Question 1206533
<font color=black size=3>
Answer: <font color=red size=4>40 degrees</font>


Explanation


This is likely what the diagram looks like.
*[illustration UploadedScreenshot_45.png]
The diagram was made with <a href="https://www.geogebra.org/calculator">GeoGebra</a>


Let D be the center of the circle. Draw segments DA and DB which are radii.
Focus only on quadrilateral DATB.
For any quadrilateral, the four inside angles must always add to 360 degrees.
D+A+T+B = 360
D+90+54+90 = 360
D+234 = 360
D = 360-234
D = 126
This is the measure of angle ADB. It is also the measure of minor arc AB. Central angles subtending an arc will have the same measure.
Note that angles A and B are 90 degrees each since they are at points of tangency.


Let E be some point on the circle that's not on minor arc AB. Refer to the diagram below.
*[illustration UploadedScreenshot_46.png]
Major arc AEB and minor arc AB glue together to form the entire circle. There are no gaps and no overlaps between the arcs. The arcs partition the circle's circumference.


Because of this we know that:
(major arc AEB) + (minor arc AB) = 360
(major arc AEB) + 126 = 360
major arc AEB = 360 - 126
major arc AEB = 234


Now we'll use the inscribed angle theorem to determine inscribed angle ACB.
angle ACB = (1/2)*(major arc AEB)
angle ACB = (1/2)*(234)
angle ACB = 117
We'll use this later after a slight detour in the next section.


-------------------------------


Draw a segment connecting points D and T.
Quadrilateral DATB is split into triangle DAT and triangle DBT.
They are both right triangles due to the points of tangency.
We can use the hypotenuse leg (HL) theorem to prove those right triangles are congruent, and it consequently means AT = TB.


Since AT = TB, we determine that triangle BAT is isosceles.


Furthermore, it means angle ABT = angle BAT since they are the base angles. The congruent base angles are opposite the congruent sides.


Focus only on triangle ATB.
The three inside angles must add to 180 degrees.
A+T+B = 180
x+54+x = 180
2x+54 = 180
2x = 180-54
2x = 126
x = 126/2
x = 63


Base angles ABT and BAT are 63 degrees each.


Then,
(angle CBA) + (angle CBT) = angle ABT
angle CBA = (angle ABT) - (angle CBT)
angle CBA = (63) - (23)
angle CBA = 40


-------------------------------


The conclusions of the first two sections are
angle ACB = 117
angle CBA = 40
These represent angles C and B in triangle ABC. 


Focus only on triangle ABC.
A+B+C = 180
A+117+40 = 180
A+157 = 180
A = 180-157
A = 23
This is the measure of angle CAB.


Then we have one last set of steps.
(angle CAB)+(angle CAT) = angle BAT
(23)+(angle CAT) = 63
angle CAT = 63-23
<font color=red>angle CAT = 40</font> is the final answer.


My response is a bit long-winded. Another tutor may provide a much more efficient pathway.
</font>