, ∠B, and ∠C. Example #2: Solve for ∠B.

Example #3: Solve for ∠A and ∠B.

You can also use the Pythagorean Theorem to solve for a missing side.

a2 + b2 = c2

Example #4. Solve for c. Example #5. Solve for a.

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Assignment #6: More Triangles

1a. Is an equilateral triangle acute or obtuse? Explain your reasoning.

b. Can a right triangle be an isosceles triangle? Why or why not?

c. Explain why it is or is not possible for an obtuse triangle to have more than one

angle larger than 90 .

2. Use the angle measures to calculate the interior and exterior angles in each

triangle.

120

105

20

32

55

A

B

C

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3. Label each triangle as acute, obtuse, right angle, isosceles, equilateral, and

scalene.

4. Given that ∆ABC is a right angled triangle, side a = 13, side b = 12, and side c is

the hypotenuse, use the Pythagorean Theorem to solve for side c.

5. Given that ΔRST is a right angled triangle, r = 7, s = 11. Use the Pythagorean

Theorem to solve for t.

s = 11

r = 7

t = ?

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Lesson #7: Properties of Quadrilaterals

A quadrilateral is any four sided figure (straight lines). All sides join to form a

closed path, and the interior angles always add up to 360°.

Figure A Figure B

Figure C Figure D

Figure E

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Property

Figure A

Parallelogram

Figure B

Rectangle

Figure C

Rhombus

Figure D

Square

Figure E

Isosceles

Trapezoid

Opposite sides are equal

All sides are equal

Opposite angles are equal

Opposite angles

are right-angles

Consecutive angles

are supplementary

Diagonals are equal

Diagonals cut each other

in half (bisect)

Diagonals cut opposite

angles in half (bisect)

Diagonals are

perpendicular to each

So, what did we learn??

A rectangle has _________ 90° angles.

A square is a rectangle with 4 ____________.

A parallelogram has opposite sides that are _____________ and

________________.

A rhombus is a parallelogram with _______ equal sides.

A trapezoid has only one pair of ____________ sides.

angles add up to 0.

angles add up to 0.

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Example 1: ABCD is a parallelogram. ABC = 1000. Find the measure of the

following and explain how you arrived at your answer:

a) CD =

b) DE =

3 E

c) BCD =

d) ADC =

e) ∠ADB =

Example 2: A given quadrilateral has the following properties:

the opposite sides have equal length

the measures of consecutive (or adjacent) angles are not equal

A) Draw the quadrilateral with these properties.

B) State the name of this quadrilateral.

B A 5

C D

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Assignment #7: Properties of Quadrilaterals

1. Using what you know about different types of parallelograms, complete the

diagram below, which will demonstrate how parallelograms are related.

2. Using your knowledge of properties of parallelograms and trapezoids, give

the requested information.

a. ∠D and ∠C

42°

122° __

15 ft b. BC and ∠D

58° 58°

parallelogram

All angles are right angles

All sides are equal

____________

____________

____________

____________

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3. Use the diagram to answer the following questions.

A E

D

C G

B F

a. Identify the shape of each piece A to G?

A ______________________ B ______________________

C ______________________ D ______________________

E ______________________ F ______________________

G ______________________

b. What are the interior angles of shape A?

c. What are the interior angles of shapes B and D?

d. What are the interior angles of shape E?

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4. Identify all the possible types of quadrilaterals described by each

statement:

a) A quadrilateral has opposite sides parallel, and at least one interior angle

is 35º.

___________________________________

b) A quadrilateral has one pair of opposite sides parallel, but the parallel

sides are not the same length.

___________________________________

c) A quadrilateral has one pair of opposite sides parallel and the same

length.

___________________________________

d) A quadrilateral has opposite sides parallel, and at least one 90º interior -

angle.

___________________________________

e) A quadrilateral has both pairs of opposite sides parallel, at least two

consecutive sides the same length, and at least one 90º angle.

