Question 1206533
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Angle ATB is 54 degrees.<br>
Since AT and BT intersect the circle, the measure of angle ATB is half the difference between the measures of major arc AB and minor arc AB.  The sum of the measures of those major and minor arcs is 360 degrees.  Simple calculations show minor arc AB is 126 degrees and major arc AB is 234 degrees.<br>
Angle CBT is 23 degrees.  Since its vertex is on the circle, its measure is half the measure of the arc it cuts off.  So the measure of arc CB is 46 degrees.<br>
The measure of arc AB is 126 degrees, and the measure of arc CB is 46 degrees; so the measure of arc AC is 80 degrees.<br>
Angle CAT has its vertex on the circle, so its measure is half the measure of the arc AC that it cuts off.  That arc measure is 80 degrees, so the angle measure is 40 degrees.<br>
ANSWER: 40 degrees<br>
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And here is an alternate path to the final answer, once we have determined that the measure of arc AB is 126 degrees.<br>
Angles CAT and CBT together cut off minor arc AB.  Since the measure of arc AB is 126 degrees, the sum of the measures of angles CAT and CBT is 63 degrees.  Then, since the measure of angle CBT is 23 degrees, the measure of angle CAT is 40 degrees.<br>