This Angle Is Called Any One of the Following

Angles 2

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An angle has two arms and a vertex.

Vertex

This angle is called any one of the following

ABC, CBA, B,

Special Angles

(Signpost Mathematics Year 8)

Name / Diagram / Explanation
Adjacent angles / Common arm

Vertex / 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°.
Angles at a point /
125°

143° / If angles meet at a point, then their sum is 360° or one revolution. e.g. 92°+125°+143° =360°
Vertically
opposite
angles /
33° 147°

33°
147° / When two straight lines intersect, the vertically opposite angles are equal.
Transversal / / A transversal is a line which cuts two or more other lines. In the diagram AB is a transversal.
Corresponding angles / / There are four pairs of corresponding angles.
These are equal when the lines are parallel.
Alternate angles / / 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.)
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 / / •Two pairs of parallel sides.
•Opposite sides equal.
•Opposite angles equal.
•Diagonals bisect one another.
Rhombus / / 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.
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.

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?

6.Calculate the marked angles.

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.

Diagram / a / b / c / d
A
B
C
D

Triangles

Calculate C.

1. Calculate the marked angles.

p =_____ q =______r =____

u =_____s =______t = ______

Equilateral triangleIsosceles triangles.

All sides and angles are equal2 sides and 2 angles are equal

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 = ______

4.Exterior angle of a triangle.

6.Complete the table for the following triangles.

Triangle / a / b / c /
A
B
C

What do you notice about the exterior angles?

7.Calculate the marked angles.

Quadrilaterals

Calculate angle D.

1.Calculate the marked angles.

3.Complete the following table.

Quadrilaterall / a / b / c / d / a + b + c + d
A
B
C

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 / 4 / 360° / 90° / 90°
pentagon / 5
hexagon / 6
heptagon / 7
octagon / 8
nonagon / 9
decagon / 10
dodecagon / 12
20-gon / 20
100-gon / 100
360-gon / 360
n-gon / n
Polygon / Number of sides / Number of triangles / Interior angle sum / Pattern
triangle / 3 / 1 / 180° = 1 x 180° / (3 - 2) x 180°
quadrilateral / 4 / 2 / 360° = 2 x 180° / (4 - 2) x 180°
pentagon / 5 / 3
hexagon / 6 / 4
heptagon / 7
octagon / 8
nonagon / 9
decagon / 10
dodecagon / 12
20-gon / 20
1 00-gon / 100
360-gon / 360
n-gon / n

Using the symmetry of the diagram, calculate the value of x.

Angles and parallels

QUESTIONS

1. aWhat 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?

2.Calculate the marked angles.

3.Find the marked angles.

p and q are called co-interior angles. Complete the rule ______.

4.Find the marked angles.

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.