MATHEMATICS
MORE EXERCISES
Angles in the same segment.

Question  1
O is the centre of the circle.
Find, giving resons, the value of x, y and z
in each diagram :

1.1                                                a = 70°                           [ A 1.1 ]

1.2                                                a = 34°                           [ A 1.2 ]

1.3                                                     a = 40°                           [ A 1.3 ]

1.4                                                     O = 90°                           [ A 1.4 ]

1.5                                                     a = 41°                           [ A 1.5 ]

1.6                                                     a = 150°                           [ A 1.6 ]

1.7                                                     a = 52°                           [ A 1.7 ]

1.8                                                     a = 66°                           [ A 1.8 ]

Question  2

O is the centre of the circle.
∠DAC = 38° and ∠BDC = 66°
Calculate, giving reasons, the size of

2.1   ∠DOC                                   [ A 2.1 ]
2.2   ∠DBC                                   [ A 2.2 ]
2.3   ∠ODC                                   [ A 2.3 ]
2.4   ∠BAC                                   [ A 2.4 ]

Question  3

O is the centre of the centre.
AOB is a diameter of the circle. ∠OCA = 28°
Calculate, giving reasons, the size of

3.1   ∠A                                   [ A 3.1 ]
3.2   ∠ACB                              [ A 3.2 ]
3.3   ∠B                                   [ A 3.3 ]

Question  4

O is the centre of circle.
∠KML = 72° and ∠MKN = 30°
Calculate, giving reasons, the size of

4.1   ∠N                                   [ A 4.1 ]
4.2   ∠KOL                              [ A 4.2 ]
4.3   ∠MLN                             [ A 4.3 ]

Question  5

O is the centre of the circle.
∠CDB = 25° and ∠BEC = 130°
Calculate, giving reasons, the size of

5.1   ∠CAB                                      [ A 5.1 ]
5.2   ∠ABD                                     [ A 5.2 ]
5.3   ∠AOD                                     [ A 5.3 ]

#### 1.1

\begin{align*} a &= 2x \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ a &= 70\degree \\ \tag{given} \\ x &= 35\degree \\ 2y &= (180\degree - a) \\ \tag{int. \angle's \Delta and radii} \\ &= (180\degree - 70\degree) \\ &= 110\degree \\ y &= 55\degree \\ \end{align*}
[ Q 1.1 ]

#### 1.2

\begin{align*} x &= 2a \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ &= 2 \times 34\degree \\ \tag{a = 34 ° given} \\ x &= 68\degree \\ 2y &= (180\degree - x) \\ \tag{int. \angle's \Delta and radii} \\ &= (180\degree - 68\degree) \\ &= 112\degree \\ y &= 56\degree \\ \end{align*}
[ Q 1.2 ]

#### 1.3

\begin{align*} x &= a = 40\degree \\ \tag{∠'s in same segment; given} \\ y &= 2a\\ \tag{∠ at centre = 2 × ∠ at circumference} \\ &= 2 \times 40\degree \\ \tag{a = 40° given} \\ y &= 80\degree \\ 2z &= (180\degree - y) \\ \tag{int. \angle's \Delta and radii} \\ &= (180\degree - 80\degree) \\ &= 100\degree \\ z &= 50\degree \\ \end{align*}
[ Q 1.3 ]

#### 1.4

\begin{align*} 2x &= 90\degree \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ x &= 45\degree \\ x &= y = 45\degree \\ \tag{∠'s in same segment} \\ 2z &= (180\degree - \angle O) \\ \tag{int. \angle's \Delta and radii} \\ &= (180\degree - 90\degree) \\ &= 90\degree \\ z &= 45\degree \\ \end{align*}
[ Q 1.4 ]

#### 1.5

\begin{align*} x &= a = 41\degree \\ \tag{∠'s in same segment; gegee} \\ y &= 2a\\ \tag{∠ at centre = 2 × ∠ at circumference} \\ &= 2 \times 41\degree \\ y &= 82\degree \\ 2z & (360\degree - y) \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ &= 360\degree - 82\degree \\ &= 278\degree \\ z &= 139\degree \\ \end{align*}
[ Q 1.5 ]

