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