Angle is one of the measurement of rotation. The measure of the angle LPN is 57°
What is the relation between central angle and angle on circumference?
Consider an arc of a circle.
Suppose it subtends θ angle on the rest of the part of the circumference (at any point). Then it subtends 2θ angle on the center of that circle.
Consider the figure attached below.
We're given that:
- Central angle made by arc LP is of measure 102 degrees
- Central angle made by arc PN is of measure 144 degrees
Thus, we get the angle subtended on center by arc LMN on the center as:
[tex]\angle LON + \angle LOP + \angle PON = 360^\circ\\\angle LON = 360 - 102 - 144 = 114^\circ\\[/tex]
Thus, arc LMN subtends angle 114 on center. Therefore, it will subtend half of this angle on any part of the rest of the circumference of this circle.
Thus, we get:
[tex]\angle LPN = \dfrac{\angle LON}{2} = 114/2 = 57^\circ[/tex]
Thus, the measure of the angle LPN is 57°
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