1

x340o

Goal ­ I can determine missing measures of angles using angle properties.

5.1All About Angles

360o 180o 90o

110o30ox 750x

OAT "X"

x xy

y

60o x

Transversals with Parallel Lines

Alternative Angles "Z"

Corresponding Angles

"F"

2

2x+1150o

y

z

a

b

75o

x

Co­interior Angles

"C"

Examples: Solve for the unknowns

1

Angles in Shapes

The Six TrianglesACUTE OBTUSE RIGHT ANGLE

ISOSCELES SCALENE EQUILATERAL

a

b

c

a

a

a

a a

b

bbb

bc

c

c

c

c

How to Label a Triangle- vertices are always labeled with UPPERCASE LETTERS

- sides are labeled with lower case letters that correspond to the vertex opposite

Goal ­ I can solve unknowns angle measurements in triangles and quadrilaterals.

5.2

2

Properties of a Triangle Properties of a Quadrilateral

Exterior Angles

Examples: Solve for all unknowns

eg.

45

6075 x

3

Internal Angles in a Polygon

Development of a formula:

# of sides # of internal degrees

Examples:

1. What is the sum of the internal angles of a 20­sided figure.

2. What is the sum of the interior angles of a 12­sided figure?

1

Goal ­ I can use the Pythagorean Theorem to solve missing sides of right triangles.

5.3 Pythagorean Theorem

Sides of right angled triangles have a special relationship.

c2 = a2 + b2

Steps:

1. Label the hypotenuse

2. Write down the formula

3. Enter values and solve

4. Do not forget units

Examples: Solve for the indicated side

1.

x10

3

2.

y

117

3.

m

8

5

1

5.4

Median ­ is a line that connects the centre (or midpoint) of a side in a triangle to the opposite vertex.

­ is the point where all 3 medians meet.

Properties1. The centroid is the balance point or centre of gravity.

2. The median cuts the area of a triangle in half.

3. The centroid divides the median in a ration of 1:2

1:2 = midpoint to centroid : centroid to vertex

Centroid

Triangle Geometry Properties

Altitude

Orthocentre

­ is a line perpendicular to the side of a triangle passing through the opposite vertex.

­ is the point where the altitudes intersect.

2

For an acute triangle the orthocentre is INSIDE the triangle.

For an obtuse triangle, the orthocentre is OUTSIDE the triangle.

3

Angle Bisectors and the Incentre

For a triangle that will contain a circle with each side tangent to the circle, the centre of the circle is the incentre.

If the incentre has a line drawn from it to the vertex, we find the line is

BISECTING the angle.

1

Measurement and Geometry Familiar 2D Shapes

Rectangle- a square is a special rectangle where all sides are equal

Parallelogram Triangle

Pythagorean Theorem

Trapezoid Circle

phi (Greek letter -we use it as a number ~3.14

Goal ­ I can determine the perimeter and area of regular and composite figures

6.1

P = 2( l + w)

A = lw

2

Composite Shape

A composite shape is a shape containing 2 or more 2D or 3D shapes.

Strategies for finding area:

1) Break shape into known shapes

2) Label and calculate the area for each

Remember that perimeter is the distance AROUND a shape ­ Note the "dotted lines"

Example:

3. Find the area of the following figure.

6.2

1

Surface Area of 3D Shapes

The CylinderNet Diagram

SA = 2πr2 + 2πrhπr2

πr2

2πrh

V= πr2h

v= area of the base x height

The Cone Net Diagram

SA = πr2 + πrs

V

The Square-Based Pyramid

SA = b2 + 4(1/2 bs)

= b2 + 2bs

V = 1/3 cube

πrs

πr2

Net Diagram

Goal ­ I can find the surface area and volume of 3D objects.

6.2

6.2

2

The Sphere

Examples:

Determine the volume and surface area of the following.

Net Diagram

4m

9m

1.

5cm

7cm

2.

6.2

3

4.

3m10m

6m

3.

6.3 Word problems

1

Word Questions

1. Find the height if SA = 1200 m2

20m

5m

h

2. Determine the surface area if the volume is 100cm3

5cm

8cm

Goal ­ I can problem solve geometry/measurement situations.

6.3

The following questions use many of the concepts learned throughout this course and applies them to measurement situations.

Gives the "answer" so we need to find a variable (measurement).

6.3 Word problems

2

4. Three cylinders are being compared. Determine the ratio of their volumes if:

Cylinder A has a radius of 'r' and height of 'h'.

Cylinder B has three times the radius and half the height.

Cylinder C has half the radius and four times the height.

3. The volume of a cone is 500m3. If the radius is 8 cm, determine the surface area.

6.4 Optimization

1

Goal ­ I can solve optimization measurement problems.

6.4 Optimization

Optimization means either of the following:

Maximize area while minimizing perimeter.

Maximize volume while minimizing surface area.

Essentially in this process, we are attempting to choose the dimensions that give us the most desirable results.

In general,

For 2D shapes the circle most optimal shape; however, it is not always the most practical. The next best shape is the square.

We try to square everything up.

For 3D shapes the sphere is the most optimal shape; however it is not always the most practical. The next best shape is the cube.

ie. cylinder: you want diameter to equal height