Question

determining unknown angle measures
$overleftrightarrow{ln}$ is tangent to circle o at point m and $overline{qm}$ is a diameter.
determine the measure of the following angles.
the measure of $angle qml$ is dropdown degrees.
the measure of $angle pmn$ is 27, 42, 69, 90 degrees.
image of circle o with tangent ln at m, diameter qm, points p, q, n, l, angles 27° at m, 42° at q

Explanation:

Step1: Analyze ∠QML

A tangent to a circle is perpendicular to the radius at the point of tangency. Since \( \overleftrightarrow{LN} \) is tangent to circle \( O \) at \( M \) and \( QM \) is a diameter (so \( OM \) is a radius, and \( QM \) is a straight line through the center), \( \angle QML = 90^\circ \) (tangent - radius perpendicularity).

Step2: Analyze ∠PMN

First, find \( \angle QMP \). In triangle \( QMP \) (or using inscribed angle and tangent - chord angle), the measure of an angle formed by a tangent and a chord is equal to the measure of the inscribed angle on the opposite side of the chord. The inscribed angle \( \angle QPM \) (wait, actually, the angle between tangent \( LN \) and chord \( PM \) is equal to the inscribed angle subtended by \( PM \). Wait, alternatively, since \( \angle QML = 90^\circ \), and \( \angle QMP = 27^\circ+ 42^\circ? \) Wait, no, looking at the diagram, \( \angle QMP \): the inscribed angle at \( Q \) is \( 42^\circ \), so the arc \( PM \) is \( 84^\circ \)? Wait, no, maybe better: \( \angle QML = 90^\circ \), \( \angle QMP = 27^\circ + 42^\circ? \) Wait, no, the angle between tangent \( LN \) and chord \( PM \) is \( \angle PMN \), and the inscribed angle over arc \( PN \) or \( QM \)? Wait, actually, \( \angle QML = 90^\circ \), and \( \angle QMP = 27^\circ + 42^\circ? \) Wait, no, the given angle at \( M \) between \( QM \) and \( PM \) is \( 27^\circ + 42^\circ? \) Wait, the diagram shows \( \angle QMP \) has a part of \( 27^\circ \) and the angle at \( Q \) is \( 42^\circ \). Wait, maybe using the fact that \( \angle QML = 90^\circ \), so \( \angle PMN=90^\circ - (27^\circ + 42^\circ)? \) No, wait, \( \angle QMP \): the inscribed angle \( \angle PQM = 42^\circ \), so the arc \( PM \) is \( 84^\circ \), but maybe simpler: \( \angle QML = 90^\circ \), and \( \angle QMP = 27^\circ+ 42^\circ? \) Wait, no, the angle between tangent \( LN \) and chord \( PM \) is \( \angle PMN \), and \( \angle QML = 90^\circ \), \( \angle QMP = 27^\circ + 42^\circ = 69^\circ? \) Wait, no, let's recast:

