Mathematics

Key unit competence

By the end of the unit, learners should be able to construct mathematical arguments
about circles and discs. Use circle theorems to solve related problems.

Unit outline

• Elements of circles and discs.

• Circle theorem.

Introduction

Unit Focus Activity

1. Research from reference books or Internet the angle theorems for circles.

2. Apply the theorems to find the values of angles given in Fig. 9.1
     below given that K, L, M and N are points on the circumference of
    a circle centre O. The points K, O, M and P are on a straight line.

3. State the reasons and theorem supporting for your in each case.

4. Compare your solutions with those of other classmates in a class
    discussion.

In P5, we already learned some basics concepts about circles, their properties and
how to get their area and circumference. We also learnt types of angles and he
angle sum in a triangle. In this unit, we will investigate the relationships between
angles when they are drawn in a circle. Let us begin this by refreshing our knowledge
on elements of a cricle.

9.1 Elements of a circle and disk

Activity 9.1

1. Use a pair of compass and a pencil to draw any circle in your exercise
    books.

The basic elements of a circle are:
(a) Centre – Is a point inside the circle and is at an equal distance from the
point on the circumference.

2. Draw a straight line that passes through the centre of the circle
    and measure the distance from the centre of both ends.

3. Cut off the circle you have drawn using a razor blade or scissors.

4. Label its parts.

9.2 Circle theorem

9.2.1 Angles at the centre and the circumference of a circle

Activity 9.2

1. Draw a circle of centre O with any convenient radius.

2. Mark two points A and B on the circumference such that AB is a
    minor arc.

3. Mark another point P on the major arc AB.

4. Draw angles AOB and APB.

5. Measure ∠AOB and ∠APB.

6. What is the relationship between the two angles?

7. Compare your observations with those of other members of
    your class. What do you notice?

Theorem 1
Angle subtended at the centre of a circle is twice the angle subtended
by the same chord or arc on the circumference of the same circle in
the same segment.

Proof

Note:
This theorem also relates the reflex angle subtended at the centre by a chord with
the angle subtended by the same chord in the minor segment as shown in Fig 9.16
below.

Exercise 9.1

1. In Fig. 9.22 below, O is the centre and AC is the diameter.

(a) If a = 24°, find x.

(b) If x = 62°, find b.

(c) If a = 52°, find b.

(d) If b = 28°, find a.

(e) Express x in terms of a.

(f) Express b in terms of a.

2. In Fig. 9.23, O is the centre of the circle. If c = 47°, find d and p.

3. AB is the diameter of a circle and C is any point on the circumference. If
    ∠BAC = 2∠CBA, find the size of angle BAC.

4. ABC is a triangle such that points A, B and C lie on circumference of a
   circle, centre O. If AC = BC and angle AOB = 72°, find angle BAC.

9.2.2 Angle in a semicircle

     Activity 9.3

1. Draw a circle, centre O, using any convenient radius.

2. On your circle draw a diameter AB.

3. Mark another point C on the circumference and join A to C and
    B to C.

4. Measure angle ACB.

5. Compare your result with those of other members of your class. What
    do you notice?
   We can apply what we have just learned about the angle at the centre of a circle
   and prove that the angle in a semicircle is always a right angle.

Theorem 2
The angle subtended by the diameter at any point on the circumference of
a circle is a right angle.
Proof

9.2.3 Angles in the same segment

Activity 9.4

1. Draw a circle centre O, with any convenient radius.

2. Mark off a minor arc AB.

3. On the major arc AB, mark distinct points P, Q, R and S.

4. Join each of the points in (3) above  so as to form angles APB, AQB,
    ARB and ASB.

5. Measure the angles.

6. What do you notice about the sizes of the four angles?
    Do your classmates have the same observation?

We have already learnt that, a chord divides
the circumference into two arcs, a major and
a minor arc.

If AB is a minor arc and P is any point on the major arc (Fig. 9.30), then ∠APB is the
angle subtended by the minor arc AB at the circumference of the circle.

If Q is another point on the major arc AB (Fig. 9.30), ∠APB and ∠AQB are both
subtended at the circumference by the minor arc AB or chord AB. Such angles
are said to be in the same segment. Consider Fig. 9.31 below which ∠AEB
and ∠AGB are subtended by chord AB onto the circumference on the major
segement.

