Question

In the given figure, O is the centre of the circle whose diameter is MN. Measures of some central angles are given in the figure. 

i. m∠AOB and m∠COD 

ii. Show that arc AB ≅ arc CD 

iii. Show that chord AB ≅ chord CD

Answer 1

i. Seg MN is the diameter of the circle. … [Given] 

∴ m∠AOM + m∠AON = 180° … [Angles in a linear pair] 

∴ m∠AOM + (m∠AOB + m∠BON) = 180° …[Angle addition property] 

∴ 100° + m∠AOB + 35° = 180° …[∵ m∠AOM = 100°, m∠BON = 35°] 

∴ m∠AOB + 135° = 180° 

∴ m∠AOB = 180°- 135° 

∴ m∠AOB = 45° …(i) 

Also, m∠DOM + m∠DON = 180° … [Angles in a linear pair] 

∴ m∠DOM + (m∠COD + m∠CON) = 180° … [Angle addition property] 

∴ 100° + m∠COD + 35° = 180° …[∵ m∠DOM = 100°, m∠CON = 35°] 

∴ m∠COD + 135° = 180° 

∴ m∠COD = 180°- 135° 

∴ m∠COD = 45° …(ii)

ii. m(arc AB) = m∠AOB = 45° … [From (i)] 

m(arc DC) = m∠DOC = 45° .. .[From (ii)] 

∴ m(arc AB) = m(arc DC) …[From (i) and (ii)] 

∴ arc AB ≅ arc CD

… [If the measures of two arcs of a circle are same, then the two arcs are congruent] 

iii. arc AB ≅ arc CD 

∴ chord AB ≅ chord CD ….[The chords corresponding to congruent arcs are congruent]