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Given AB is perpendicular to BC

<ABO= (2x + y)degrees
<OBC= (6x + 8)deg.
<AOB= (23 + 90)deg
<BOC= (4x + 4)degrees

Find <ABO

(don't know how to draw it but a triangle is within a circle. The triangle name is ABC and segment BO goes through the triangle.)

Confusing question.

You say "a triangle is within a circle"
Are the vertices on the triangle ON the circle?
Where does O come in ? Is O the centre of the circle?

also the statement
<AOB= (23 + 90)deg looks like a typo, all the others contain variable names.

no the vertices is on the point. the 0 i in the center of the circle.

To find the measure of angle ABO, we need to use the given information about the angles in triangle ABC.

We know that AB is perpendicular to BC, which means angle ABC is a right angle (90 degrees). And since angle AOB is given as (23 + 90) degrees, we can calculate the measure of angle ABO.

First, let's use the information given to create an equation:

Angle ABC + Angle OBC + Angle ABO + Angle BOA = 360 degrees (the sum of angles in a triangle)

Substituting the given values into the equation:

90 degrees + (6x + 8) degrees + ABO + (2x + y) degrees = 360 degrees

Simplifying the equation:

(6x + 8 + 2x + y) degrees + ABO = 360 degrees - 90 degrees

Combining like terms:

8x + y + ABO = 270 degrees

Since we know that the sum of angles ABO, OBA, and OBC is equal to 180 degrees (the sum of angles in a straight line), we can rewrite the equation as:

ABO + (23 + 90) degrees + (4x + 4) degrees = 180 degrees

Substituting the given values:

ABO + 113 degrees + (4x + 4) degrees = 180 degrees

Simplifying:

(4x + 4 + 113) degrees + ABO = 180 degrees

Combining like terms:

4x + 117 + ABO = 180 degrees

Now we have two equations that represent the same value of ABO:

8x + y + ABO = 270 degrees

and

4x + 117 + ABO = 180 degrees

We can solve this system of equations to find the values of x, y, and ABO.

However, since we do not have enough information about the values of x and y, we cannot find the exact measure of ABO at this point. Additional information is needed to solve the problem.