Answer: 111 degrees
You may need to input the number only without the "degrees".
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Step-by-step explanation:
First we'll focus on triangle DEH. Ignore the other points and values.
We have the sides of EH = 32 and DH = 49. Let x be the measure of angle DHE, which is one of the pieces of angle DHG. We'll use the cosine rule to find x.
cos(angle) = adjacent/hypotenuse
cos(H) = EH/DH
cos(x) = 32/49
x = arccos(32/49)
x = 49.227195
The value is approximate and rounded to 6 decimal places. Make sure your calculator is in degree mode. Arccos is the same as inverse cosine.
We'll come back to this value later.
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Now focus on triangle HFG. Ignore the other points.
We have the sides of FG = 38 and GH = 43
Let y be the measure of angle FHG, which is the missing piece to form angle DHG. Based on how x and y are set up, we can say x+y = angle DHG.
To find y, we'll need the sine rule this time
sin(angle) = opposite/hypotenuse
sin(H) = FG/GH
sin(y) = 38/43
y = arcsin(38/43)
y = 62.094527 which is approximate to 6 decimal places
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The last step is to add the values of x and y we found earlier
angle DHG = (angle DHE) + (angle FHG)
angle DHG = (x) + (y)
angle DHG = (49.227195) + (62.094527)
angle DHG = 111.321722
angle DHG = 111 degrees after rounding to the nearest whole degree.
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If you wanted to do this all in one step, you could say:
arccos(32/49)+arcsin(38/43) = 111.3217 which rounds to 111 degrees.