What are the common ways to use the word 'wager'?
“How about a (24) ________?” he said to Michael. “You make a house and I’ll make a little boat”. In a couple of hours the pair had cut, crafted and glued the matchsticks together and the task was complete. “I’m going to call the boat Hazel,” Brian said proudly. That was when his (25) ________ really started.
#24
A challenge (the correct answer)
B strife
C rivalry
D wager

Question

Answer

“Let’s make a wager”.  Which mean let’s make a bet.  We also use it in a construction like this.  “I wager X is the case“ or “ I wager X is true”.  This practically means “I think X is true”, but literally means that we would be willing take the bet that X is true.

Answer

To me that question could have more than one correct answer in position #24 or #25.

And wager to me makes a very good and natural sentence.

"I'll challenge you to X" is a more natural way to begin that text. 
"How about a wager" (bet is a more common word) is perfectly valid.

#25 could be A  C  D or possibly C.

If we knew what the section of the book was teaching we could give a better answer.

Answer

challenge is correct or much better with the options for #25 you provided. With "wager" there would be an amount/prize/consequence of winning or loosing the "wager" = a bet, in the fist part.

Answer

Thank you very much, Mark!

Answer

Wager can be used as a verb or a noun. It means a bet (noun) or to bet (verb).
Verb example: I wager $30 on Galloping Giant to win the final horse race today.

Noun example: I made a wager with my friend that Tony Ferguson would win UFC 239 on 9 May 2020.

You can therefore use this word in any betting conversation. Synonyms are bet, stake, gamble, lay (a bet).

I hope this assists you.
Teacher Mark