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# Find the range of values for $z$(FIGURE CANNOT COPY)

## $3<z<7$

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are two triangles. They're set up to use hinge here. Um, which states off we have corresponding sides that are congruent and we know the included angle than the side opposite of the largest angle is gonna be the longest side. So this allows us to set up the algebraic equation. 16 is greater than four Z minus 12. Using some algebra. I'm gonna add 12 to both sides, which gives me 20 a is greater than for Z and dividing by four. I get seven is greater than Z. But this is not the only condition we need to consider. Remember four Z minus 12 represents the smallest side and it has to be greater than zero. So we also have to consider for Z minus 12 Must be greater than zero if the shorter side is greater than zero 16 is obviously greater than zero. So again, using a little bit of algebra, I'm gonna add 12 both sides and get four Z is greater than 12 and dividing by four I get Z is now greater than three. So this means for all values Z z must be greater than three. Oh, excuse me. Z must be greater than three. But less than seven

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