Question: Find m∠LN. F 38 G 56 H 58 J 76 K 96° L 20° M P N

Solution

The problem asks you to find the measure of angle \( \angle LNM \). Given: - \( \angle LPM = 20^\circ \) as an inscribed angle. - \( \angle LKM = 90^\circ \) as the right angle of the semicircle. \( \angle LPM \) is half of the angle \( \angle LNM \) because \( \angle LPM \) is an inscribed angle subtending the same arc as \( \angle LNM \). Therefore, the measure of angle \( \angle LNM \) is: \[ \angle LNM = 2 \times \angle LPM \] \[ \angle LNM = 2 \times 20^\circ \] \[ \angle LNM = 40^\circ \] It seems like my calculation contradicts the options, so let’s consider the semicircle property. Since \( \angle LNM \) is an external angle to the triangle \( LPM \), it is equal to the sum of the opposite internal angles: \[ \angle LNM = \angle LKP + \angle LPM \] \[ \angle LNM = 90^\circ + 20^\circ \] \[ \angle LNM = 110^\circ \] If this also doesn’t match, ensure you correctly interpreted the arc or consider checking if the \( \angle LNM \) directly subtends a semicircle, which is why I might have considered arc options. It’s crucial to ensure that other angles and positions were accurately marked or missing interpretations from the image. Thus, you can look through the options again, given your context.