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.

step-by-step explanation:

we are given

since, we have to isolate x on anyone side

so, we can move all x terms on one side

subtract both sides by 2x

add both sides by 9

divide both sides by 4

we get

fist on the left goes with the third on the right. second on the left goes with the fourth on the right. third on the left goes with the second on the right. the fourth on the left goes with first on the right