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:

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

or

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:

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

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

or 

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 . The fractions convert as follows:

  is the greatest of the three, so for this triangle to be possible it must hold that

or, equivalently,

This is false, so a triangle with these sidelengths cannot 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 . The fractions convert as follows:

 is the greatest of the three, so for this triangle to be possible it must hold that

or, equivalently,

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

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 . The measures of two interior angles can be calculated by subtracting the exterior angle measures from :

The triangle has two interior angles of measure .

By the Converse of the Isosceles Triangle Theorem, since the triangle has two congruent angles, their opposite sides are congruent. Also, since an equilateral triangle has three angles of measure , the triangle is not equilateral. The triangle is isosceles, but not equilateral.

The sum of the measures of the interior angles of a triangle is , so 

Substitute the given two angle measures and solve for :

Subtract  from both sides:

Therefore, 

By the Isosceles Triangle Theorem, if , their opposite sides are also congruent - that is, . Since this is not the case, .

In any triangle, the sum of the lengths of any two sides is greater than the length of the third side; the statement  is a specific example. This is a direct result of the Triangle Inequality Theorem.

We are given that two sides of a triangle, sides  and  of , are congruent to two sides of another triangle, sides  and  of ; we are also given that the included angle of the former, , has greater degree measure than that of the latter, . It is a consequence of the Hinge Theorem (also known as the Side-Angle-Side Inequality Theorem) that the side opposite the former is longer than that opposite the latter - that is, .

In addition to the fact that  and , we also have that , since, by the Reflexive Property of Congruence, any segment is congruent to itself. We can restate this in a more usable form as .

Therefore, two sides of  are congruent to two corresponding sides of , but the included angle of the former has greater measure than that of the latter. It follows from the Hinge Theorem, or Side-Angle-Side Inequality Theorem, that the third side of the former is longer than the third side of the latter - that is, . The statement is true.

The triangle cannot be constructed because the sum of the lengths of the two shorter sides does not exceed the length of the longest side. This violates the conditions of the Triangle Inequality Theorem.