Use the pythagorean theorem:

\(\displaystyle a^2+b^2=c^2\)

\(\displaystyle c^2-b^2=a^2\)

\(\displaystyle 12^2-9^2=x^2\)

\(\displaystyle 144-81=63=x^2\)

\(\displaystyle \sqrt{63}=x\)

The triangle has two sides of the same length, 18.4, so, by definition, it is isosceles.

Convert each of the three measures to the same unit; we will choose the smallest unit, millimeters.

One meter is equivalent to 1,000 millimeters, so 12 meters can be converted to millimeters by multiplying by 1,000:

\(\displaystyle 12\textrm{ m }\times 1,000 \textrm{ mm / m}= 12,000 \textrm{ mm}\)

One centimeter is equivalent to ten millimeters, so 1,200 cenitmeters can be converted to millimeters by multiplying by 10:

\(\displaystyle 1,200\textrm{ cm }\times 10 \textrm{ mm / cm}= 12,000 \textrm{ mm}\)

These two sides have the same length. However, the third side, which has length 12 millimeters, is of different length. Since the triangle has exactly two congruent sides, it is by definition isosceles, but not equilateral.

Convert each of the three measures to the same unit; we will choose the smallest unit, inches.

One foot is equal to twelve inches, so \(\displaystyle 1\frac{1}{2}\) feet can be converted to inches by multiplying by 12:

\(\displaystyle 1\frac{1}{2} \textrm{ ft} \times 12 \textrm{ in / ft } = 18 \textrm{ in}\)

One yard is equal to 36 inches.

The lengths of the sides in inches are 18, 24, and 36. Since no two sides have the same measure, the triangle is by definition scalene.

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 \(\displaystyle t\):

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

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

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

By the Isosceles Triangle Theorem, if two sides of a triangle have the same length, their opposite angles have the same measure. Since no two angles have the same measure, no two sides have the same length. This makes the triangle scalene.

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 

\(\displaystyle t + 25^{\circ }+ 130^{\circ } = 180 ^{\circ }\)

Solve for \(\displaystyle t\):

\(\displaystyle t + 155^{\circ } = 180 ^{\circ }\)

\(\displaystyle t + 155^{\circ } - 155^{\circ } = 180 ^{\circ } - 155^{\circ }\)

\(\displaystyle t = 25^{\circ }\)

The triangle has two congruent angles - each with measure \(\displaystyle 25^{\circ }\). As a consequence, by the Converse of the Isosceles Triangle Theorem, the triangle has two congruent sides, making it, by definition, isosceles.

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 

\(\displaystyle t + 40^{\circ }+ 40^{\circ } = 180 ^{\circ }\)

Solve for \(\displaystyle t\):

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

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

\(\displaystyle t = 100^{\circ }\)

The triangle therefore has an angle with measure greater than \(\displaystyle 90^{\circ }\) - an obtuse angle. The triangle is by definition an obtuse triangle,

First, convert all of the given sidelengths to the same unit; here, we choose the smallest unit, millimeters.

One meter is equal to 1,000 millimeters.

One centimeter is equal to 10 millimeters, so convert 250 centimeters to millimeters by multiplying by 10:

\(\displaystyle 250 \textrm{ cm } \times 10 \textrm{mm / cm} = 2,500 \textrm{ mm}\)

The measures of the sides of the triangle, in millimeters, are 1,000, 1,200, and 2,500.

By the Triangle Inequality, the sum of the measures of the shortest two sides of a triangle must exceed the length of the longest side, so for this triangle to be possible it must hold that

\(\displaystyle 1,000 + 1,200 > 2,500\)

or

\(\displaystyle 2,200 > 2,500\)

This is false, so a triangle with these sidelengths cannot exist.

First, convert all of the given sidelengths to the same unit; here, we choose the smallest unit, inches.

One foot is equal to 12 inches, so to convert feet to inches, multiply by 12:

\(\displaystyle 5 \textrm{ ft} \times 12 \textrm{ in / ft} = 60\textrm{ in}\)

One yard is equal to 36 inches, so to convert yards to inches, multiply by 36:

\(\displaystyle 1\frac{1}{2}\textrm{ yd} \times 36 \textup{in / yd}= 54 \textrm{ in}\)

The measures of the sides of the triangle, in inches, are 50, 54, and 60. 

By the Triangle Inequality Theorem, the sum of the measures of the shortest two sides of a triangle must exceed the length of the longest side, so for this triangle to be possible it must hold that

\(\displaystyle 50+ 54 > 60\)

or 

\(\displaystyle 104 > 60\)

This is indeed the case, so a triangle with these sidelengths can exist.

By the Triangle Inequality Theorem, the sum of the measures of the shortest two sides of a triangle must exceed the length of the longest side. 

Write each length in terms of a common denominator; this is \(\displaystyle LCM (2, 4, 8) = 8\). The fractions convert as follows:

\(\displaystyle \frac{1}{2} = \frac{1 \times 4 }{2 \times 4} = \frac{4}{8}\)

\(\displaystyle \frac{1}{4} = \frac{1 \times 2 }{4 \times 2} = \frac{2}{8}\)

\(\displaystyle \frac{1}{2}\)  is the greatest of the three, so for this triangle to be possible it must hold that

\(\displaystyle \frac{1}{8} + \frac{1}{4} > \frac{1}{2}\)

or, equivalently,

\(\displaystyle \frac{1}{8} +\frac{2}{8}> \frac{4}{8}\)

\(\displaystyle \frac{3}{8}> \frac{4}{8}\)

This is false, so a triangle with these sidelengths cannot exist.