Find the measure of the angle given below

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Explanation:

You subtract the arc measures, and then divide the result in half.

The order of subtraction is larger arc minus smaller arc.

Angle ABC = ( (arc AC) - (arc DE) )/2

Angle ABC = (90 - 40)/2

Angle ABC = 50/2

Angle ABC = 25 degrees

Side note: segments AB and BC are considered secant segments.

The next section will go into a more in-depth look. The next section is optional.

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Alternative Method:

Refer to the diagram below. I've added points D and E to the diagram.

Based on those points, let

Note how inscribed angle CAB subtends the arc CED. The measure of arc CED is 40+x. Half this is (40+x)/2, which is the measure of inscribed angle CAB. Through similar logic, inscribed angle ACB is (40+y)/2.

Also, note how the four arc pieces of the full circle are: AC = 90, EC = x, DE = 40, and AD = y.

The four arc pieces must add to 360 as they form a full circle

AC+EC+DE+AD = 360

90+x+40+y = 360

x+y+130 = 360

x+y = 360-130

x+y = 230

y = 230-x

From this, we can rewrite the expression for inscribed angle ACB

angle ACB = (40+y)/2

angle ACB = (40+230-x)/2

angle ACB = (270-x)/2

Now because the interior angles of a triangle must add to 180, we can then use this fact to find angle ABC

(angle ABC) + (angle CAB) + (angle ACB) = 180

angle ABC = 180 - (angle CAB) - (angle ACB)

angle ABC = 180 - (40+x)/2 - (270-x)/2

angle ABC = 180 - 40/2 - x/2 - 270/2 + x/2

angle ABC = 180 - 20 - x/2 - 135 + x/2

angle ABC = 25 degrees

Ultimately the x terms cancel out when we combine the -x/2 and +x/2 together.