Question

triangle inequalities
which statement is true regarding triangle tuv?
image of triangle tuv with sides: tu = 5 units, uv = 8 units, tv = 11 units
options:

• angles u and v must be equal.
• angle t is the smallest angle.
• angle v is the smallest angle.
• angles u and t must be equal.

Explanation:

Step1: Recall Triangle Angle-Side Relationship

In a triangle, the larger the side length, the larger the opposite angle, and vice versa. The sides of triangle \( TUV \) are: \( TU = 5 \) units, \( UV = 8 \) units, \( TV = 11 \) units.

Step2: Identify Opposite Angles

• Opposite \( TU \) (5 units) is angle \( V \).
• Opposite \( UV \) (8 units) is angle \( T \).
• Opposite \( TV \) (11 units) is angle \( U \).

Step3: Compare Side Lengths and Angles

Since \( TU = 5 \) is the shortest side, its opposite angle \( V \) should be the smallest angle. Let's check other options:

• Angles \( U \) and \( V \): Opposite sides are 11 and 5 (unequal), so angles not equal.
• Angles \( U \) and \( T \): Opposite sides are 11 and 8 (unequal), so angles not equal.
• Angle \( T \) as smallest: Opposite side 8 is longer than 5, so angle \( T \) is larger than angle \( V \).

Answer:

Angle V is the smallest angle.