Question

in the diagram, o is the centre of a circle which passes through the points a, b, c and d. bce and cdf are straight lines. ba is parallel to cd, adf = 80° and oda = 45°. (i) name three other angles on the diagram which are also 80°. (ii) find (a) reflex bod, (b) abo

Explanation:

Step1: Use the property of alternate - interior angles

Since $BA\parallel CD$ and $\angle ADF = 80^{\circ}$, $\angle BAD=80^{\circ}$ (alternate - interior angles). Also, angles in the same segment of a circle are equal. So, $\angle BCD = 80^{\circ}$ (angles in the same segment as $\angle BAD$) and $\angle BOD = 160^{\circ}$ (the central angle is twice the inscribed - angle subtended by the same arc $BD$), and the reflex of $\angle BOD$ is considered. The angle $\angle BED$ (where $E$ is on the extension of $BC$) is also $80^{\circ}$ (angles in the same segment as $\angle BAD$).

Step2: Find reflex $\angle BOD$

The central angle $\angle BOD$ subtended by arc $BD$ and the inscribed - angle $\angle BAD$ are related by the formula $\angle BOD = 2\angle BAD$. Since $\angle BAD=80^{\circ}$, $\angle BOD = 160^{\circ}$. The reflex of $\angle BOD=360^{\circ}-160^{\circ}=200^{\circ}$.

Step3: Find $\angle ABO$

$OA = OD$ (radii of the circle), so $\angle OAD=\angle ODA = 45^{\circ}$. Then $\angle OAB=\angle BAD-\angle OAD=80^{\circ}-45^{\circ}=35^{\circ}$. Also, $OA = OB$ (radii of the circle), so $\triangle OAB$ is isosceles and $\angle ABO=\angle OAB = 35^{\circ}$.

Answer:

(i) $\angle BAD$, $\angle BCD$, $\angle BED$
(ii) (a) $200^{\circ}$
(b) $35^{\circ}$