By measurement, ∠AEB = ∠AGB
Similarly, consider Fig. 9.32 below
∠ACB and ∠ADB that are subtended by chord AB or by the major arc AB at the
circumference. These are also angles in the same segment. Are they also equal? Check
by measuring.

Theorem 3

Angles subtended on the circumference
by the same chord in the same segment are equal.

Proof

Given: Circle centre O
           Chord BC (Fig. 9.33)

Let ∠BAC and ∠BDC be the angles
subtended by the chord BC on the major
arc BADC.
To prove: that ∠BAC = ∠BDC
Construction: Join BO, OC.
Proof: ∠BOC = 2 ∠BAC ………angle
           at the centre of a circle is twice the
           angle subtended by the same chord
           on the circumference.

Similarly, ∠ BOC = 2 ∠BDC.
∴ ∠BAC = ∠BDC.

Activity 9.5

1. Draw a circle with any convenientradius.

2. On the circumference, mark  points A, B, C and D such that
    chord AB has the same length as chord CD.

3. On major arc AB mark point P and on major arc CD mark point Q.

4. Draw angles APB and CQD.

5. Measure the angles APB andCQD.

6. What do you notice about thesizes of the angles?

7. Using a piece of sewing thread, measure the lengths of the minor
    arcs AB and CD. What do you notice about their lengths?

In this activity, we also learn that:
1. Equal arcs of the same circle subtend equal angles at the circumference.

2. Equal chords of the same circle cut off equal arcs.

Example 9.8

ABCDE is a regular pentagon inscribed in a circle (Fig. 9.37). Show
that ∠ACD = 2 ∠ACB.

Solution

With reference to Fig. 9.37 (b), ∠ACB is subtended at the circumference of the
circle by the minor arc AB. ∠ECA is subtended at the circumference
 of the circle by the minor arc AE. ∠DCE is subtended at the circumference
of the circle by the minor arc DE. But arcs AB, AE and DE are equal.
Therefore, they subtend equal angles at the circumference.
∴ ∠ACB     = ∠ECA = ∠DCE
But ∠ACD  = ∠ACE + ∠ECD

∴ ∠ACD = 2∠ACB

Exercise 9.3

1. Find x and y in Fig. 9.38 below.

9.2.4 Angles in a cyclic quadrilateral

Activity 9.6

1. Draw a circle centre O using any convenient radius.

2. On the circumference, mark points A, B, C and D in that order and join
    them to form a quadrilateral.

3. Measure angles ABC and ADC. Find their sum.

4. Measure angles BAD and BCD. Find their sum.

5. What do you notice about the two sums in 3 and 4?

6. Are the pairs of angles in 3 and 4adjacent or opposite?

7. Do the other members of your class have the same observations as you
    do?

8. Produce side AB of the quadrilateral, and measure the exterior angle so
    formed. How does the size of this angle compare with that of interior ∠ADC?

A cyclic quadrilateral is a quadrilateral
whose vertices all lie on a circle.
The distinctive property of a cyclic
quadrilateral to be looked under this
theorem is that its opposite angles are

A cyclic quadrilateral is a quadrilateral whose vertices all lie on a circle.
The distinctive property of a cyclic quadrilateral to be looked under this
theorem is that its opposite angles are suplementary.

Theorem 4.1

The opposite interior angle of a cyclic
quadrilateral are supplementary or add
upto 180º.

Proof

Given: Quadrilateral
ABCD inscribed
in a circle centre O.

To prove:

(i) ∠ADC + ∠ABC = 2 right angles

(ii) ∠BAD + ∠BCD = 2 right angles
Join OA and OC.
∠ADC = 1–
2 ∠AOC …………∠ADC and
∠AOC subtended by the arc ABC at the
circumference and centre of the circle
respectively.

Theorem 4.2

If one side of a cyclic quadrilateral is produced, the exterior angle formed is
equal to the opposite interior angle of the quadrilateral.

Proof

Given: Quadrilateral ABCD inscribed
in a circle centre O.

To prove:
(i) ∠ADC = the exterior angle at B.

(ii) ∠BAD = the exterior angle at C

(iii) ∠DCB = exterior angle at A

(iv) ∠ABC = exterior angle at D Produce line AB to a point E.

Solution

∠ADB = ∠ADC – ∠BDC
= 122° – 41°
= 81°
∴ ∠ADB = ∠ACB.
But these angles are subtended by the same
side AB at the points C and D, on the same
side of AB.
Therefore ABCD is a cyclic quadrilateral
and so, point A, B, C and D are concyclic.

