Question

which statement is true regarding triangle tuv?
○ angle t is the smallest angle.
○ angle v is the smallest angle.
○ angles u and v must be equal.
○ angles u and t must be equal.

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the length of a side, the larger the measure of the angle opposite to it, and the smaller the length of a side, the smaller the measure of the angle opposite to it. The side - angle relationship is based on the fact that in a triangle, if \(a,b,c\) are the lengths of the sides opposite to angles \(A,B,C\) respectively, then \(a > b>c\) implies \(A > B > C\).

Step2: Identify the sides and their opposite angles

In triangle \(TUV\):

• Side \(TU = 5\) units, and the angle opposite to \(TU\) is \(\angle V\) (because in triangle \(TUV\), side \(TU\) is between \(T\) and \(U\), so the angle opposite to \(TU\) is \(\angle V\)).
• Side \(UV=8\) units, and the angle opposite to \(UV\) is \(\angle T\) (side \(UV\) is between \(U\) and \(V\), so the angle opposite to \(UV\) is \(\angle T\)).
• Side \(TV = 11\) units, and the angle opposite to \(TV\) is \(\angle U\) (side \(TV\) is between \(T\) and \(V\), so the angle opposite to \(TV\) is \(\angle U\)).

Step3: Compare the side lengths and their opposite angles

We have the side lengths: \(TU = 5\), \(UV = 8\), \(TV=11\). So, \(TUAccording to the side - angle relationship, the angle opposite to the shortest side is the smallest angle. Since \(TU\) is the shortest side (\(TU = 5\)) and the angle opposite to \(TU\) is \(\angle V\), \(\angle V\) is the smallest angle.

Let's analyze the other options:

• Option "Angle \(T\) is the smallest angle": The side opposite to \(\angle T\) is \(UV = 8\), which is not the shortest side, so \(\angle T\) is not the smallest angle.
• Option "Angles \(U\) and \(V\) must be equal": For angles \(U\) and \(V\) to be equal, the sides opposite to them (\(TV\) and \(TU\) respectively) should be equal. But \(TV = 11\) and \(TU=5\) are not equal, so angles \(U\) and \(V\) are not equal.
• Option "Angles \(U\) and \(T\) must be equal": For angles \(U\) and \(T\) to be equal, the sides opposite to them (\(TV\) and \(UV\) respectively) should be equal. But \(TV = 11\) and \(UV = 8\) are not equal, so angles \(U\) and \(T\) are not equal.

Answer:

Angle \(V\) is the smallest angle.