Statement

I can't tell if the "true or false" is asking about the entire statement or about the final sentence summing up with a definitive statement.

As to the general statements, in an academic setting what's being put forth is true (that greater leverage increases risk, thus cost of debt/equity). In reality, required rates of return on debt and equity don't really move in that kind of a manner (not cleanly). If you're issued corporate debt at a 5% interest rate and then you buy back shares with the debt proceeds, your cost of debt is still 5%, even though your business has greater leverage (and risk); it's possible that next time you raise debt your interest rate could be higher due to worse debt levels. On the other hand, your use of debt could help you establish a better credit rating, thus reducing cost of debt in the future.

As far as cost of equity, that's the investor's required rate of return. Leverage is certainly an aspect of risk that an investor takes into account; on the other hand, it depends how much debt we're talking about. Moving a company's debt ratio from 15% to 20% is probably not going to actually impact required rate of return on equity; rising from 15% to 60% will likely impact cost of equity.

"Since the firm's WACC is a weighted average of required returns on debt and equity, it must also rise."

Yes, it's just simple math (see edit below: it's actually false because you don't know the original and final debt/equity ratio). If your cost of debt and equity rise then your WACC rises. If the previous statements are taken at face value (that cost of debt and equity increased) then this statement is true (although, as mentioned, I would question whether or not to take the premise at face value).

Edit: as HighlyClevered points out below, you cannot determine the delta in WACC without first knowing the debt/equity ratio.