Exercise 9.4

1. A, B, C, D and E are five points, in
that order, on the circumference of a circle (Fig. 9.52).

(a) Write down all angles in the figure equal to ∠ACB.

(b) Write down all angles in the figure supplementary to ∠BCD.

(c) If ∠ACB = 40° and ∠ACD =75°, find the size of ∠DEB.

2. In Fig. 9.53, find:

3. Fig. 9.54 consists of two intersecting
   circles. Use it to find the angles marked by letters.

4. ABCD is a cyclic quadrilateral in a circle centre O.

(a) If ∠ABD = 32°, find ∠ACD.

(b) If AOC is a straight line, write down the size of ∠ABC.

5. AB is a chord of a circle centre O. If ∠AOB = 144°, calculate the angle
    subtended by AB at a point on the minor arc AB.

9.2.5 Tangent to a circle

9.2.5.1 Definition of tangent to acircle

Activity 9.7

1. Draw a circle of radius 10 cm.

2. Using a ruler and pencil, carefully draw lines that touch the
    circumference at only one point. What is the name of such a line.

4. ABCD is a cyclic quadrilateral in a circle centre O.

(a) If ∠ABD = 32°, find ∠ACD.

(b) If AOC is a straight line, write down the size of ∠ABC.

5. AB is a chord of a circle centre O. If ∠AOB = 144°, calculate the angle
    subtended by AB at a point on the minor arc AB.

9.2.5 Tangent to a circle

9.2.5.1 Definition of tangent to a circle

Activity 9.7

1. Draw a circle of radius 10 cm.

2. Using a ruler and pencil, carefully draw lines that touch the
     circumference at only one point. What is the name of such a line.

Consider Fig. 9.55
In Fig. 9.55(a), lines KL and PQ have only one point common with the circle.
A line with at least one point common with the circle is said to meet the circle
at that point.

In Fig. 9.55(b), line RS has two distinct points common with the circle. Such a
line is said to meet and cut the circle at the two points.

In Fig. 9.55(c), the line TV has one point of contact with the circle. Line TV is said
to meet and touch the circle at that point of contact. Point U is called the point of
contact.

Note:
1. A line which cuts a circle at two distinct points (as in Fig. 9.45(b))
    is called a secant of the circle.

2. A line which has one, and only one point in contact with a circle
   (as in Fig. 9.55(c)), however far it is produced either way, is called a
   tangent of the circle.

Activity 9.8

1. Draw a circle of radius 10 cm on amanilla paper.

2. As accurately as possible, draw a tangent to the circle at a point of
    your choice.

3. Join point P to the centre ofthe circle.

4. Measure the angle between the tangent and radius.

5. Repeat this for other tangents drawn at other points on the
    circumference. What do you conclude.

Fig. 9.56(a) to (d) shows what happens when the secant PQRS moves away from
the centre of the circle. As the secant moves further away, the points Q and R
get closer to each other and the chord QR gets shorter each time. Eventually Q and
R coincide at one point (Fig. 9.56(d)). On the other hand, angles OQP and ORS
become smaller and smaller. Eventually when Q and R coincide, angles OQR and
ORS each becomes 90°. Note that in ΔOQR, since OQ = OR,
∠OQR = ∠ORQ. It follows that ∠PQO = ∠SRO.

Therefore, when Q and R coincide (Fig. 9.56(d)), ∠PQO =∠SRO = 90°.
Hence the radius is perpendicular to the tangent PS.

Theorem 5
1. A tangent to a circle is perpendicular to the radius drawn through the
    point of contact.

2. Conversely stated the perpendicular to a tangent at its point of contact
     passes through the centre of the circle.

Exercise 9.5

1. In the Fig. 9.60 below, O is the centre of the circle while A and C are points
    on the circumference of the circle. BCO is a straight line and BA is a tangent to
    the circle. AB = 8cm and OA = 6cm.

(a) Explain why ∠ OAB is a right angle.

(b) Find the length BC.

2. Fig. 9.61 shows a circle,centre O. PR is a tangent to the circle at P and PQ
    is a chord.

Calculate:
(a) ∠RPQ given that ∠POQ = 85°.

(b) ∠RPQ given that ∠PQO = 26°.

(c) ∠POQ given that ∠RPQ = 54°.

(d) ∠POQ given that ∠QPO = 17°.

