Question

enter the measure of the angle. degrees

Explanation:

Step1: Identify right angle

From the diagram, $\angle U Z L$ (or the angle with the right angle symbol) is $90^\circ$. So, $(6x - 25)^\circ+(4x)^\circ = 90^\circ$.

Step2: Solve for x

Combine like terms: $6x - 25 + 4x=90$.
Simplify: $10x - 25 = 90$.
Add 25 to both sides: $10x=90 + 25=115$? Wait, no, wait. Wait, actually, the right angle is between UZ and the other line, so $(6x - 25)+(4x)=90$? Wait, no, maybe the two angles $(6x - 25)$ and $(4x)$ add up to 90 degrees (since there's a right angle symbol). So:
$6x - 25 + 4x=90$
$10x - 25 = 90$
$10x=90 + 25=115$? Wait, that can't be. Wait, maybe the right angle is 90, so $6x - 25 = 90 - 4x$? No, let's re-express. Wait, the angle between UZ and ZL is a right angle (90 degrees), so the sum of $(6x - 25)^\circ$ and $(4x)^\circ$ is 90 degrees. So:
$6x - 25 + 4x = 90$
$10x = 90 + 25$
$10x = 115$
Wait, that gives $x = 11.5$, but then $4x = 46$, $6x -25=69 -25=44$, 44+46=90. Wait, no, 611.5=69, 69-25=44, 411.5=46, 44+46=90. Then the angle we need? Wait, maybe the angle is $(4x)^\circ$? Wait, the problem says "Enter the measure of the angle" – maybe the angle is $(4x)^\circ$ or $(6x -25)^\circ$? Wait, maybe I misread. Wait, the diagram has a right angle (the small square), so the two angles adjacent to the right angle (the ones with $(6x -25)^\circ$ and $(4x)^\circ$) add up to 90 degrees. So:
$6x - 25 + 4x = 90$
$10x = 115$
$x = 11.5$
Then, if we need the measure of, say, the $(4x)^\circ$ angle: $4x = 4*11.5 = 46$ degrees. Wait, but let's check again. Wait, maybe the right angle is 90, so $6x -25 = 90 - 4x$? No, the sum is 90. So:
$6x -25 + 4x = 90$
$10x = 115$
$x = 11.5$
Then $4x = 46$, $6x -25 = 44$. So if the angle in question is $(4x)^\circ$, then it's 46 degrees. Wait, but maybe the problem is that the two angles are complementary (sum to 90), so solving for x and then finding the angle. Let's confirm:
$6x -25 + 4x = 90$
$10x = 115$
$x = 11.5$
Then $4x = 4*11.5 = 46$. So the measure of the angle (assuming it's the $(4x)^\circ$ angle) is 46 degrees. Wait, but maybe the angle is $(6x -25)^\circ$: 6*11.5 -25=69-25=44. But 44+46=90, which matches the right angle. So depending on which angle, but the problem says "Enter the measure of the angle" – maybe the angle is $(4x)^\circ$, so 46 degrees. Wait, but maybe I made a mistake. Wait, let's re-express the equation:
$6x -25 + 4x = 90$
$10x = 115$
$x = 11.5$
Then $4x = 46$. So the measure is 46 degrees.

Answer:

46