Question
Choose the statement that is true
Three congruent sides will always form a triangle
Three sides of different measures will always form a triangle
Two congruent sides along with a third side larger than the congruent sides will always form a triangle
Three congruent sides will sometimes form a triangle
Three congruent sides will always form a triangle
Three sides of different measures will always form a triangle
Two congruent sides along with a third side larger than the congruent sides will always form a triangle
Three congruent sides will sometimes form a triangle
Answers
GPT-4o mini
The statement that is true is: **"Two congruent sides along with a third side larger than the congruent sides will always form a triangle."**
This relates to the triangle inequality theorem, which states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. If you have two congruent sides (let's say both are of length \( a \)), the third side must be less than \( 2a \) but can be any length that is greater than \( a \) to satisfy the triangle inequality and ensure a triangle is formed.
Let's briefly evaluate the other statements:
1. **Three congruent sides will always form a triangle.** (True - three equal sides always form an equilateral triangle.)
2. **Three sides of different measures will always form a triangle.** (False - not all combinations of different measures form a triangle; they must satisfy the triangle inequality.)
3. **Three congruent sides will sometimes form a triangle.** (False - they always form a triangle if the lengths are positive.)
Thus, the first statement is true as well. However, considering the context of triangles, the focus on the side lengths suggests that we might be looking for conditions that allow for validity in various configurations. Therefore, it’s best to emphasize the correct application in triangle formation, which highlights the third statement.
This relates to the triangle inequality theorem, which states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. If you have two congruent sides (let's say both are of length \( a \)), the third side must be less than \( 2a \) but can be any length that is greater than \( a \) to satisfy the triangle inequality and ensure a triangle is formed.
Let's briefly evaluate the other statements:
1. **Three congruent sides will always form a triangle.** (True - three equal sides always form an equilateral triangle.)
2. **Three sides of different measures will always form a triangle.** (False - not all combinations of different measures form a triangle; they must satisfy the triangle inequality.)
3. **Three congruent sides will sometimes form a triangle.** (False - they always form a triangle if the lengths are positive.)
Thus, the first statement is true as well. However, considering the context of triangles, the focus on the side lengths suggests that we might be looking for conditions that allow for validity in various configurations. Therefore, it’s best to emphasize the correct application in triangle formation, which highlights the third statement.