Question

In the figure, chord LM ≅ chord LN, ∠L = 35°. 

Find

1. m(arc MN)
2. m(arc LN)

Answer 1

i. `∠L = 1/2 m(arc  MN)`   ...[Inscribed angle theorem]

∴ `35^circ = 1/2 m(arc  MN)`

∴ 2 × 35° = m(arc MN)

∴ m(arc MN) = 70°

ii. In ∆LMN,

chord LM ≅ chord LN

∴ ∠M = ∠N   ...[Isosceles triangle theorem]

∴ ∠L + ∠M + ∠N = 180°   ...[Sum of the measures of angles of a triangle is 180°]

∴ 35° + ∠M + ∠M = 180°

∴ 2∠M = 180° – 35°

∴ 2∠M = 145°

∴ `∠M = (145^circ)/2`

Now, m(arc LN) = 2 × ∠M   ...[Inscribed angle theorem]

= `2 xx (145^circ)/2`

= 145°