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 two of the sides dropdown the len dropdown options: never meet, only meet if they lie on the third side, meet at a third vertex , which shows there is no way to construct a triangle in which the dropdown of

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 the given side lengths

Here, the two sides are \(3\) and \(6\), and the third side is \(9\). Let's check the sum of the two smaller sides: \(3+6 = 9\).

Step3: Relate to the figure and options

According to the triangle inequality, the sum of two sides must be greater than the third side (not equal). If the sum of two sides equals the third side, the two sides will lie on the third side (they won't form a triangle with a distinct third vertex, i.e., they never meet to form a triangle in the usual sense). So for the first drop - down (the relationship between the sum of two sides and the third side), it should be "equals" (since \(3 + 6=9\)), and for the second drop - down, the two sides with lengths \(6\) and \(3\) (whose sum is \(9\), equal to the third side) will "never meet" (to form a triangle with a distinct third vertex) or "only meet if they lie on the third side" (since when the sum of two sides equals the third side, they are colinear with the third side). And the last part is about the "sum" of the two sides (since we are checking \(3 + 6\) in relation to \(9\)).

Answer:

First drop - down: equals; Second drop - down: never meet (or only meet if they lie on the third side); Last drop - down: sum