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1
Topic 6
Angles in polygons and circles
Contents
6
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6
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2
6
6
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6
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14
14
Learning Objectives
By the end of this topic, you should be able to:
• calculate an interior angle in a polygon;
• use the relationship in a circle between its centre, a chord and a perpendicular
bisector;
• identify the relationship between a radius and a tangent to a circle;
• identify the angle in a semi-circle
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ANGLES IN POLYGONS AND CIRCLES
6
...
A regular polygon has all its sides of equal length
and can be composed of congruent isosceles triangles
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The sum of the angles in a triangle is 180 ◦ so each angle in an equilateral triangle is
60◦
...
The sum of the angles in a square is 360 ◦ so each angle is 90 ◦
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Angles round a point equal 360 ◦ so each angle at the centre of a pentagon is 360 ÷ 5 =
72◦
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The interior angle is 108 ◦
...
)
© H ERIOT-WATT U NIVERSITY
TOPIC 6
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• All the interior angles of a regular polygon will be the same size
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Problem:
Calculate the size of an interior angle of a
regular nonagon
...
)
...
Problem:
© H ERIOT-WATT U NIVERSITY
3
4
TOPIC 6
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a)
...
b)
...
c)
...
d)
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Solution:
a) Calculate the angles at the centre of the pentagon GHIJK
360 ÷ 5 = 72◦
Calculate the interior angle 180 - 72 = 108
Therefore, ∠GHI = 108 ◦
b) ∠GHI & ∠AHG are supplementary angles
...
∠GAH = 180 - (2 × 72) = 36 ◦
d) Angles AHB and GHI are vertically opposite angles
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∠BAH = (180 -108) ÷ 2 = 36 ◦
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...
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ANGLES IN POLYGONS AND CIRCLES
Q3:
Find the size of the interior angle BDF
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Q4:
Find the size of ∠ABC
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Q5:
Find the size of ∠LAB
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Angles in polygons exercise
Q6:
Calculate the size of an angle at the centre
of the regular polygon
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© H ERIOT-WATT U NIVERSITY
5
6
TOPIC 6
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a) What size is the interior angle x◦ ?
b) What size is the angle y◦ ?
c) What size is the angle z◦ ?
...
2
Using angle properties in circles
Angles of a triangle
When two parallel lines are cut by a transversal, the patterns of angles around each
point are the same
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These
are alternate angles
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Given any triangle we can always extend the sides and draw a line through the vertex
© H ERIOT-WATT U NIVERSITY
TOPIC 6
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We have formed two pairs of useful alternate angles
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The sum of
the angles in any triangle equal 180 ◦
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...
If radii are drawn to the end of a chord then the triangle
formed will be isosceles
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Properties of a circle (2)
A circle with its centre is drawn and a diameter drawn through the centre
...
not the end point of the existing diameter
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© H ERIOT-WATT U NIVERSITY
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TOPIC 6
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the configuration of the
diagonals is a rectangle
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The diameter subtends an angle of 90 ◦ at any point on the circumference
...
...
A chord that
cuts the circle in two places is drawn at right angles to the diameter
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the chord is bisected
...
the two points of intersection
are closer together and the chord is at right angles to the diameter
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ANGLES IN POLYGONS AND CIRCLES
9
to the first two is drawn further to the right
...
At some point the two points of intersection will become one
...
A tangent will be at right
angles to a radius drawn to the point of contact
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Examples
1
...
C 100o
∠BCA = 100◦
...
xo
A
Solution:
∠CAB = 40◦ (isosceles triangle: (180 - 100) ÷ 2 = 40) ⇒ x = 40
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Problem:
© H ERIOT-WATT U NIVERSITY
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TOPIC 6
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∠BAC = 38◦
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Solution:
∠ACB = 90◦ (angle in a semi-circle) ⇒ x = 90
∠CBA = 52◦ (third angle in a triangle) ⇒ y = 52
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Problem:
Find the values of x and y in the diagram
...
