The sum of the three interior angles in any triangle must equal \(\displaystyle 180\) degrees. Since this triangle has two equivalent interior angles, it must be an isosceles triangle. Additionally, the sum of the two equivalent interior angles is: \(\displaystyle 45+45=90\).

Since, \(\displaystyle 180-90=90\) degrees, the third interior angle must be a right angle--which makes this triangle an isosceles right triangle. 

A scalene triangle doesn't have any equivalent sides or equivalent interior angles. Since this triangle has one obtuse interior angle, it must be an obtuse scalene triangle. 

Acute scalene triangles must have three different acute interior angles--which always have a sum of \(\displaystyle 180\) degrees. 

Thus, the solution is:

\(\displaystyle 78+47=125\)

\(\displaystyle 180-125=55\)

\(\displaystyle 78+47+55=180\)

An obtuse isosceles triangle has one obtuse interior angle and two equivalent acute interior angles. Since the sum total of the interior angles of every triangle must equal \(\displaystyle 180\) degrees, the solution is:

\(\displaystyle 108+36=144\)

\(\displaystyle 180-144=36\)

Therefore, this triangle has one interior angle of \(\displaystyle 144\) degrees and two equivalent interior angles of \(\displaystyle 36\) degrees--making this an obtuse isosceles triangle. 

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

\(\displaystyle 18.4^{2} + 18.4 ^{2} = 338.56 + 338.56 = 677.12\)

Calculate the square of the length of the longest side:

\(\displaystyle 26 .0^{2} = 676\)

The former is greater than the latter. This indicates that the triangle is acute.

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

\(\displaystyle 19.5 ^{2}+ 46.8^{2} = 380.25+2,190.24 =2,570.49\)

Calculate the square of the length of the longest side:

\(\displaystyle 50.7^{2} =2,570.49\)

The two quantities are equal, so by the Converse of the Pythagorean Theorem, the triangle is right.

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

\(\displaystyle 9^{2} + 12^{2} = 81 + 144 = 225\)

Calculate the square of the length of the longest side:

\(\displaystyle 18^{2} = 324\)

The former is less than the latter. This indicates that the triangle is obtuse.

Given the lengths of its three sides, a triangle can be identified as acute, right, or obtuse by the following process:

Calculate the sum of the squares of the lengths of the two shortest sides:

\(\displaystyle 14^{2}+ 18 ^{2} = 196 + 324 = 520\)

Calculate the square of the length of the longest side:

\(\displaystyle 2 0 ^{2} = 400\)

The former is greater than the latter. This indicates that the triangle is acute.

The measures of the interior angles of a triangle add up to \(\displaystyle 180^{\circ }\). If \(\displaystyle t\) is the measure of the third angle, then 

\(\displaystyle t + 45 ^{\circ }+ 80^{\circ } = 180 ^{\circ }\)

Solve for \(\displaystyle t\):

\(\displaystyle t + 125^{\circ } = 180 ^{\circ }\)

\(\displaystyle t + 125^{\circ } - 125^{\circ } = 180 ^{\circ } - 125^{\circ }\)

\(\displaystyle t = 55^{\circ }\)

Each of the three angles has measure less than \(\displaystyle 90^{\circ }\), so each angle is, by definition, acute. This makes the triangle acute.

At a given vertex, an exterior angle and an interior angle of a triangle form a linear pair, making them supplementary - that is, their measures total \(\displaystyle 180^{\circ }\). The measures of two interior angles can be calculated by subtracting the exterior angle measures from \(\displaystyle 180^{\circ }\):

\(\displaystyle 180^{\circ } - 100 ^{\circ } = 80 ^{\circ }\)

The triangle has two interior angles of measure \(\displaystyle 80 ^{\circ }\).

The measures of the interior angles of a triangle add up to \(\displaystyle 180^{\circ }\). If \(\displaystyle t\) is the measure of the third angle, then 

\(\displaystyle t + 80^{\circ } + 80^{\circ } = 180^{\circ }\)

\(\displaystyle t +160^{\circ } = 180^{\circ }\)

\(\displaystyle t +160^{\circ }- 160 ^{\circ } = 180^{\circ } - 160 ^{\circ }\)

\(\displaystyle t = 20 ^{\circ }\)

All three interior angles measure less than \(\displaystyle 90^{\circ }\), making them acute. The triangle is, by definition, acute.