The Greek prefix di- means two. So, a "dilemma" is a problem with two possible solutions, a predicament, a quandary. To provide additional help, a "notion" is an idea or a feeling; a "suggestion" is an idea expressed to help someone with a problem. 

The Greek prefix arch- can have two meanings. It either means chief as in "monarch" or ancient, old-fashioned as in "archaic." "Archaic" means old-fashioned and out of date, obsolete, no longer needed or used. To provide additional help, "impressive" means inspiring admiration, making people impressed; "domineering" means bossy, dominating, in-charge; "confounding" means confusing, bewildering; "recessive" means (of a gene) not dominant, hidden. 

The Greek prefix mis- means wrong, bad, as in "mistake" which means error. A "misnomer" is an inaccurate name, an error in naming something. So, the best possible answer is "error." You might also notice the root word -nom- which often means name. So, simply a "misnomer" is a wrong name, an error in naming. 

The Greek prefix melan- means black, dark, so "melancholy" means a dark mood, sadness, depression, misery. Additionally, the other four answer choices all mean deep happiness. 

The Greek prefix opt- means eyes, sight, or related to sight. So, "optical" means of or relating to sight. The synonym here is not perfect, but sometimes you have to find the answer that is "closest" to the word in question. 

The Greek prefix pater- means father, so "paternity" means fatherhood. The Greek prefix mater- means mother, so "maternity" means motherhood. Another example of a pater- word is "patriarch" which refers to a man in charge of a group.

The Greek prefix idio- means unusual, weird, distinct, as in "idiot." So, an "idiosyncrasy" is an unusual personality trait, a quirk. Additionally, a "hint" is a clue; "patriotism" is love of one's country; "xenophobia" is fear of foreigners; "compassion" is warmth, generosity, and kindness. 

The Greek prefix ethno- means race, so an "ethnicity" is a race of people. Other examples of ethno- words include "ethnic" which means racial and "ethnology" the study of different races. It is worth noting that the suffix -ology means study of.

The Greek prefix a- means without, so "abyss" means without a bottom, bottomless, a void, a chasm. To provide further help, "timeless" means not fixed in any era, always fashionable and appropriate; "thankless" refers to a task that no-one wants to do because it is little appreciated.

The Greek prefix dia- means through, between, across, so "dialogue" means a conversation between two people. Another example of a dia- word is "diameter" which refers to a line that goes from one side of a circle to another, the passes across and through the circle. To provide additional help, "commiseration" is showing sympathy for someone's sadness, joining someone in sadness; "condescension" is talking down to someone in an arrogant way; a "confrontation" is a fight.