Page 140 - Mathematics_Form_3

Circles

2. Label the vertices of the cyclic Theorem 5.4 is described geometrically
quadrilateral formed in task 1. as follows.

3. Extend one line segment of the Consider Figure 5.15, where angle
quadrilateral and identify the RQX is an exterior angle of the cyclic
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external angle formed together quadrilateral PSRQ.
with its opposite interior angle. Its opposite interior angle is P S R . Hence,
̂
̂
̂
4. Use a protractor to measure the using Theorem 5.4, P S R = R Q X

interior angle and its corresponding
opposite exterior angle. S
R
5. Repeat tasks 1 to 4 using two cyclic
quadrilaterals in different circles
with different but convenient radii.

6. What have you observed from the P Q X
angles you have measured in each
quadrilateral? Figure 5.15: Exterior angle of a cyclic

7. Write down a general rule which quadrilateral
summarizes your findings. Theorem 5.4 is proved as follows.

8. Share your findings with other Proof: Let O be the centre of a circle
students for further discussion. in Figure 5.15. If PSRQ is a

cyclic quadrilateral with PQ
The conclusion you have drawn in extended to x, it follows that
Activity 5.7 is another circle theorem ˆ ˆ
which describes the relationship between PSR RQP 180+ = ° (opposite angles in
an interior angle in a cyclic quadrilateral a cyclic quadrilateral). (1)
ˆ
and its opposite exterior angle. These But, RQX RQP 180+ ˆ ˆ = ° ( PQX is a
angles are always equal. straight angle). (2) Mathematics for Secondary Schools

Comparing equations (1) and (2)
Theorem 5.4
gives,
The interior angle of a cylic quadrilateral
is equal to its opposite exterior angle. PSR RQP+ ˆ ˆ = RQX RQP+ ˆ ˆ

This is reffered to as “exterior angle of Thus, PSR = RQX.
ˆ
ˆ
cyclic quadrilaterals.”
ˆ
ˆ
Therefore, PSR = RQX.

Student\s Book Form Three 133

18/09/2025 09:59:43
MATHEMATIC F3 SB.indd 133 18/09/2025 09:59:43
MATHEMATIC F3 SB.indd 133