Lessons notes for students preparing for WASSCE, NECO and JAMB EXAMS in West Africa.

1. Worked
Examples
Multiple
Choice
Practice
Questions
Lesson
Notes EXIT

2. < 𝑄𝑃𝑅 =
1
2
< 𝑅𝑂𝑄
< 𝑂𝑅𝑄 = 32°
(𝑏𝑎𝑠𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑎𝑛 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑂𝑄𝑅 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙)
< 𝑅𝑂𝑄+ < 𝑂𝑅𝑄 +< 𝑂𝑄𝑅 = 180°
(𝑠𝑢𝑚 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑂𝑄𝑅)
< 𝑅𝑂𝑄 + 32° + 32° = 180°
< 𝑅𝑂𝑄 + 64° = 180°
< 𝑅𝑂𝑄 = 180° − 64°
< 𝑅𝑂𝑄 = 116°
∴ < 𝑄𝑃𝑅 =
1
2
116°
∴< 𝑄𝑃𝑅 = 58°
In the diagram, O is the centre of the circle,
< 𝑂𝑄𝑅 = 32° and
< 𝑀𝑃𝑄 = 15°.
Calculate:
i) < 𝑄𝑃𝑅,
ii) < 𝑀𝑄𝑂
𝑊𝐴𝑆𝑆𝐶𝐸 𝐽𝑈𝐿𝑌, 2006.
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diagram
More

3. From 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑂𝑀𝑄, < 𝑂𝑀𝑄 +< 𝑀𝑄𝑅 +< 𝑄𝑂𝑀 = 180°
But
< 𝑄𝑂𝑀 = 2 < 𝑄𝑃𝑀, 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑖𝑠 𝑡𝑤𝑖𝑐𝑒 𝑡ℎ𝑒 𝑜𝑛𝑒 𝑜𝑛 𝑡ℎ𝑒
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
⇒ < 𝑄𝑂𝑀 = 2 15°
< 𝑄𝑂𝑀 = 30°
⇒< 𝑂𝑀𝑄 +< 𝑀𝑄𝑂 + 30° = 180°
Also, < 𝑂𝑀𝑄 =< 𝑀𝑄𝑂, 𝑏𝑎𝑠𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑂𝑀𝑄
⇒< 𝑂𝑀𝑄 +< 𝑂𝑀𝑄 + 30° = 180°
2 < 𝑂𝑀𝑄 + 30° = 180°
2 < 𝑂𝑀𝑄 = 180° − 30°
2 < 𝑂𝑀𝑄 = 150°
< 𝑂𝑀𝑄 =
150°
2
∴ < 𝑂𝑀𝑄 = 75°
In the diagram, O is the centre of the circle,
< 𝑂𝑄𝑅 = 32° and
< 𝑀𝑃𝑄 = 15°.
Calculate:
i) < 𝑄𝑃𝑅,
ii) < 𝑀𝑄𝑂
𝑊𝐴𝑆𝑆𝐶𝐸 𝐽𝑈𝐿𝑌, 2006.
Show
diagram
Flip to see ii
Worked
Example 2

4. From 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑂𝑀𝑄, < 𝑂𝑀𝑄 +< 𝑀𝑄𝑅 +< 𝑄𝑂𝑀 = 180°
But
< 𝑄𝑂𝑀 = 2 < 𝑄𝑃𝑀, 𝑎𝑛𝑔𝑙𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑖𝑠 𝑡𝑤𝑖𝑐𝑒 𝑡ℎ𝑒 𝑜𝑛𝑒 𝑜𝑛 𝑡ℎ𝑒
𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒
⇒ < 𝑄𝑂𝑀 = 2 15°
< 𝑄𝑂𝑀 = 30°
⇒< 𝑂𝑀𝑄 +< 𝑀𝑄𝑂 + 30° = 180°
Also, < 𝑂𝑀𝑄 =< 𝑀𝑄𝑂, 𝑏𝑎𝑠𝑒 𝑎𝑛𝑔𝑙𝑒𝑠 𝑜𝑓 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑂𝑀𝑄
⇒< 𝑂𝑀𝑄 +< 𝑂𝑀𝑄 + 30° = 180°
2 < 𝑂𝑀𝑄 + 30° = 180°
2 < 𝑂𝑀𝑄 = 180° − 30°
2 < 𝑂𝑀𝑄 = 150°
< 𝑂𝑀𝑄 =
150°
2
∴ < 𝑂𝑀𝑄 = 75°
In the diagram, 𝑂 is the centre of the
circle PQRS and < 𝑃𝑄𝑅 = 65°. Find the
value of angle 𝑥.
𝑊𝐴𝑆𝑆𝐶𝐸 𝐽𝑈NE, 2012.
Show
diagram

