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Angles 2
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An angle has two arms and a vertex.
Vertex
Vertex
This angle is called any one of the following
ABC, CBA, B,
Special Angles
(Signpost Mathematics Year 8)
Name
Diagram
Common arm
Adjacent angles
Vertex
Explanation
Adjacent angles:
(1) have a common arm
(2) have a common vertex
(3) and lie on opposite sides of the
common arm.
Complementary
angles
Any two angles whose sum is 90°
are complementary angles.
e.g. 46° + 44° = 90°
Supplementary
angles
Any two angles whose sum is 180°
are supplementary angles.
e.g. 128°+52°= 180°.
Adjacent
complementary
angles
These are formed when a right
angle is cut into two parts. The
angles must add up to 90°.
Adjacent
supplementary
angles
These are formed when a straight
angle is cut into two parts. The
angles must add up to 180°.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
Angles at a
point
If angles meet at a point, then their
sum is 360° or one revolution. e.g.
92°+125°+143° =360°
125°
143°
Vertically
opposite
angles
33°
When two straight lines intersect,
the vertically opposite angles are
equal.
147°
33°
147°
A
Transversal
A transversal is a line which cuts
two or more other lines. In the
diagram AB is a transversal.
B
Corresponding
angles
Alternate angles
There are four pairs of corresponding angles.
These are equal when the lines are
parallel.
There are two pairs of alternate
angles.
These are equal when the lines
are parallel.
Cointerior
angles
There are two pairs of cointerior
angles.
These are supplementary when
the lines are parallel.
(They add up to 180°.)
Equilateral
triangle
All sides are equal. All angles are
equal to 60°.
Isosceles
triangle
Two sides are equal.
Angles opposite equal sides are
equal.
(These are called base angles.)
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
Scalene triangle
No sides are equal. All angles are
different.
Acute-angled
triangle
All angles are acute (less than 90°).
Right-angled
triangle
One angle is a right angle (90°).
Obtuse-angled
triangle
One angle is an obtuse angle
(greater than 90°). The longest side
is opposite the largest angle.
Quadrilateral
Four-sided figure
Trapezium
• One pair of opposite sides parallel
Parallelogram
Rhombus
•
•
•
•
Two pairs of parallel sides.
Opposite sides equal.
Opposite angles equal.
Diagonals bisect one another.
A rhombus has all the properties of
a parallelogram and...
All sides are equal.
Diagonals bisect each other at right
angles.
Diagonals bisect the angles through
which they pass.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
Rectangle
• A rectangle has all the properties
a parallelogram and...
• All angles are right angles.
• Diagonals are equal.
Square
A square has all the properties of a
rhombus and a rectangle.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
Calculating Angles
Making Sense with Mathematics: Murray Britt and Peter Hughes
Three identical regular
hexagons surround each
point without gaps or
overlap.
1.
What is the
angle of one revolution?
2.
What fraction of
a revolution is each
angle of each hexagon?
3.
What is the size
of the shaded angle?
4.
These three
pentagons don’t
surround a point. What
is the angle of the gap?
5.
These two
octagons also leave a
gap.
What is the angle of the gap?
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
5
6.
Calculate the marked angles.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
7. a and b are supplementary angles.
Calculate the marked angles.
8.
Copy and complete
the table for the diagrams.
The pairs a, b and c, d are called vertically
opposite angles.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
Diagram
a
b
c
A
B
C
D
9
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d
Angles 2
Triangles
Calculate C.
1. Calculate the marked angles.
p =_____
u =_____
Equilateral triangle
All sides and angles are equal
q =_______
s =______
Isosceles triangles.
2 sides and 2 angles are equal
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r =____
t = ______
Angles 2
2.
Find the marked angle. (If the letters are the same the angles are the same.)
3.
a and b are complementary because they must add up to 90 degrees.
Calculate the marked angles.
x = ________
a = _________________
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
4.
Exterior angle of a triangle.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
6.
Complete the table for the following triangles.
Triangle
a
b
c
A
B
C
a + b+c =
What do you notice about the exterior angles?
7.
Calculate the marked angles.
Quadrilaterals
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Angles 2
Calculate angle D.
1.
Calculate the marked angles.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
3.
Complete the following table.
Quadrilateral
l
A
a
b
c
d
B
C
Making Sense with Mathematics – Murray Britt and Peter Hughes.
a+b+c+d
Angles 2
The turtle completes a journey around the quadrilateral when it returns to its starting position.
QUESTIONS
1. a How many revolutions will the turtle make to complete a journey?
b What angle will it rotate during a journey?
Polygon
Number of sides
Total exterior angle
Exterior angle
Interior angle
equilateral triangle
3
360°
120°
60°
square
pentagon
hexagon
heptagon
octagon
nonagon
decagon
dodecagon
20-gon
100-gon
360-gon
n-gon
4
5
6
7
8
9
10
12
20
100
360
n
360°
90°
90°
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
Polygon
Number of sides
Number of triangles
Interior angle sum
Pattern
triangle
3
1
180° = 1 x 180°
(3 - 2) x 180°
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
dodecagon
20-gon
1 00-gon
360-gon
n-gon
4
5
6
7
8
9
10
12
20
100
360
n
2
3
4
360° = 2 x 180°
(4 - 2) x 180°
z =__________
k = _____
t =__________
z =_________
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
Using the symmetry of the diagram, calculate the value of x.
Angles and parallels
QUESTIONS
1. a What is the size of the shaded angle B? Explain
how you worked it out.
b. AP is parallel to BQ. Why?
c.
Calculate the marked angles.
Light rays are bent (refracted) as they enter and leave a
glass medium. The rays in and out are parallel. What is
the size of r?
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
2.
Calculate the marked angles.
3.
Find the marked angles.
p and q are called co-interior angles. Complete the rule p + q =_____________.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
4.
Find the marked angles.
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
Making Sense with Mathematics – Murray Britt and Peter Hughes.
Angles 2
MEP Book 7 pages 15 to 27, Bearings Book 8 pages 9 to 16.
Topic 3 Angle Geometry.
Making Sense with Mathematics – Murray Britt and Peter Hughes.