How many unique triangles can be made where one angle measures 60 degrees and another angle is an obtuse angle?

Answer 1

Infinitely many can be made.

With one angle 60°, that leaves 180° - 60° = 120° as the sum of the other two angles.

An obtuse angle is one greater than 90° and less than 180°.

Thus as long as one of the other angles is greater than 90° and less than 120°, the other will be 120° less it.

Some possible other angles:

100° and 20°

100.5° and 19.5°

100.05° and 19.95°

100.025° and 19.975°

There are infinitely many numbers between 100 and 101, therefore there are infinitely many pairs of number whose sum is 120 where one of them lies between 100 and 101 (inclusive, or exclusive - it matter not which).

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If you meant triangles with angles which are WHOLE numbers, then there are:

Smallest obtuse angle = 91°

Largest obtuse angle = 119°

→ there are 119 - 91 + 1 = 29 such triangles.

Answer 2

Infinitely many. The obtuse angle can have any value in the open interval (90, 120) degrees and since the angular measure is a continuous variable, there uncountable infinite number of possible values for the obtuse angle. Furthermore, for value of the obtuse angle, there are infinitely many possible lengths for the side between the 60 degree angle and the obtuse angle and so infinitely many similar triangles.

Answer 3

the obtuse angle cannot be greater than 120 degrees but there is an infinite number of unique triangles that can be made in the range from 90 to 120 degrees