- The measure of arc LN is 92°, so the central angle ∠LON is also 92°.
- The inscribed angle ∠LMN subtends the same arc LN, so its measure is half of the central angle: 92° / 2 = 46°.
- In triangle LMN, the sum of interior angles is 180°. Given ∠MLN = 68° and ∠LMN = 46°, then ∠MNL = 180° - 68° - 46° = 66°.
- The inscribed angle ∠MNL subtends arc ML, so the measure of arc ML is twice the angle: 2 * 66° = 132°.
- The total circumference of the circle is 360°. The sum of arcs LN, ML, and the remaining arc (which includes NP and PM) must equal 360°.
- Arc LN = 92°, arc ML = 132°, so the remaining arc (NP + PM) = 360° - 92° - 132° = 136°.
- Since no additional information is given about arcs NP and PM individually, we cannot determine their separate measures. However, if the question implies finding a specific angle or arc based on the diagram’s context (e.g., angle at P or arc MP), further clarification may be needed. But based on standard interpretation and given values, the key computed values are:
- ∠LMN = 46°
- ∠MNL = 66°
- Arc ML = 132°
- Remaining arc (NP + PM) = 136°
Parent Tip: Review the logic above to help your child master the concept of central angles and inscribed angles worksheet.
