An illustration shows three rays extending from a common origin. The angle measures between them are labeled (x + 4) degrees and (3x + 2) degrees. What are the two unknown angle measures that form the 90 degree angle?
The sum of the angle measures in a triangle is 180 degrees. In this case, we have three angles, (x + 4) degrees, (3x + 2) degrees, and the unknown angle that forms a 90-degree angle with the other two angles.
So, (x + 4) + (3x + 2) + unknown angle = 180.
Simplifying the equation, we get 4x + 6 + unknown angle = 180.
Subtracting 6 from both sides, we have 4x + unknown angle = 174.
We know that the unknown angle forms a 90-degree angle with the other two angles, so we can write another equation: unknown angle + (x + 4) + (3x + 2) = 180.
Simplifying, we get unknown angle + 4x + 6 = 180.
Subtracting 6 from both sides, we have unknown angle + 4x = 174.
Now we have a system of equations:
4x + unknown angle = 174,
unknown angle + 4x = 174.
To solve this system, we can subtract the second equation from the first:
(4x + unknown angle) - (unknown angle + 4x) = 174 - 174,
4x - 4x + unknown angle - unknown angle = 0,
0 = 0.
This means that the value of the unknown angle does not matter, and the two unknown angle measures that form the 90-degree angle are both unknown.