Solution:
∠CAT = 90◦ (radius/tangent) ⇒ y = 90
∠CBT = 90◦ (radius/tangent)
∠BTA = 80◦ (fourth angle in a quadrilateral: 360 - 90 - 90 - 100 = 80) ⇒ x = 80
...
Problem:
Find the values of x, y and z in the
diagram
...
Solution:
∠ACT = 50◦ (symmetry) ⇒ x = 50
∠CAT = 90◦ (radius/tangent) ⇒ y = 90
∠CTA = 40◦ (third angle in triangle: 180 - 50 - 90) ⇒ z = 40
...
ANGLES IN POLYGONS AND CIRCLES
Key Point
• Look for isosceles triangles formed with two radii and a chord
...
• The diameter and tangent are perpendicular and we call this property 'A
tangent is at right angles to the radius at the point of contact'
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∠BCA = 80◦
...
Write down your working
...
Q10:
AB is a diameter and C is a point
on the circumference of the circle
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Calculate the size of ∠ACB and ∠CBA
...
...
BT and AT are tangents, AC and BC are
radii
...
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ANGLES IN POLYGONS AND CIRCLES
Q12:
Find the values of x, y and z in the diagram
below
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Write down your working
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Q13:
The circle opposite has centre O
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OC is a radius
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Calculate the size of angles OCM, OCA
and ABC if ∠COM = 38 ◦
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AC and BC are radii
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a) What is the value of x (in degrees)?
b) What is the value of y (in degrees)?
c) What is the value of z (in degrees)?
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AC and BC are radii
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© H ERIOT-WATT U NIVERSITY
TOPIC 6
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Q16:
AB is a diameter and C is a point on the
circumference of the circle
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Calculate the size of angles ACB and
CBA
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Q17:
AC and BC are radii of the circle
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What is angle ACB (in degrees)?
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SR is a tangent to the circle, parallel to PQ
and perpendicular to OR
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a) Calculate ∠PRO
b) Calculate ∠POR
c) Calculate ∠OQR
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ANGLES IN POLYGONS AND CIRCLES
6
...
• A star formed in a polygon will also have a smaller polygon within it
...
• Supplementary angles add up to 180 ◦
...
• Corresponding angles are equal (look for an "F" shape)
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Angle properties in circles
• A triangle formed by 2 radii and a chord is an isosceles triangle
...
• A tangent is at right angles to the radius at the point of contact
...
4
End of topic test
End of topic 6 test
Q19: Angles in polygons
Calculate the size of an angle at the centre
of the regular polygon
...
...
ANGLES IN POLYGONS AND CIRCLES
Q20: Angles in polygons
Calculate the size of an interior angle of a
regular polygon
...
Q21: Angles in polygons
a) Calculate the angle a ◦
...
c) Calculate the angle c◦
...
Q22: Angle properties in circles
AC and BC are radii of the circle
...
What is ∠ABC (in degrees)?
...
a) What is ∠ACB (in degrees)?
b) What is ∠CAB (in degrees)?
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ANGLES IN POLYGONS AND CIRCLES
Q24: Angle properties in circles
Find the values of x and y in the diagram
below
...
AC and BC are
radii
...
Q25: Angle properties in circles
Find the values of x, y and z in the diagram
below
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AC and BC are
radii
...
Q26: Angle properties in circles
SR is a tangent to the circle and is parallel
to the diameter PQ
...
a) Find ∠PSQ
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c) Find ∠SPQ
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© H ERIOT-WATT U NIVERSITY
GLOSSARY
Glossary
congruent
Two figures or objects are said to be congruent if they have exactly the same shape
and size, or if one has the same shape and size as the mirror image of the other
...
vertically opposite angles
Vertically opposite angles are equal and can be spotted from an X shape and are
sometimes known as X angles
...
y = 144◦
c)
The supplement of y = 36 ◦
z = 180 - (2 × 36) = 108◦ (isosceles triangle)