Concept,
Two chords with a shared termination point on the circle make an inscribed angle in a circle. The vertex of the angle is this shared terminal point. An inscribed angle is equal to half the length of the intercepted arc.
Given,
We have been given a figure in which the line [tex]BD[/tex] is the tangent to the circle at the point [tex]B[/tex] and the measure of the arc [tex]AC[/tex] is [tex]108^{\circ}}[/tex]. And also we have some options:
A. [tex]38^{\circ}}[/tex]
B. [tex]18^{\circ}}[/tex]
C. [tex]118^{\circ}}[/tex]
D. [tex]72^{\circ}}[/tex]
To find,
We have to choose the correct option which tells the measure of the angle of [tex]CBD[/tex].
Solution,
In the figure, we can see that [tex]\angle ABC[/tex] is an inscribed angle and we know that the inscribed angle is half of the measure of the intercepted arc.
And from the figure arc [tex]AC[/tex] is the intercepted arc.
Thus, we can write
[tex]\angle ABC=\frac{\widehat{AC}}{2}[/tex]
[tex]\angle ABC=\frac{108}{2}[/tex]
[tex]\angle ABC=54^{\circ}[/tex]
So, the measure of [tex]\angle ABC=54^{\circ}[/tex].
Now given that [tex]BD[/tex] is a tangent to the circle at the point [tex]B[/tex].
Thus, we will get
[tex]\angle ABC+\angle CBD=90^{\circ}[/tex]
[tex]54^{\circ}+\angle CBD=90^{\circ}[/tex]
[tex]\angle CBD=90^{\circ}-54^{\circ}[/tex]
[tex]\angle CBD=36^{\circ}[/tex]
Thus, the measure of the angle [tex]CBD=36^{\circ}[/tex].
So, the correct option is A. [tex]36^{\circ}[/tex].