Wait, \( \angle QML = 90^\circ \) (tangent - diameter perpendicularity). Then, \( \angle PMN = 90^\circ - \angle QMP \). What is \( \angle QMP \)? The angle between \( QM \) and \( PM \): from the diagram, the angle at \( M \) between \( QM \) and \( PM \) is \( 27^\circ + 42^\circ? \) Wait, the angle at \( Q \) is \( 42^\circ \), so the inscribed angle over arc \( PM \) is \( 42^\circ \), so arc \( PM \) is \( 84^\circ \), so the central angle would be \( 84^\circ \), but maybe not. Alternatively, since \( \angle QML = 90^\circ \), and \( \angle QMP = 27^\circ + 42^\circ = 69^\circ? \) No, wait, the answer for \( \angle PMN \): since \( \angle QML = 90^\circ \), and \( \angle QMP = 27^\circ + 42^\circ = 69^\circ \)? Wait, no, \( \angle PMN = 90^\circ - (27^\circ + 42^\circ)? \) No, \( 27 + 42 = 69 \), \( 90 - 69 = 21 \), no. Wait, maybe I messed up. Wait, the options for \( \angle PMN \) are 27, 42, 69, 90. And \( \angle QML \) is 90. Then \( \angle PMN \): let's see, \( \angle QML = 90^\circ \), and \( \angle QMP = 27^\circ + 42^\circ = 69^\circ \)? No, \( \angle PMN = 90^\circ - 27^\circ - 42^\circ? \) No, 90 - 27 - 42 = 21, not an option. Wait, maybe the angle between tangent and chord: the measure of the angle between a tangent and a chord is equal to the measure of the inscribed angle on the opposite side of the chord. So \( \angle PMN \) (tangent \( LN \) and chord \( PM \)) is equal to the inscribed angle over arc \( PN \) or \( QM \). Wait, the inscribed angle at \( Q \) is \( 42^\circ \), so arc \( PM \) is \( 84^\circ \), and the angle between tangent and chord \( PM \) is equal to the inscribed angle over arc \( PM \)'s complement? Wait, no, the formula is: the measure of an angle formed by a tangent and a chord is equal to half the measure of its intercepted arc. So \( \angle PMN \) intercepts arc \( PN \)? Wait, maybe better: \( \angle QML = 90^\circ \) (tangent - diameter), so \( \angle PMN = 90^\circ - \angle QMP \). \( \angle QMP \) is composed of \( 27^\circ \) and the angle equal to \( 42^\circ \) (inscribed angle). Wait, the angle at \( Q \) is \( 42^\circ \), so the angle at \( M \) between \( QM \) and \( PM \) is \( 42^\circ + 27^\circ = 69^\circ \)? Then \( \angle PMN = 90^\circ - 21^\circ? \) No, this is confusing. Wait, the options for \( \angle PMN \) are 27, 42, 69, 90. And \( \angle QML = 90^\circ \). So if \( \angle QMP = 21^\circ \), no. Wait, maybe \( \angle PMN = 69^\circ \)? Wait, no, let's start over:

1. \( \angle QML \): tangent to circle at \( M \), \( QM \) is diameter. So tangent is perpendicular to diameter, so \( \angle QML = 90^\circ \).

2. \( \angle PMN \): the angle between tangent \( LN \) and chord \( PM \). The measure of this angle is equal to the measure of the inscribed angle on the opposite side of the chord \( PM \). The inscribed angle at \( Q \) is \( 42^\circ \), and the angle between \( QM \) and \( PM \) is \( 27^\circ \), so the total angle \( \angle QMP = 27^\circ + 42^\circ = 69^\circ \)? Wait, no, \( \angle PMN = 90^\circ - (90^\circ - 69^\circ)? \) No, I think I made a mistake. Wait, the correct approach:

- Tangent - radius (or diameter) perpendicularity: \( \angle QML = 90^\circ \).

- The angle between tangent \( LN \) and chord \( PM \) is \( \angle PMN \), and this angle is equal to the measure of the inscribed angle subtended by arc \( PM \). Wait, the inscribed angle at \( Q \) is \( 42^\circ \), so arc \( PM \) is \( 84^\circ \), so the angle between tangent and chord \( PM \) is \( \frac{1}{2} \) arc \( PM \)? No, the formula is that the angle between tangent and chord is equal to the inscribed angle on the opposite side of the chord. So \( \angle PMN = \angle PQM + \angle QMP? \) No, \( \angle PQM = 42^\circ \), and \( \angle QMP = 27^\circ \), so \( \angle PMN = 42^\circ + 27^\circ = 69^\circ \). Yes, that makes sense. So \( \angle PMN = 69^\circ \), and \( \angle QML = 90^\circ \).

Answer:

The measure of \( \angle QML \) is \( 90 \) degrees.
The measure of \( \angle PMN \) is \( 69 \) degrees.