Question

In the adjoining figure O is the centre of the circle whose diameter is MN. Measures of some central angles are given in the figure. hence find the following

(1) m∠ AOB and m∠ COD
(2) Show that arc AB ≌ arc CD.
(3) Show that chord AB ≌ chord CD

Answer 1

(1) MN is the diameter of the circle with centre O.
$$\begin{gathered} \angle \mathrm{MOA}+\angle \mathrm{AOB}+\angle \mathrm{BON}=180° \\ \Rightarrow 100°+\angle \mathrm{AOB}+35°=180° \\ \Rightarrow \angle \mathrm{AOB}=180°-100°-35°=45° \end{gathered}$$
Similarly, $\angle \mathrm{COD}=45°$
(2) Eqaual arcs subtend equal angles at the centre.
$\angle \mathrm{AOB}=\angle \mathrm{COD}=45°$
So, arc AB ≌ arc CD
(3) Corresponding chords of congruent arcs are congruent.
arc AB ≌ arc CD
So, chord AB ≌ chord CD