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SCHOLAR Study Guide
National 5 Mathematics
Course Materials
Topic 6: Angles in polygons and
circles
Authored by:
Margaret Ferguson
Reviewed by:
Jillian Hornby
Previously authored by:
Eddie Mullan
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
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Distributed by the SCHOLAR Forum.
SCHOLAR Study Guide Course Materials Topic 6: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
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1
Topic 1
Angles in polygons and circles
Contents
6.1
Determining an angle in a polygon . . . . . . . . . . . . . . . . . . . . . . . .
3
6.2
6.3
Using angle properties in circles . . . . . . . . . . . . . . . . . . . . . . . . . .
Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
16
6.4
End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
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.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
1.1
Determining an angle in a polygon
A polygon is a many sided 2D shape. A regular polygon has all its sides of equal length
and can be composed of congruent isosceles triangles.
Look at these regular polygons.
The sum of the angles in a triangle is 180 ◦ so each angle in an equilateral triangle is
60◦ . The interior angle is 60 ◦ .
The sum of the angles in a square is 360 ◦ so each angle is 90 ◦ . The interior angle is
90◦ .
Angles round a point equal 360 ◦ so each angle at the centre of a pentagon is 360 ÷ 5 =
72◦ . All triangles are isosceles so the other two angles are equal and (180 − 72) ÷ 2 =
54◦ . The interior angle is 108 ◦ . (Note that 180 − 72 = 108.)
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
Key point
Remember:
•
Angles round a point will equal 360 ◦ .
•
All the interior angles of a regular polygon will be the same size.
Example
Problem:
Calculate the size of an interior angle of a
regular nonagon.
Solution:
The angles at the centre are 360 ÷ 9 = 40 ◦
The interior angle is (180 − 40) ÷ 2 × 2 = 180 − 40 = 140◦
(Remember the interior angle = 180 - angle at centre.)
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
5
Example
Problem:
Calculate the size. . .
a) . . .of ∠GHI, the interior angle in the
little pentagon.
b) . . .of ∠AHG.
c) . . .of ∠GAH.
d) . . .of ∠BAH.
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.
∠AHG = 180 − 108 = 72◦
c) Triangle AGH is isosceles so angles AHG and AGH are equal.
∠GAH = 180 − (2 × 72) = 36◦
d) Angles AHB and GHI are vertically opposite angles.
since angle GHI = 108◦ , ∠AHB = 108◦
Triangle ABH is isosceles so angles BAH and ABH are equal.
∠BAH = (180 − 108) ÷ 2 = 36◦
..........................................
Determining angles in polygons practice
Q1:
Go online
Calculate the size of an angle at the centre
of a regular octagon.
..........................................
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
Q2:
Calculate the size of an interior angle of a
regular hexagon.
..........................................
Q3:
Find the size of the interior angle BDF.
..........................................
Q4:
Find the size of ∠ABC.
..........................................
Q5:
Find the size of ∠LAB.
..........................................
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
7
Angles in polygons exercise
Q6:
Go online
Calculate the size of an angle at the centre
of the regular polygon.
..........................................
Q7:
Calculate the size of an interior angle of
the regular polygon .
..........................................
Q8: This is a regular decagon.
a) What size is the interior angle x◦ ?
b) What size is the angle y◦ ?
c) What size is the angle z◦ ?
..........................................
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
1.2
Using angle properties in circles
Angles of a triangle
Go online
When two parallel lines are cut by a transversal, the patterns of angles around each
point are the same.
We are particularly interested in this pair of equal angles (x ◦ ) and this pair (y ◦ ). These
are alternate angles. . . the ones that form ’Z-bends’.
Given any triangle we can always extend the sides and draw a line through the vertex
parallel to the opposite side.
We have formed two pairs of useful alternate angles. Notice in the following diagram
that the three angles of the triangle come together to form a straight line. The sum of
the angles in any triangle equal 180 ◦ . The angles which form a straight angle are called
supplementary angles and equal 180 ◦◦ .
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
9
Properties of a circle (1)
All radii of a circle are equal. If radii are drawn to the end of a chord then the triangle
formed will be isosceles.
Go online
..........................................
Properties of a circle (2)
A circle with its centre is drawn and a diameter drawn through the centre. A point is
chosen on the circumference of the circle. . . not the end point of the existing diameter.
A diameter can always be drawn from this point.
These two diameters are equal and bisect each other. . . the configuration of the
diagonals is a rectangle. So given any diameter, any point on the circumference is a
corner of a rectangle using the diameter as a diagonal.
The diameter subtends an angle of 90 ◦ at any point on the circumference. The right
angle is always on the circumference of the circle opposite the diameter and we call this
property "The angle in a semi-circle is a right angle".
© H ERIOT-WATT U NIVERSITY
Go online
10
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
..........................................
Properties of a circle (3)
Go online
A circle with its centre is drawn and a diameter drawn through the centre. A chord that
cuts the circle in two places is drawn at right angles to the diameter. The diameter is an
axis of symmetry. . . the chord is bisected.
