Question

select the correct answer from each drop-down menu. how does the figure help verify the triangle inequality theorem? the two sides with lengths of 6 and 3 will drop - down two of the sides drop - down the length of the third side. , which shows there is no way to construct a triangle in which the drop - down of reset next

Explanation:

Step1: Recall Triangle Inequality Theorem

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. For sides \(a\), \(b\), \(c\), we have \(a + b>c\), \(a + c>b\), and \(b + c>a\).

Step2: Analyze Given Side Lengths

We are given two sides with lengths \(6\) and \(3\), and a third side with length \(9\). Let's check the sum of the two smaller sides: \(3+6 = 9\). But according to the triangle inequality theorem, the sum should be greater than the third side, not equal.

Step3: Interpret the Figure's Role

The figure (with the arcs not intersecting appropriately) shows that when we try to construct a triangle with sides \(3\), \(6\), and \(9\), the two sides with lengths \(6\) and \(3\) will not be able to form a triangle with the third side of length \(9\) because \(3 + 6\) is not greater than \(9\) (it's equal). This verifies the triangle inequality theorem by showing that a triangle cannot be constructed when the sum of two sides is not greater than the third side.

Answer:

First drop - down: sum to (or "add up to" - but more precisely, in the context, the two sides with lengths 6 and 3 will "sum to" a value equal to the third side length, violating the inequality).
Second drop - down: not be able to form a triangle (or similar phrasing about the construction failure).
Third drop - down: sum of two sides is less than or equal to (in this case equal to) the third side (to complete the statement about the triangle inequality violation).

(Note: Since the exact drop - down options are not fully visible, the above is based on the triangle inequality concept. If the drop - down options were, for example, for the first blank: "sum to", second: "not form a triangle", third: "sum of two sides is equal to", then those would be the answers.)