3. In Fig. 9.62, ABC is a tangent andBE is a diameter to the circle.

Calculate:

(a) ∠EBD if ∠CBD = 33°.

(b) ∠BED if ∠ABD =150°.

(c) ∠DBC if ∠DEB = 65°.

(d) ∠ABD if ∠BED = 38°.

4. Two circles have the same centre O, but different radii. PQ is a chord
    of the bigger circle but touches thesmaller circle at A. Show that PA =
    AQ.

5. Two circles have the same centre O  and radii of 13 cm and 10 cm. AB
     is a chord of the bigger circle, but a tangent to the small circle. What is
     the length of AB?

6. A tangent is drawn from a point 17 cm away from the centre of a circle
    of radius 8 cm. What is the length of the tangent?

9.2.5.2 Constructing a tangent at any given point on the circle

Activity 9.9

1. Draw a circle, centre O, usingany radius.

2. Draw a line OB through any point A on the circumference,
    with B outside the circle.

3. At A, construct a line PQ perpendicular to OB.
    To construct a tangent to a circle, we use
    the fact that a tangent is perpendicular to
     the circle at the point of contact.

The line PQ (Fig. 9.63) is a tangent tothe circle at A.

9.2.5.3 Constructing tangents to a circle from a common point

Activity 9.10

1. Draw a circle of any radius and centre O.

2. Mark a point T outside the circle.

3. Join OT. Construct the perpendicular bisector of TO to
    meet TO at P.

4. With centre P, radius PO, construct arcs to cut the circle
    at A and B.

5. Join A to T and B to T. What do you notice?

Activity 9.11

1. Draw a circle of any radius, centreO.

2. Choose any points A and B on the circle. Construct tangents at
    A and B

3. Produce the tangents till they meet at a point T.

4. Join OA, OB and OT.

5. Measure:      (a) AT, BT.
                        (b) ∠ATO, ∠BTO,
                        (c) ∠AOT, ∠BOT

6. What do you notice?

7. Which points on a circle would have tangents that do not
    meet?

If two tangents are drawn to a circlefrom a common point:
(a) the tangents are equal;

(b) the tangents subtend equal angles at the centre;

(c) the line joining the centre to the common point bisects the angles
     between the tangents.

Exercise 9.6

1. In Fig 9.67 below, A, B and D are points on the circumference of a
    circle centre O. BOD is the diameter of the circle while BC and AC are the
    tangents of the circle. Angle OCB
    = 34°. Work out the size of angle DOA.

2. In Fig 9.68 A and B are points on the circumference of a circle, centre X. PA
   and PB are tangents to the circle. Angle APB = 86°. Find the size of angle y.

3. In Fig. 9.69, O is the centre of the circle. PT and RT are tangents to the
   circle.

Calculate:
(a) ∠POT if ∠OTR = 34°.

(b) ∠PRO if ∠PTR = 58°.

(c) ∠TPR if ∠PRO = 15°.

(d) ∠RTO if ∠POR = 148°.

4. Draw a circle, centre O, and radius 2.5 cm. Mark points A and B on
    the circle such that ∠AOB = 130°. Construct tangents at A and B.
Measure:
(a) The lengths of the tangents.

(b) The angle formed where the tangents meet.

5. In Fig. 9.70, O is the centre of the circle. If BO = 19.5 cm, BQ = 18 cm,
    QC = 8.8 cm and AO = 9.9 cm, what are the lengths of:

   (a) AB          (b) BC       (c) AC?

6. A tangent is drawn to a circle of radius 5.8 cm from a point 14.6 cm
    from the centre of the circle. What is the length of the tangent?

7. Tangents are drawn from a point 10 cm away from the centre of a circle

of radius 4 cm. What is the length of the chord joining the two points of
contact?

8. Tangents TA and TB each of length 8 cm, are drawn to a circle of radius
   6 cm. What is the length of the minor arc AB?

9. Construct two tangents from a point A which is 6 cm from the centre of a
   circle of radius 4 cm.

(a) What is the length of the tangent?

(b) Measure the angle subtended at the centre of the circle.

10. Draw a line KL= 6 cm long. Construct a circle centre K radius 3.9 cm such
     that the tangent LM from L to the circle is 4.5 cm. Measure ∠KLM.

9.2.6 Angles in alternate segment

In Fig. 9.71, ABC is a tangent to the circle at B. Thechord BD divides the circle into
two segments BED and BFD. We say that BFD is the alternate segment
to ∠ABD.