___________________________________

f) A quadrilateral has opposite sides parallel, consecutive angles equal, and

consecutive sides not the same length.

___________________________________

g) A quadrilateral has diagonals that are perpendicular to each other, but

not the same length.

___________________________________

h) A quadrilateral has diagonals that are the same length, but not

perpendicular to each other.

___________________________________

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Lesson #8: Regular Polygons

Polygon: A polygon is a straight-sided, closed-path figure.

Regular Polygon: The sides of a regular polygon are all equal in length and its

internal angles are all the same size. Therefore, the angles and sides are all

congruent. If a regular polygon is cut into pieces from the centre, the angles at

the center equal 360°.

Polygon Number of Sides

Triangle 3

Quadrilateral 4

Pentagon 5

Hexagon 6

Heptagon 7

Octagon 8

Nonagon 9

Decagon 10

Undecagon 11

Dodecagon 12

A polygon with even sides has opposite sides that are parallel, a polygon with

uneven sides has no parallel sides.

Example 1: A coin is in the shape of a regular polygon with 11 sides. State the

measure of a central angle in degrees.

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Example 2:

a. What is the internal angle at each corner of a

regular pentagon?

b. What is the sum of the internal angles of a regular pentagon?

Note: A pentagon can be divided into 3 triangles by

drawing all possible diagonals from one of the corners

(vertices). Remember, the sum of the interior angles in

a triangle is 180º.

3 triangles each 180º = 540º

Therefore: The interior angles of a polygon can be found with this formula:

What happens to the interior angles as n increases?

What shape begins to occur once n increases?

?

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Assignment #8

1. Explain why the following polygons are not regular polygons?

2a. Draw a hexagon. How many complete hexagons will fit around a single point?

b. Will hexagons sharing a common point lie flat on a surface if each one exactly

touches its two neighbors? Explain your answer.

3. Determine (by illustration or calculation) the total number of diagonals in a

regular six-sided polygon.

Hint: In order to calculate the number of diagonals,

the formula is

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4. Anne is making a window casing (frame) for the following nonagonal window.

a. What is the shape of each piece of casing?

b. What are the internal angles of each piece of casting?

5. Sigrid is building a hexagonal gazebo. The gazebo is to be 10 feet across the

diagonals.

a. How long is each edge of the gazebo?

b. What are the interior angles of the

outside edges and the central angles?

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6. A land surveyor surveys a property and plots the following angles on the plan

shown below. Has the surveyor done the work correctly? How do you know?

98°

91° 138°

130°

120° 146°

7. Find the missing information.

a. b.

60 20

12 m 6 cm 6 cm

60

c. d.

X ? 85

110

e. ?

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Lesson #9: EXAM QUESTIONS:

Applications of Polygon Properties

Standard POLYGONS Exam Question:

Polygons are often used in construction, commercial, industrial, or artistic applications.

A) Demonstrate one use of the various properties of polygons in the real world by performing the

following two steps: (2 marks)

• State a specific example where the various properties of polygons are used.

• Support your example with a written explanation of how the various properties of polygons are used.

B) Sketch a reasonably neat picture or diagram (not necessarily to scale) that supports your example in

Part A. (1 mark)

Exemplars of Full Mark Student Answers:

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Assignment #9: EXAM QUESTIONS:

Applications of Polygon Properties

1. Create one response to the following exam question that is your own:

Polygons are often used in construction, commercial, industrial, or artistic applications.

A) Demonstrate one use of the various properties of polygons in the real world by performing

the following two steps: (2 marks)

• State a specific example where the various properties of polygons are used.

• Support your example with a written explanation of how the various properties of polygons

are used.

B) Sketch a reasonably neat picture or diagram (not necessarily to scale) that supports your

example in Part A. (1 mark)

geometry and trigonometry unit€¦ · geometry and trigonometry unit: # assignment completed?...

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