Question

if ( mwidehat{ny}=96^{circ} ) and ( mangle niy = 109^{circ} ), find ( mwidehat{lb} ).

Explanation:

Step1: Recall the property of vertical angles

Vertical angles are equal. So, \(m\angle N I Y=m\angle L I B = 109^{\circ}\).

Step2: Use the formula for the measure of an arc in a circle

The sum of arcs around a circle is \(360^{\circ}\). Also, in a circle, the measure of an arc is related to the central - angle. If two chords intersect at the center \(I\), and we know that \(m\overset{\frown}{NY} = 96^{\circ}\), and \(m\angle N I Y=m\angle L I B\).
The formula for the measure of an arc \(\overset{\frown}{LB}\):
We know that the central - angle \(\angle LIB\) subtends the arc \(\overset{\frown}{LB}\). Also, if we consider the relationship between the central - angle and the arc.
Let \(x=m\overset{\frown}{LB}\). Since the central - angle \(\angle LIB\) and the arc \(\overset{\frown}{LB}\) have the relationship \(m\angle LIB=\frac{1}{2}(m\overset{\frown}{LB}+m\overset{\frown}{NY})\) (this is incorrect, actually, when two chords intersect at the center of a circle, the measure of the central - angle is equal to the measure of the arc it subtends).
Wait, correct approach: In a circle, the measure of a central - angle is equal to the measure of the arc it subtends. But we also know that the sum of central - angles around a point is \(360^{\circ}\). However, if we assume that the two non - overlapping arcs \(\overset{\frown}{NY}\) and \(\overset{\frown}{LB}\) and their corresponding central - angles.
Since \(m\angle N I Y\) and \(m\angle L I B\) are vertical angles (\(m\angle N I Y = m\angle L I B\)), and the measure of an arc subtended by a central - angle \(\theta\) is \(\theta\) (in degrees).
We use the property that the sum of arcs \(\overset{\frown}{NY}+\overset{\frown}{LB}+ \text{other arcs}=360^{\circ}\). But if we assume that the two arcs \(\overset{\frown}{NY}\) and \(\overset{\frown}{LB}\) and their vertical - angle - related central - angles.
Another way: The measure of an arc \(s\) in a circle is given by \(s = r\theta\) (in radians), but in degree measure, for a unit - circle (or when radius is not relevant in degree measure of arc), the measure of a central - angle \(\theta\) (in degrees) is equal to the measure of the arc it subtends.
We know that \(m\angle N I Y=\frac{1}{2}(m\overset{\frown}{NY}+m\overset{\frown}{LB})\) (no, that's the formula for the angle formed by two chords intersecting inside the circle, but when the intersection is at the center, \(m\angle N I Y\) (central - angle) \(=\frac{1}{2}(m\overset{\frown}{NY}+m\overset{\frown}{LB})\) is wrong. The correct formula for the angle formed by two chords intersecting at the center is \(m\angle N I Y=\frac{1}{2}(m\overset{\frown}{NY}+m\overset{\frown}{LB})\) is incorrect. The correct formula: When two chords intersect at the center of a circle, \(m\angle N I Y\) (central - angle) and the arcs:
If we consider the two vertical angles \(\angle N I Y\) and \(\angle L I B\).
We know that \(m\angle N I Y = 109^{\circ}\), and \(m\overset{\frown}{NY}=96^{\circ}\)
By the formula \(m\angle N I Y=\frac{1}{2}(m\overset{\frown}{NY}+m\overset{\frown}{LB})\) (angle formed by two chords intersecting inside the circle: \(m\angle=\frac{1}{2}(m\text{arc}_1 + m\text{arc}_2)\))

$$109=\frac{1}{2}(96 + m\overset{\frown}{LB})$$

Step3: Solve the equation for \(m\overset{\frown}{LB}\)

Multiply both sides of the equation \(109=\frac{1}{2}(96 + m\overset{\frown}{LB})\) by \(2\):

$$2\times109=96 + m\overset{\frown}{LB}$$

$$218=96 + m\overset{\frown}{LB}$$

Subtract \(96\) from both sides:

$$m\overset{\frown}{LB}=218 - 96$$

$$m\overset{\frown}{LB}=122$$

Answer:

\(122\)