#### 1.6

\begin{align*} 2x &= a \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ 2x &= 150\degree \\ \tag{given} \\ x &= 75\degree \\ x &= y =75\degree \\ \end{align*}
[ Q 1.6 ]

#### 1.7

\begin{align*} x &= 2a \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ x &= 2 \times 52\degree \\ \tag{given} \\ &= 104\degree \\ y &= a = 52\degree \\ \tag{∠'s in same segment; given} \\ z &= y = 52\degree \\ \end{align*}
[ Q 1.7 ]

#### 1.8

\begin{align*} 2x &= a = 66\degree \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ x &= 33\degree \\ y &= x = 33\degree \\ \tag{∠'s in same segment} \\ 2z &= (360\degree - a) \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ &= 360\degree - 66\degree \\ &= 294\degree \\ z &= 147\degree \\ \end{align*}
[ Q 1.8 ]

#### 2.1

\begin{align*} \angle DOC &= 2 \times \angle DAC \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ &= 2 \times 38\degree \\ &= 76\degree \\ \end{align*}
[ Q 2.1 ]

#### 2.2

\begin{align*} \angle DBC &= \angle DAC \\ \tag{∠'s in same segment, chord DC} \\ &= 38\degree \\ \end{align*}
[ Q 2.2 ]

#### 2.3

\begin{align*} 2 \times \angle ODC &= (180\degree - \angle DOC) \\ \tag{int. \angle's \Delta ODC, OD = OC} \\ &= 180\degree - 76\degree \\ &= 104\degree \\ \angle ODC &= 52\degree \\ \end{align*}
[ Q 2.3 ]

#### 2.4

\begin{align*} \angle BAC &= \angle BDC \\ \tag{∠'s in same segment, chord BC} \\ &= 66\degree \\ \end{align*}
[ Q 2.4 ]

#### 3.1

\begin{align*} \angle A &= \angle OCA \\ \tag{OC = OA} \\ &= 28\degree \\ \end{align*}
[ Q 3.1 ]

#### 3.2

\begin{align*} \angle ACB &= 90\degree \\ \tag{∠ in semi-circle} \\ \end{align*}
[ Q 3.2 ]

#### 3.3

\begin{align*} \angle B &= (180\degree - (\angle BCA + \angle A)) \\ \tag{int. \angle's \Delta ABC} \\ &= (180\degree - (90\degree + 28\degree)) \\ &= 62\degree \\ \end{align*}
[ Q 3.3 ]

#### 4.1

\begin{align*} \angle N &= \angle KML \\ \tag{∠'s in same segment, chord KL} \\ &= 72\degree \\ \end{align*}
[ Q 4.1 ]

#### 4.2

\begin{align*} \angle KOL &= 2 \times \angle KML \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ &= 2 \times 72\degree \\ &= 144\degree \\ \end{align*}
[ Q 4.2 ]

#### 4.3

\begin{align*} \angle KML &= \angle MKN \\ \tag{∠'s in same segment, chord MN} \\ &= 30\degree \\ \end{align*}
[ Q 4.3 ]

#### 5.1

\begin{align*} \angle CAB &= \angle CDB \\ \tag{∠'s in same segment, chord BC} \\ &= 25\degree \\ \end{align*}
[ Q 5.1 ]

#### 5.2

\begin{align*} \angle ABD &= \angle BEC - \angle BAC \\ \tag{ext. ∠ of Δ ABE} \\ &= 130\degree - 25\degree \\ &= 105\degree \\ \end{align*}
[ Q 5.2 ]

#### 5.3

\begin{align*} \angle AOD &= 2 \times \angle ABD \\ \tag{∠ at centre = 2 × ∠ at circumference} \\ &= 2 \times 105\degree \\ &= 210\degree \\ \end{align*}
[ Q 5.3 ]