Making Sentences with New Words I Learned.

1. Autonomous
Korean students need to autonomously choose the subjects they want to learn.

-It sounds more natural to put the adverb in front of the verb in this sentence.

-
2. Obstacle
A hot temper can be an obstacle to becoming a good teacher.

-If it is singular an article is necessary.-

3. Furious
I was furious when I was told that I am ugly. -this is good, but I would put "was ugly", but "am ugly" is not wrong- 

4. Arbitrary
The public (is/was) strongly opposed to the president's arbitrary decision.

-You could remove the (is/ was), but in that case, you need to remove the "to": -

The public strongly opposed the president's arbitrary decision.

1. Autonomous 

Korean students need to choose their own subjects autonomously what they want to learn.

Korean students must autonomously choose the subjects they wish to learn.

2. Obstacle
Hot temper can be an obstacle to become a good teacher.

A hot temper is an obstacle on the path of becoming a good teacher.  

    (In this case I think 'obstacle' and 'path' are words that complement eachother and provide the reader with better imagery).

3. Furious
I was furious when I was told that I am ugly.

I was furious when told I was ugly. (tense agreement)

4. Arbitrary
The public strongly opposed to the arbitrary decision of the president.

The public strongly opposed the arbitrary decision of the president.