A second chord parallel to the first is drawn to the right. . . the two points of intersection
are closer together and the chord is at right angles to the diameter. A third chord parallel
to the first two is drawn further to the right. . . the two points of intersection are even
closer together and the chord is at right angles to the diameter.
At some point the two points of intersection will become one. . . the chord will become
a tangent and the tangent is at right angles to the diameter. A tangent will be at right
angles to a radius drawn to the point of contact.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
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..........................................
Examples
1.
Problem:
B
AC and BC are radii.
C 100o
∠BCA = 100◦ .
Calculate the size of ∠CAB.
xo
A
Solution:
∠CAB = 40◦ (isosceles triangle: (180 − 100) ÷ 2 = 40) ⇒ x = 40
..........................................
2.
Problem:
AB is a diameter and C is a point
on the circumference of the circle.
∠BAC = 38◦ .
Calculate the size of ∠ACB and ∠CBA.
Solution:
∠ACB = 90◦ (angle in a semi-circle) ⇒ x = 90
∠CBA = 52◦ (third angle in a triangle) ⇒ y = 52
..........................................
3.
Problem:
Find the values of x and y in the diagram.
BT and AT are tangents, AC and BC are
radii.
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
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
..........................................
4.
Problem:
Find the values of x, y and z in the
diagram.
BT and AT are tangents, AC and BC are
radii.
Solution:
∠ACT = 50◦ (symmetry) ⇒ x = 50
∠CAT = 90◦ (radius/tangent) ⇒ y = 90
∠CTA = 40◦ (third angle in triangle: 180 - 50 - 90) ⇒ z = 40
..........................................
Key point
•
Look for isosceles triangles formed with two radii and a chord.
•
The right angle is always on the circumference of the circle opposite the
diameter and we call this property ’The angle in a semi-circle is a right
angle’.
•
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’.
Properties of a circle practice
Q9:
Go online
AC and BC are radii.
∠BCA = 80◦ .
Calculate the size of ∠CAB.
Write down your working.
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
Q10:
AB is a diameter and C is a point
on the circumference of the circle.
∠BAC = 42◦ .
Calculate the size of ∠ACB and ∠CBA.
Write down your working.
..........................................
Q11:
Find the values of x and y in the diagram
below.
BT and AT are tangents, AC and BC are
radii.
Write down your working.
..........................................
Q12:
Find the values of x, y and z in the diagram
below.
BT and AT are tangents, AC and BC are
radii.
Write down your working.
..........................................
Q13:
The circle opposite has centre O.
AD is a diameter. OC is a radius.
BC is a chord and is perpendicular to AD.
Calculate the size of angles OCM, OCA
and ABC if ∠COM = 38 ◦ .
Write down your working.
..........................................
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
Properties of a circle exercise
Q14:
Go online
BT and AT are tangents.
AC and BC are radii.
Find the values of x, y and z in the
diagram.
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)?
..........................................
Q15:
BT and AT are tangents.
AC and BC are radii.
Find the values of x and y in the diagram.
a) What is the value of x (in degrees)?
b) What is the value of y (in degrees)?
..........................................
Q16:
AB is a diameter and C is a point on the
circumference of the circle.
∠CAB is 26◦ .
Calculate the size of angles ACB and
CBA.
a) What is ∠ACB (in degrees)?
b) What is ∠CBA (in degrees)?
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
Q17:
AC and BC are radii of the circle.
∠CAB is 46◦ .
What is angle ACB (in degrees)?
..........................................
PO, QO and OR are radii of the circle.
SR is a tangent to the circle and
perpendicular to OR.
∠PRS is 51◦ .
Q18: Calculate ∠PRO
..........................................
Q19: Calculate ∠POR
..........................................
Q20: Calculate ∠OQR
..........................................
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
1.3
Learning points
Angles in a polygon
•
An angle at the centre of an n sided polygon = 360 ÷ n
•
An interior angle is an angle at a vertex on a polygon.
•
A star formed in a polygon will also have a smaller polygon within it.
Angle properties
•
Complementary angles add up to 90 ◦ .
•
Supplementary angles add up to 180 ◦ .
•
Vertically opposite angles are equal (look for an "X" shape).
•
Corresponding angles are equal (look for an "F" shape).
•
Alternate angles are equal (look for a "Z" shape).
Angle properties in circles
•
A triangle formed by 2 radii and a chord is an isosceles triangle.
•
The angle in a semi-circle is a right angle.
•
A tangent is at right angles to the radius at the point of contact.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
1.4
17
End of topic test
End of topic 6 test
Angles in polygons
Go online
Q21:
Calculate the size of an angle at the centre
of the regular polygon.
Give your answer correct to 1 decimal
place.
..........................................
Q22:
Calculate the size of an interior angle of a
regular polygon.
..........................................
Q23:
a) Calculate the angle a ◦ .
b) Calculate the angle b ◦ .
c) Calculate the angle c◦ .
..........................................
Angle properties in circles
Q24:
AC and BC are radii of the circle.
