3. INTRODUCTION
The study of geometry contributes to helping students develop the
skills of:
• Visualisation,
• Critical thinking,
• Intuition,
• Perspective,
• Problem-solving,
• Conjecturing, deductive reasoning, logical argument and proof.
4. THE CIRCLE
• A set of points that are the same distance from a central point.
5. Theorem 1(A)
The perpendicular drawn from the centre of a circle to a
chord bisects the chord. (perp. from centre to cord)
7. Theorem 1(A): Converse
The line segment joining the centre of a circle to the
midpoint of a chord is perpendicular to the chord.
(line from centre bisects chord)
12. Theorem 2 23 FEBRUARY 2022
Subtended Angles
In circle geometry, an angle is subtended by an arc or chord when its
two rays pass through the endpoints of that arc or CHORD.
14. Theorem 2
The angle subtended by an arc at the centre of a circle is
twice the angle subtended by the same arc at the
circumference of the circle.
(∠ at the centre = 2×∠ at circumference)
18. Theorem 3
The angle subtended at the
circle by a diameter is a
right angle. We say that the
angle in a semi-circle is 90° .
(∠ in a semi-circle)
19. Converse of theorem 3
If the angle subtended by a chord at the circumference of the circle is
90°, then the chord is a diameter. (chord subtends 90°)
23. Theorem 4
An arc or chord of a circle
subtends equal angles at the
circumference of the circle. We
say that the angles in the same
segment of the circle are equal.
(∠ s in the same segment)
25. Theorem 4:
Converse
If a line segment joining two points subtends equal angles
at two other points on the same side of the line segment,
then these four points are concyclic (converse of angles in
same segment)
32. Theorem 5:
converse
If the opposite angles of a quadrilateral are supplementary, then
the quadrilateral is a cyclic quadrilateral.
(opp ∠s supplementary)
35. Theorem 6
An exterior angle of a cyclic
quadrilateral is equal to the
interior opposite angle.
(ext ∠ of cyclic quad)
36. Theorem 6:
converse
If an exterior angle of a
quadrilateral is equal to the
interior opposite angle, then
the quadrilateral is a cyclic
quadrilateral.
(converse, ext ∠ of cyclic quad)
51. Theorem 7
A tangent to a circle is perpendicular to the radius at the point of
contact. (tan is perp. to rad)
52. Converse of Theorem 7
If a line is drawn perpendicular to a radius at the point where the
radius meets the circle, then the line is a tangent to the circle.
60. Theorem 9
The angle between a tangent to
a circle and a chord drawn from
the point of contact is equal to
an angle in the alternate
segment.
(Tan-chord)
62. Converse of
theorem 9
If a line is drawn through the endpoint of a chord, making
with the chord an angle equal to an angle in the alternate
segment, then the line is a tangent to the circle.
(converse, tan-chord)