Answer:
- ∠g and ∠h are complementary angles
- ∠g and ∠h are acute angles
Step-by-step explanation:
Use the given information to determine what the angles can be.
g = 2x -90 . . . . given
g > 0 . . . . . . . . . given
2x -90 > 0
2x > 90 . . . . . add 90
x > 45 . . . . . . divide by 2
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h = 180 -2x
h > 0
180 -2x > 0
180 > 2x
90 > x
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The requirement that both angles be greater than zero puts limits on x:
45 < x < 90
We can put this back into the given relations for g and h:
g = 2x -90
x = (g +90)/2
45 < (g +90)/2 < 90 . . . . substitute for x
0 < g/2 < 45 . . . . . . . . . . subtract 45
0 < g < 90 . . . . . . . . . . . . g is an acute angle
Similarly, ...
h = 180 -2x
x = (180 -h)/2 = 90 -h/2
45 < (90 -h/2) < 90 . . . . substitute for x
-45 < -h/2 < 0 . . . . . . . . . subtract 90
90 > h > 0 . . . . . . . . . . . multiply by -2; h is an acute angle
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We can add the angle measures to see if they are supplementary or complementary:
g + h = (2x -90) +(180 -2x)
g + h = 90 . . . . . simplify; the angles are complementary
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The relevant observations are ...
- ∠g and ∠h are complementary angles
- ∠g and ∠h are acute angles