Similarly, BED is the alternate segment to ∠CBD.

Activity 9.12

1. Draw a circle of any radius.

2. Draw a tangent at any point B.

3. Draw a chord BD.

4. Mark points P, Q, R on the circumference in the same
    segment. Join BP, BQ, BR, DP, DQ and DR.

5. Measure angles BPD, BQD and BRD.
   What do you notice?

Theorem 6

If a tangent to a circle is drawn, and from the point of contact a chord is
drawn, the angle which the chord makes with the tangent is equal to
the angle the chord subtends in the alternate segment of the circle. This
is called the alternate segment theorem.

Proof

(a) We use Fig. 9.73(a) to show that ∠RQT = ∠QST.

Exercise 9.7
1. If < BAD = 19°, find <ACB. in Fig 9.77.

2. In Fig. 9.78, AC is a tangent to the circle and BE//CD.

(a) If ∠ABE = 42°, ∠BDC = 59°,find∠BED.

(b) If ∠DBE = 62°, ∠BCD = 56°, find ∠BED.

3. In Fig. 9.79, PR is a tangent to the circle.

(a) If ∠PQT = 66°, find ∠QST.

(b) If ∠QTS = 38° and ∠QRS =30°, find ∠QST.

(c) If ∠QTS = 35° and ∠TQS =58°, find ∠QRS.

(d) If ∠PQT = 50° and ∠PRS = 30°,find ∠SQT.

4. In Fig. 9.80, AB, BC and AC are tangents to the circle. If ∠BAC = 75° and
   ∠ABC = 44°, find ∠EDF, ∠DEF and ∠EFD.

5. In Fig. 9.81, KLM is a tangent to the
    circle. If ∠LPN = 38° and ∠KLP = 85°, find ∠PQN.

6. In Fig. 9.82, DC is a tangent to the circle. Show that ∠CBD = ∠ADC.

7. In Fig. 9.83, AB and DE are tangents to the circle. ∠ABC = 40°
    and ∠BCD = 38°. Find ∠CDE.

8. In Fig. 9.84, ABC is a tangent to the circle at B and ADE is a straight line.
    If ∠BAD = ∠DBE, show that BE is a diameter.

9. In Fig. 9.85, AD is a tangent to the circle. BC is a diameter of the circle
    and ∠BCD = 30°. Find ∠DAB.

10. In Fig. 9.86, AD is a tangent to the circle at D, ∠DAB = 28° and ∠ADC
      = 112°. Find the angle subtended at the centre of the circle by the chord
      BC.

11. Points A, B and C are on a circle such that ∠ABC = 108°. Find the angle
      between the tangents at A and C.

12. In Fig. 9.87, O is the centre of the circle. AB and CD are chords that
      meet at X. XT is a tangent to the circle.

Show that (a) XT2 = XA . XB
                (b) XT2 = XC . XD.

9.2.7 Properties of chords Perpendicular bisector of a chord

Activity 9.13
1. Draw a circle of any radius r cm, centre O. Draw a chord AB (not
    a diameter). From O draw a line perpendicular to AB, cutting
    AB at N. Measure AN and NB. What do you notice?

2. Draw a circle of any radius r
    cm, centre O. Draw a chord CD (not a diameter). Construct a
    perpendicular bisector of CD. What do you notice about the
    bisector and centre of the circle?

3. Repeat step 2 for any of the same circle.

Theorem 7

1. A perpendicular drawn from the centre of a circle to a chord bisects the chord.

2. A perpendicular bisector of a chord passes through the centre
    of the circle.

Using the properties noted above, we can calculate the length of a chord.

Exercise 9.8

1. A chord of a circle of radius 13 cm is at a distance of 5 cm from the centre.
    Find the length of the chord.

6. A circle has a chord whose length is 9 cm. The chord is 4 cm from the
    centre of the circle. The same circle  has a chord which is 3.5 cm from the
  centre, what is its length?

7. A chord of a circle is 12 cm long and 5 cm from the centre. What is the
    length of a chord which is 3 cm from the centre?

8. A chord of a circle of radius 5.5 cm subtends an angle of 42° at the
    centre. Find the difference in length between the chord and the minor arc.

