which of the following statements is true? answer options: ( mangle ghj = 2(mangle ihj) ), ( mangle ghi = mangle ghj ), ( \frac{1}{2}(mangle ghj) = mangle ihj ), ( mangle ihj = \frac{1}{2}(mangle ghi) )

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# Brief Explanations:
From the diagram, we can see that ray $ HJ $ bisects $ \angle GHI $, which means it divides $ \angle GHI $ into two equal angles. So, $ \angle IHJ $ is half of $ \angle GHI $, or $ m\angle IHJ=\frac{1}{2}(m\angle GHI) $. Let's check the other options:
- For $ m\angle GHJ = 2(m\angle IHJ) $, we don't know the relationship between $ \angle GHJ $ and $ \angle IHJ $ from the diagram (since $ HJ $ bisects $ \angle GHI $, not necessarily $ \angle GHJ $).
- For $ m\angle GHI=m\angle GHJ $, $ \angle GHI $ and $ \angle GHJ $ are not the same as $ \angle GHJ $ is a larger angle (or not, but from the bisecting, we know $ \angle GHI $ is split into two, so this is false).
- For $ \frac{1}{2}(m\angle GHJ)=m\angle IHJ $, there's no indication that $ HJ $ bisects $ \angle GHJ $, so this is not supported.
- For $ m\angle IHJ=\frac{1}{2}(m\angle GHI) $, since $ HJ $ bisects $ \angle GHI $, this is true as an angle bisector divides an angle into two equal - measure angles.

# Answer:
$ \boldsymbol{m\angle IHJ=\frac{1}{2}(m\angle GHI)} $ (the option with this statement)

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