Hey there! I'm happy to help you out!
SOLVING THE TRIANGLE
The sum of all interior angles of a triangle equals 180 degrees.
Since at least two angles have to measure sixty degrees, that means that the other one has to equal sixty as well, because the two sixties we already have equal 120, so the other one has to be 60 degrees!
So, we can see that all three angles are 60 degrees. Since all of the angles are congruent, we can deduce that all of the sides are congruent as well, so this is going to be an equilateral triangle (all sides and angles the same)!
PICKING THE ANSWER
Let's look at each of the answer options and see which one fits.
OPTION A: MORE THAN ONE TRIANGLE EXISTS WITH THE GIVEN CONDITION, AND ALL INSTANCES MUST BE ISOSCELES TRIANGLES.
This answer option is incorrect because we have already figured out that the only angle that will work with the two angles is sixty degrees, so there is only one triangle that follows this condition, which is our equilateral triangle. Our triangle is isosceles (at least two sides are the same), but equilateral is more exact in terms of classification and the first part of the option was incorrect.
OPTION B: MORE THAN ONE TRIANGLE EXISTS WITH THE GIVEN CONDITION, AND ALL INSTANCES MUST BE ISOSCELES TRIANGLES.
This is answer option is incorrect as well because it says that there is more than one triangle that exists with this condition. We have already found that there is only one.
OPTION C: ONE AND ONLY ONE TRIANGLE EXISTS WITH THE GIVEN CONDITION, AND IT MUST BE AN EQUILATERAL TRIANGLE.
This is correct because we have found that our triangle is equilateral and we cannot make any other triangle with at least two sixty degree angle.
THE SOLUTION
Therefore, the correct option is Option C.
I hope that this helps! Have a wonderful day! :D