9. P, Q and R are points on the circumference of a circle. If PQ =
   12cm, PR = 12cm and QR = 8cm, what is the radius of the circle?
   We can also extend the theorem of on perpendicular bisector of a chord to
    parallel and equal chords.

Activity 9.14

1. Draw a circle, centre O, radius 4 cm. Draw chords PQ and RS
   such that PQ//RS. Construct a perpendicular bisector of PQ and
   let it cut RS at point T. (Fig. 9.95).

In general:
1. If two chords of a circle are parallel, then the perpendicular bisector of
    one is also perpendicular bisector of the other.

2. The midpoints of parallel chords of a circle lie on a diameter.

3. If two chords of a circle are equal, then they are equidistant from the
    centre.

4. If two chords of a circle are equidistant from the centre, then
    their lengths are equal

5. If two chords of a circle are equal, then the angles they subtend at the
    centre are equal.

6. If two angles at the centre of a circle are equal, then they are subtended
    by equal chords.

Example 9.23

Two parallel chords of a circle are of lengths 8 cm and 12 cm respectively and
are 10 cm apart. What is the diameter of the circle?

Note:
Equal chords of a circle are equidistant
from the centre of circle. Conversely, chords which are
equidistant from the centre of a circle are equal.

Exercise 9.9

1. Two parallel chords of a circle are of
    lengths 3 cm and 5 cm respectively. What is the radius of the circle if the
   chords are;
    (a) 1 cm apart            (b) 8 cm apart.
    (State your answers to 2 d.p.)

    Unit Summary

• Angles in the same segment i.e. subtended by the same arc, are equal.
  In Fig. 9.101, p = q. x = 2p = 2q

• The angle subtended at the centre of a circle by an arc is double that
   subtended by the same arc on the remaining part of the circumference.
   In Fig. 9.101, x = 2p or x = 2q.

• The angle subtended by the diameter on the circumference (i.e. the angle
   in a semicircle) is a right angle. In Fig. 9.102, x = y = 90°.

• Opposite angles of a cyclic quadrilateral (a quadrilateral with its four vertices
  lying on the circumference of a circle)  are supplementary (i.e. they add
  up to 180°). If one side of a cyclic quadrilateral is produced, the exterior
  angle thus formed equals the interior opposite angle (Fig. 9.103).

• The perpendicular bisector of a chord passes through the centre of
   the circle. It also bisects the angle that the chord subtends at the centre.

• Equal chords of a circle are equidistant from the centre of the circle.

• If two or more chords of a circle are parallel, then the perpendicular
  bisector of one bisects the others.

• The midpoints of parallel chords lie on a diameter.

• Equal chords subtend equal angles at the centre.

• The tangent to a circle at a point is  perpendicular to the radius of the
   circle at that point.

• Two tangents to a circle, drawn from the same point, have the same length
  (Fig. 9.104).

• The angle formed by a tangent ofa circle and a chord drawn from
  the point of contact is equal to the angle that the chord subtends in the
  alternate segment (Fig. 9.105).

Unit 9 Test

1. In Fig. 9.106, find the angles marked with letters a, b, c, d and e.

2. In Fig. 9.107, find the angles marked by the letters a, b, c and d.

3. In Fig. 9.108, AB is a diameter. Find the sizes of the angles marked x and y.

4. Fig 9.109 shows a rectangle ABCD which is inscribed in a circle, centre
   O and radius 10 cm. Given that AB = 16 cm, calculate
   (taking p = 3.142)

a) the width BC of the rectangle,

(b) the central angle BOC,

(c) the length of the minor arc BC,

(d) the shaded area.

5. In Fig. 9.110, PQ and PT are tangents to the circle. Given that ∠OQU = 28° and
    OQS = 15°, find the sizes of angles PQU, QSU, QUS and QRS.

6. A chord of a circle of length 15 cmsubtends an angle of 120° at the
   centre. Calculate the radius of the circle and the length of the minor arc.

7. Find the marked angles in the quadrilaterals in

    Fig. 9.111 (a) and (b).

8. The angles of a cyclic quadrilateral are 6x, 3x, x +y and 3x + 4y in that
    order. Determine the values of x and y, and hence the sizes of the angles of
    the quadrilateral.

9. Find the size of the angle marked x in Fig. 9.112.

10. In Fig. 9.113, BOE is a diameter,
     ∠CAE = 45°, ∠BEA = 50°,
     ∠BEC = 25°, ∠DCE = 20° and
     ∠DEF = 130°. Find: