Line AB is perpendicular to line BC. Point E is in the interior of angle ABC. If angle ABE = 2x+3 and angle EBC = 4x-9, find x. Then find the measure of both angles.

Their sum must equal 90°

2x+3 + 4x-9 = 90
6x = 96
x = 16

one angle is 2(16)+3 or 35°
the other is 4(16)-9 or 55°

(notice 35+55=90)

To solve this problem, we can use the fact that the sum of the angles in a triangle is always 180 degrees.

We know that line AB is perpendicular to line BC, which means that angle ABC is a right angle and measures 90 degrees.

We are given that angle ABE = 2x + 3 and angle EBC = 4x - 9.

To find x, we'll first set up an equation using the fact that the sum of the angles in triangle ABE is 180 degrees:

(2x + 3) + (4x - 9) + 90 = 180

Simplifying the equation, we have:
2x + 3 + 4x - 9 + 90 = 180

Combining like terms, we have:
6x + 84 = 180

Next, we'll subtract 84 from both sides of the equation:
6x = 96

Dividing both sides of the equation by 6, we get:
x = 16

Now that we have found the value of x, we can substitute it back into the expressions for angle ABE and angle EBC to find their measures.

Angle ABE = 2x + 3 = 2(16) + 3 = 32 + 3 = 35 degrees

Angle EBC = 4x - 9 = 4(16) - 9 = 64 - 9 = 55 degrees

So, x = 16 and the measures of angle ABE and angle EBC are 35 degrees and 55 degrees, respectively.

To find the value of x, we can use the fact that the sum of the angles of a triangle is 180 degrees. Since angle ABC is a right angle (perpendicular lines form right angles), we have:

angle ABC + angle ABE + angle EBC = 180 degrees.

Since angle ABE = 2x + 3 and angle EBC = 4x - 9, we can substitute these values in the equation:

90 degrees + (2x + 3) + (4x - 9) = 180 degrees.

Let's simplify the equation:

90 + 2x + 3 + 4x - 9 = 180.

Combine like terms:

6x + 84 = 180.

Subtract 84 from both sides:

6x = 96.

Divide both sides by 6:

x = 16.

Now that we have found the value of x, we can substitute it back into the angle measurements to find their values:

angle ABE = 2x + 3 = 2(16) + 3 = 35 degrees.

angle EBC = 4x - 9 = 4(16) - 9 = 55 degrees.

Therefore, x = 16, angle ABE = 35 degrees, and angle EBC = 55 degrees.