∠BCA is 90◦ .
What is ∠ABC (in degrees)?
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
..........................................
Q25:
AB is a diameter and C is a point on the
circumference of the circle.
a) What is ∠ACB (in degrees)?
b) What is ∠CAB (in degrees)?
..........................................
Q26:
Find the values of x and y in the diagram
below.
BT and AT are tangents. AC and BC are
radii.
a) What is the value of x (in degrees)?
b) What is the value of y (in degrees)?
..........................................
Q27:
Find the values of x, y and z in the diagram
below.
BT and AT are tangents. AC and BC are
radii.
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)?
..........................................
Q28:
TR is a tangent to the circle.
OS is a radius and QSR = 46 ◦ .
a) Find ∠PSQ.
b) Find ∠SQO.
c) Find ∠SPQ.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ANGLES IN POLYGONS AND CIRCLES
..........................................
..........................................
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20
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.
supplementary angles
Supplementary angles add up to 180 ◦ .
vertically opposite angles
Vertically opposite angles are equal and can be spotted from an X shape and are
sometimes known as X angles.
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 6
Answers to questions and activities
6 Angles in polygons and circles
Determining angles in polygons practice (page 5)
Q1: 45 ◦
Q2: 120 ◦
Q3: 120 ◦
Q4: 120 ◦
Q5: 60 ◦
Angles in polygons exercise (page 7)
Q6: 360 ÷ n = 360 ÷ 11 = 32 · 7◦
Q7: 180 − (360 ÷ n) = 180 − (360 ÷ 8) = 135◦
Q8:
a) An angle at the centre = 360 ÷ 10 = 36 ◦
The interior angle is (180 − 36) ÷ 2 × 2 = 180 − 36 = 144◦
x = 144◦
b) y is vertically opposite an interior angle x.
y = 144◦
c) The supplement of y = 36◦
z = 180 − (2 × 36) = 108◦ (isosceles triangle).
z = 108◦
Properties of a circle practice (page 12)
Q9: ∠CAB = 50◦ (isosceles triangle) ⇒ x = 50
Q10:
∠ACB = 90◦ (angle in a semi-circle) ⇒ x = 90
∠CBA = 48◦ (third angle in a triangle) ⇒ y = 48
Q11:
∠CAT = 90◦ (radius/tangent) ⇒ y = 90
∠CBT = 90◦ (radius/tangent)
∠BTA = 70◦ (fourth angle in a quadrilateral) ⇒ x = 70
Q12:
∠ACT = 60◦ (symmetry) ⇒ x = 60
∠CAT = 90◦ (radius/tangent) ⇒ y = 90
∠CTA = 30◦ (third angle in triangle) ⇒ z = 30
© H ERIOT-WATT U NIVERSITY
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ANSWERS: TOPIC 6
Q13:
∠OCM = 52◦ (right-angled triangle: 180 − 90 − 38 = 52)
∠COA = 142◦ (supplementary angles: 180 − 38)
∠OCA = 19◦ (isosceles triangle: (180 − 142) ÷ 2)
∠ACB = 71◦ (52 + 19 = 71)
∠ABC = 71◦ (Symmetry)
Properties of a circle exercise (page 14)
Q14:
a) 90◦
b) 180◦ − 90◦ − 37◦ = 53◦
c) 53◦
Q15:
a) 90◦
b) 360◦ − 90 ◦ − 90◦ − 63◦ = 117◦
Q16:
a) 90◦
b) 180◦ − 90◦ − 26◦ = 64◦
Q17: 180◦ − (2 × 46◦ ) = 88◦
Q18:
Hint:
•
The tangent meets the radius at right angles.
Answer: 90◦ − 51◦ = 39◦
Q19:
Hint:
•
Triangle POR is isosceles.
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 6
Answer: 180◦ − (2 × 39)◦ = 102◦
Q20:
Hint:
•
Triangle QOR is isosceles and ∠PRQ is an angle in a semi-circle therefore is equal
to 90◦ .
Answer: 180◦ − 90◦ − 39◦ = 51◦
End of topic 6 test (page 17)
Q21: (360 ÷ n =) 360 ÷ 7 = 51 · 4◦
Q22: 180 − (360 ÷ n) = 180 − (360 ÷ 6) = 120◦
Q23:
a) 180 − (360 ÷ 8) = 135◦
b) 180 − a = 180 − 135 = 45◦
c) 180 − 2 × b = 180 − 2 × 45 = 90◦
Q24: (180◦ − 90◦ ) ÷ 2 = 45◦
Q25:
a) 90◦
b) 180◦ − 90◦ − 64◦ = 26◦
Q26:
a) 90◦
b) 360◦ − 90◦ − 90◦ − 70◦ = 110◦
Q27:
a) 90◦
b) 180◦ − 90◦ − 20◦ = 70◦
c) 70◦
Q28:
a) 90◦
b) 44◦
c) 180◦ − 90◦ − 44◦ = 46◦
© H ERIOT-WATT U NIVERSITY
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