5. Find the value of a in the diagram below, if AB
and BC are tangents, and O is the centre of the
circle.
A . 74°
B .64°
C . 104°
D .174°
E.124°
EXIT
MCQ

6. In the diagram below, O is the centre of the
circle and AB is a tangent to the circle at A.
Calculate the value of a and b.
B .𝑎 = 90°, 𝑏 = 65°
A .𝑎 = 155°, 𝑏 = 90°
C 𝑎 = 65°, 𝑏 = 90°
D .𝑎 = 90°, 𝑏 = 75°
E .𝑎 = 90°, 𝑏 = 85°
EXIT
MCQ

7. In the diagram below, P, Q, R and S are four points on a
circle. PR and QS intersect at T, such that
<STR=140° and <PRQ=30°. Find <SPR.
B 110°
A . 280°
C .90°
D .70°
•
•°
E .40°
EXIT
MCQ

8. The diagram shows a circle centre O. A, B and C are
points on the circumference. DBO is a straight line. DA
is a tangent to the circle. Calculate < 𝐵𝐶𝐴.
E .124°
A .64°
D .62°
B .14°
C .31°
EXIT
MCQ

9. The diagram shows a circle centre O. A, B and C are
points on the circumference. DBO is a straight line. DA
is a tangent to the circle. Calculate < 𝐴𝑂𝐷.
D.62°
A .64°
E .124°
B .14°
C .31°
EXIT
MCQ

10. The diagram shows a circle centre O. A, B and C are points on
the circumference. DBO is a straight line. DA is a tangent to
the circle at A and < 𝐵𝐴𝐷 = 38°. Find the value of 𝑧.
C .76°
A .38°
D .104°
B .52°
E .164° EXIT
MCQ

11. The diagram shows a circle centre O. A, B and C are points on
the circumference. DBO is a straight line. DA is a tangent to
the circle at A and < 𝐵𝐴𝐷 = 38°. Which theorem can be used to
find 𝑧?
A . Angles in alternate
segment
C . Alternate angles
D . Angles in the same
segment
B . Cyclic quadrilaterals
E . Angles about the centre
EXIT
MCQ

12. The diagram shows a circle centre O. A, B and C are points on
the circumference. DBO is a straight line. DA is a tangent to
the circle at A and < 𝐵𝐴𝐷 = 38°. Find the value of 𝑥.
B .52°
A .38°
D .104°
C .76°
E .164° EXIT
MCQ

13. The diagram shows a circle centre O. A, B and C are points on
the circumference. DBO is a straight line. DA is a tangent to
the circle at A and < 𝐵𝐴𝐷 = 38°. Find the value of 𝑤 − 𝑦.
D .270°
A .38°
B .90°
C .60°
E .180°
EXIT
MCQ

14. In the diagram, O is the centre of the circle. BD and CD
are tangents. Which of the following statements is NOT
true about the diagram?
D .< 𝑂𝐵𝐷 =< 𝑂𝐸𝐵
A .< 𝑂𝐵𝐷+ < 𝑂𝐶𝐷 = 180°
C .< 𝐶𝑂𝐵+ < 𝐵𝐷𝐶 = 180°
B .< 𝑂𝐵𝐸 =< 𝑂𝐸𝐷
C . < 𝑂𝐵𝐷 =< 𝑂𝐶𝐷

15. Semi-circles
Semi-circles
𝑟
𝑂 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟
The circle is a locus of points equidistant
from a fixed point called the centre, O.
The radius is 𝑟.
The diameter divides the circle into two
semi-circles. The diameter is a special
chord.
A chord divides the circle into two semi-
circles.
EXIT
MCQ

16. The tangent to a circle meets the radius at an angle
of 90°. The same thing applies to the diameter
EXIT
MCQ

17. The angle between a chord and a tangent is equal to the
angle created in the alternate segment
EXIT
MCQ

18. Angle 𝑎 and 𝑏 are equal because they are all created in the same
Segment by the same arc or chord
EXIT
MCQ

19. Angle at the centre
𝜃
The angle subtended at the centre is twice the one
subtended at the circumference by an arc.
𝑂
EXIT
MCQ

20. A quadrilateral that has its four vertices on the circumference of a circle is calle
a cyclic quadrilateral
The opposite angles of a cyclic quadrilateral will always sum up to 180°