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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 |
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Answer» 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] |
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