4. SUM OF INTERIOR ANGLES IN
POLYGONS
• We have learnt that the sum of angles in a
polygon can calculated by the formula
180°(𝑛 − 2), where 𝑛 is the number of sides
of the polygon. Now complete the table
below.
•
POLYGON No. of sides Sum of angles
Triangle 3 180°
Quadrilateral 4 360°
Pentagon
Hexagon
.
.
.
6. THE CIRCLE THEOREMS
• 1 Angle at the centre
2 Angle in a semicircle
3 Angles in same segment
4 Cyclic quadrilateral
5 Tangent lengths
6 Tangent/radius angle
7 Alternate segment
8 Perpendicular
• 9 Equal chords
7. Circle Theorem 1
The angle a chord subtends at the centre of a circle is twice the
angle it subtends at the circumference of the circle. 𝒂 = 𝟐𝒃
Fig. 1
8. EXAMPLE
• In the diagram below, O is the centre of the
circle, |AO|=|BO| and ∠𝐴𝑂𝐵 = 68°. Find the
value of 𝑥.
O
A B
C
𝑥
68°
19. Circle Theorem 5
• The lengths of the two tangents from a point to a circle are
equal.
• A radius and a tangents forms 90 degrees at their meeting
point
21. SOLUTION
• ∠𝐴𝐵𝑂 = ∠𝐴𝐶𝑂 = 90°
• The sum of the interior angles of a
quadrilateral is 360°
• ∴ 𝑋 + 90° + 90°+ 115° = 360°
• ⟹ 𝑋 + 295° = 360°
• ⟹ 𝑋 = 360° − 295°
• = 65°
22. Circle Theorem 6
• The angle between a tangent and a radius in
a circle is 90°. <ADC = <AEC.
23. Circle Theorem 7
• Alternate segment theorem:
The angle (α) between the tangent and the
chord at the point of contact (D) is equal to
the angle (β) in the alternate segment*.
24. EXAMPLE
• In the diagram below, 𝑃𝑅 the diametre , 𝑃𝑇 is
the tangent. ∠𝑄𝑃𝑅 = 𝑎° and ∠𝑄𝑃𝑇 = 54°.
Find the value of 𝑎.
O
𝑎°
𝑃
𝑄
𝑅
𝑆
T
54°
30. SOLVED EXAMPLE
• In the diagram bellow, 𝑇𝐵 touched the circle
at B and 𝐵𝐷 is the diametre, ∠𝑇𝐵𝐴 = 31°.
Calculate
a. ∠𝐴𝐷𝐶
b. ∠𝐴𝐵𝐶
c. ∠𝐶𝐴𝐷
T
B
A
D
C
310
690
32. Real life Application of Circle Theorem
• Imagine you are standing on the surface of the
earth!
33. Real life Application of Circle Theorem
• You can use Pythagoras’ Theorem to calculate
the distance from the top of your head to the
Centre of the earth, the radius of the earth
and your height. Amazing, all by the
application of the Circle theorem!
• HERE YOU ARE!
34. Real life Application of Circle Theorem
• John Dalton reconstructed chemistry at the start of the 19th
century on the basis of atoms, which he regarded as tiny spheres,
and in the 20th century, models of circular orbits and spherical
shells were originally used to describe the motion of electrons
around the spherical nucleus. Thus circles and their geometry have
always remained at the heart of theories about the microscopic
world of atoms and theories about the cosmos and the universe.
• Geometry continues to play a central role in modern mathematics,
but its concepts, including many generalizations of circles, have
become increasingly abstract. For example, spheres in higher
dimensional space came to notice in 1965, when John Leech and
John Conway made a spectacular contribution to modern algebra
by studying an extremely close packing of spheres in 24-
dimensional space.