Download tpc maths (part a) - nswtmth307a

TPC MATHS (PART A) - NSWTMTH307
TOPIC 3: ANGLES AND SHAPES
3.1 Measuring and Constructing Angles
The following diagram shows an angle and names the parts of an angle.
Angles are sometimes named with one letter, or with three letters and a symbol of an
angle to avoid confusion.
Angle ‘E’, or
<FED or <DEF
Note: The apex must be the middle letter
The following diagram names the different types of angles according to the size of the
angle.
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To measure or construct an angle, we use a protractor.
To measure an angle with a protractor:
1. Line up the base line of the protractor with one arm of the angle (the arms of the
angle may have to be extended).
2. Line up the vertical centre line with the apex of the angle.
3. Read off the scale in degrees – check whether the angle is acute or obtuse and make
sure that you read off the correct scale (0-90o for acute, 90-180o for obtuse).
4. Note that the two scales run in different directions, so be careful if you are
interpolating.
The following diagram shows a protractor being used to measure an angle.
To construct an angle with a protractor:
1. Draw a base line first – this will be one arm of the angle
2. Mark a point on this base line – this will be the apex of the angle
3. Measure and mark the required angle size and join up the mark to the apex to draw
the other arm of the angle
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3.2 Angle Sum of Triangles and Quadrilaterals
Adjacent Angles forming a right angle add up to 90o – these are called complementary
angles
Adjacent Angles forming a straight line add up to 180o – these are called supplementary
angles
Angles forming a full revolution add up to 360o
b
a
a + b = 360o
Vertically Opposite Angles
Two pairs of equal vertically opposite angles are formed when two straight lines intersect.
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Angle sum of a Triangle
The angles of a triangle always add up to 180o.
Angle sum of a Quadrilateral
The angles of a quadrilateral always add up to 360o.
3.3 Circle Geometry
There are some special angles in circles.
The angle between a radius of a circle and a tangent is 90o.
The angle formed inside a semi-circle is 90o.
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3.4 Perpendicular and Parallel Lines
There are three types of angles formed when a pair of parallel lines is crossed by a
transversal.
Alternate Angles
Two pairs of alternate angles are formed as shown.
Alternate Angles are EQUAL
Corresponding Angles
Four pairs of corresponding angles are formed as shown.
Corresponding Angles
are EQUAL
Co-Interior Angles
Two pairs of corresponding angles are formed as shown.
Co-Interior Angles ADD
UP TO 180o
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3.5 Symmetry
Line Symmetry
The simplest symmetry is Line Symmetry (sometimes called Reflection Symmetry or Mirror
Symmetry or just Symmetry). It is easy to recognise, because one half is the reflection of
the other half. The line of reflection is called the Line of Symmetry.
These figures are symmetrical in relation to the
dashed line.
The line is called a symmetry line.
This means that one half of the figure
is the mirror image of the other half.
Imagine that you folded the figure along the symmetry line. Then both sides would exactly meet.
Or, place a mirror along the symmetry line. You see the other half of the figure reflected in the mirror.
Some shapes you can fold two different ways
so that the sides meet. The cross-shape
on the right has two different symmetry lines.
Look at this flower shape.
It has four different symmetry lines.
Check them by using the mirror.
Any line that you draw through the circle's
center point is a symmetry line.
So, we can't even count how many symmetry
lines a circle has!
Some shapes have only
one symmetry line,
like this arrow shape.
Many figures are not symmetrical at
all.
Everyday items can have symmetry as well as geometrical shapes.
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Rotational Symmetry
With Rotational Symmetry, the image is rotated (around a central point) so that it appears
2 or more times. How many times it appears as the same shape is called the Order.
Here are some examples:
Order
Example Shape
Artwork
... and there is Order 4, 5, etc ...
Point Symmetry
Point symmetry is a special type of rotational symmetry where the object has the same
shape if it is rotated 180o.
Point symmetry is the same as
Rotational Symmetry of Order 2
The test for Point Symmetry is that every part must have a matching part:


the same distance from the central point
but in the opposite direction.
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3.6 Solving Geometry Problems
If you
-
are trying to solve geometric problems, here are some hints of what to look for:
vertically opposite angles
right angles
straight line angles
angle sum of triangles and quadrilaterals
special angles in a circle
the three types of parallel line angles formed by a transversal – alternate,
corresponding and co-interior
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