The argument presents a conditional relationship (if you have a dog, you know the true value of companionship), gives an example of someone who confirms the necessary condition (Alicia, who knows the true value of companionship), and then moves to a conclusion (Alicia has a dog). This commits the same confusing necessary and sufficient conditions flaw as the stimulus. Confirming the necessary condition (that you know the true value of companionship) tells us nothing about the sufficient condition (that you own a dog).

Wrong flaw. This denies the sufficient condition (Fran has never discovered something new) and, as a result, denies the necessary condition (Fran hasn’t explored all the possible solutions to a problem). Denying the sufficient condition in a conditional relationship tells you nothing about the necessary condition. The stimulus, meanwhile, didn’t deny its sufficient condition.

No flaw. This denies the necessary condition (Jill is a person who accomplishes everything) and, as a result, denies the sufficient condition (Jill is a person who faces sufficient challenges). This is a valid use of the contrapositive.

No flaw. Any closed plane figure bounded by straight lines is a polygon. The figure on the board is a closed plane figure bounded by straight lines, so it’s a polygon.

Wrong flaw. (E) errs because it triggers neither the sufficient (people who haven’t lost something they can’t afford to lose) nor the necessary (will be lax about keeping their property secure) conditions, yet still tries to draw a conclusion. Jon being lax about keeping property secure when it’s something he can afford to lose isn’t the same as the necessary condition, and Jon having never lost anything isn’t the same as the sufficient condition. The stimulus, meanwhile, doesn’t fail to trigger its necessary condition.