If the measures of the exterior angles of the acute angles of a right triangle are (6x+23) and (4x+17), find the measures of the acute angles.
Let x = acute angle 1 and 4x + 17 ext angle
Let y = acute angle 2 and 6x + 23 ext angle
x + 4x + 17 = 180
X=32.6
y =90-x = 90-32.6 = 57.4
4(32.6) + 17 = 147.4
Note: Cannot use x value for ext. 6x+23. Y is different acute angle. Vice versa.
To find the measures of the acute angles, we can use the fact that the sum of the measures of the exterior angles of any polygon is always 360 degrees.
In a right triangle, one of the acute angles is 90 degrees. Therefore, the sum of the measures of the other two exterior angles should be equal to 360 - 90 = 270 degrees.
Let's set up an equation to solve for x:
(6x + 23) + (4x + 17) = 270
Now, combine like terms:
10x + 40 = 270
Subtract 40 from both sides:
10x = 230
Divide both sides by 10:
x = 23
Now that we have the value of x, we can substitute it back into the expressions for the measures of the exterior angles to find their values:
For the first exterior angle:
6x + 23 = 6(23) + 23 = 138 + 23 = 161 degrees
For the second exterior angle:
4x + 17 = 4(23) + 17 = 92 + 17 = 109 degrees
Therefore, the measures of the acute angles in the right triangle are 161 degrees and 109 degrees.