Question

In the figure below, $\angle L = 35^\circ$. Find: 
(i) $m(\text{arc } MN)$ 
(ii) $m(\text{arc } MLN)$ 

Hint:

In a circle, the measure of an inscribed angle is half the measure of its intercepted arc. The sum of the measures of the major and minor arcs is always $360^\circ$.

Answer 1

(i)
\[ \angle L = \dfrac{1}{2} \, m(\text{arc } MN) \quad \text{(By the Inscribed Angle Theorem)} \]
\[ 35^\circ = \dfrac{1}{2} \, m(\text{arc } MN) \]
\[ 2 \times 35^\circ = m(\text{arc } MN) \]
\[ \boxed{m(\text{arc } MN) = 70^\circ} \]
(ii)
\[ m(\text{arc } MLN) = 360^\circ - m(\text{arc } MN) \quad \text{(By definition of measure of arc)} \]
\[ m(\text{arc } MLN) = 360^\circ - 70^\circ \]
\[ \boxed{m(\text{arc } MLN) = 290^\circ} \]
Final Answers:
(i) $m(\text{arc } MN) = 70^\circ$
(ii) $m(\text{arc } MLN) = 290^\circ$