Question
Carrie is asked to draw a triangle with the following specification.
at least two angles measuring 60°
Which of the following statements about this triangle is true?
A.
One and only one triangle exists with the given condition, and it must be an isosceles triangle.
B.
More than one triangle exists with the given condition, and all instances must be isosceles triangles.
C.
More than one triangle exists with the given condition, and all instances must be equilateral triangles.
D.
One and only one triangle exists with the given condition, and it must be an equilateral triangle.
at least two angles measuring 60°
Which of the following statements about this triangle is true?
A.
One and only one triangle exists with the given condition, and it must be an isosceles triangle.
B.
More than one triangle exists with the given condition, and all instances must be isosceles triangles.
C.
More than one triangle exists with the given condition, and all instances must be equilateral triangles.
D.
One and only one triangle exists with the given condition, and it must be an equilateral triangle.
Answers
GPT-4o mini
To analyze the problem, let's review the properties of the angles in a triangle. The angles in a triangle must sum up to \(180^\circ\). If at least two angles measure \(60^\circ\), we have the following scenarios:
1. **Case 1:** If exactly two angles are \(60^\circ\), let's say angles \(A\) and \(B\) are \(60^\circ\):
- The third angle \(C\) can be calculated as:
\[
C = 180^\circ - A - B = 180^\circ - 60^\circ - 60^\circ = 60^\circ
\]
- This means all three angles are \(60^\circ\), making the triangle equilateral.
2. **Case 2:** If one angle is \(60^\circ\) and the other is unique, it still does not satisfy the condition of "at least two angles measuring \(60^\circ\)" unless the second angle is also \(60^\circ\).
Given that any triangle with two angles measuring \(60^\circ\) must have its third angle also \(60^\circ\), we conclude that:
- The triangle must be equilateral.
Therefore, based on the information provided, only **one** triangle (the equilateral triangle) satisfies the condition of having at least two angles of \(60^\circ\).
From the given options, the correct answer is:
**D. One and only one triangle exists with the given condition, and it must be an equilateral triangle.**
1. **Case 1:** If exactly two angles are \(60^\circ\), let's say angles \(A\) and \(B\) are \(60^\circ\):
- The third angle \(C\) can be calculated as:
\[
C = 180^\circ - A - B = 180^\circ - 60^\circ - 60^\circ = 60^\circ
\]
- This means all three angles are \(60^\circ\), making the triangle equilateral.
2. **Case 2:** If one angle is \(60^\circ\) and the other is unique, it still does not satisfy the condition of "at least two angles measuring \(60^\circ\)" unless the second angle is also \(60^\circ\).
Given that any triangle with two angles measuring \(60^\circ\) must have its third angle also \(60^\circ\), we conclude that:
- The triangle must be equilateral.
Therefore, based on the information provided, only **one** triangle (the equilateral triangle) satisfies the condition of having at least two angles of \(60^\circ\).
From the given options, the correct answer is:
**D. One and only one triangle exists with the given condition, and it must be an equilateral triangle.**