9514 1404 393
Answer:
your choice is the best
Explanation:
If you simply look at the segments marked with expressions in z, you have* ...
z+23 > 0 ⇒ z > -23
3z +9 > 0 ⇒ z > -9/3 = -3
That is, the side lengths must be positive. We must also have the side labeled z+23 be longer than the side labeled 3z+9:
z +23 > 3z +9
14 > 2z . . . . . . . subtract z+9
7 > z
So, the limits on the marked side lengths require ...
-3 < z < 7 . . . . matches the 4th choice
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If you actually work out the possible sizes of these triangles such that the angles are the given measures, you find that the minimum length of the legs with the hash mark is about 15.8649, and the minimum value of z is ...
z > (12√6 -7)/5 ≈ 4.47878
This value is obtained when the triangles are isosceles. Apparently, you're not supposed to think that deeply about this question.
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* The "or equal to" case is not included, because that would allow the triangles to have zero area. That is, the triangles would be a single point.