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 sides of triangle \(TUV\) are: \(TU = 5\) units, \(UV=8\) units, and \(TV = 11\) units.

Step2: Identify the angles opposite each side

• The side opposite angle \(V\) is \(TU\) (length \(5\) units).
• The side opposite angle \(T\) is \(UV\) (length \(8\) units).
• The side opposite angle \(U\) is \(TV\) (length \(11\) units).

Step3: Compare the side lengths and their opposite angles

Since \(5<8 < 11\), the side \(TU\) (length \(5\)) is the shortest side. The angle opposite to \(TU\) is angle \(V\). So, angle \(V\) is the smallest angle.

Let's analyze the other options:

• Option 1: Angle \(T\) is opposite to side \(UV = 8\) units. Since \(8>5\), angle \(T\) is not the smallest angle.
• Option 3: Angles \(U\) and \(V\) are not necessarily equal. For angles \(U\) and \(V\) to be equal, the sides opposite to them (\(TV\) and \(TU\)) should be equal, but \(TV = 11\) and \(TU=5\), so they are not equal.
• Option 4: Angles \(U\) and \(T\) are not necessarily equal. For angles \(U\) and \(T\) to be equal, the sides opposite to them (\(TV\) and \(UV\)) should be equal, but \(TV = 11\) and \(UV = 8\), so they are not equal.

Answer:

Angle V is the smallest angle.