Let's Break it Down
Every triangle has three sides and three angles. Of course, not all side lengths and angle measures result in a triangle. You can use the triangle inequality theorem to identify when a triangle does not exist due to its side lengths and you can use the triangle sum theorem to identify when a triangle does not exist due to its angle measurements. Click each tab to learn about each theorem.
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
So, when given a triangle with side lengths a, b, c, as seen here:
\(\text{a}\ +\ \text{b}\ \gt\ \text{c}\)
\(\text{a}\ +\ \text{c}\ \gt\ \text{b}\)
\(\text{b}\ +\ \text{c}\ \gt\ \text{a}\)
Take a look at this example.
Mathalio was tasked in constructing a tortilla chip with side lengths \(0.5\) inch, \(1.0\) inch, and \(3.0\) inches. Upon further review, he realizes that a triangle with these side lengths cannot exist, which means that he cannot construct this tortilla chip.
Explain how Mathalio could have used the triangle inequality theorem to determine that a triangle with side lengths \(0.5\) inch, \(1.0\) inch, and \(3.0\) inch cannot be constructed.
When tasked to construct a chip with side lengths \(0.5\) inch, \(1.0\) inch, and \(3.0\) inch, Mathalio realized that the lengths of the sides do not satisfy the triangle inequality theorem. The sum of the lengths of two sides must be greater than the length of the third side.
\(1.0\ \text{in}. +\ 3.0\ \text{in}. =\ 4.0\ \text{in}.\), which is greater than \(0.5\ \text{in}\).
\(3.0\ \text{in}. +\ 0.5\ \text{in}. =\ 3.5\ \text{in}.\), which is greater than 1.0 in.
But
\(0.5\ \text{in}. +\ 1.0\ \text{in}. = 2.5\ \text{in}.\), which is not greater than \(3.0\ \text{in}\).
This means that a triangle with side lengths \(0.5\) inch, \(1.0\) inch, and \(3.0\) inch, cannot exist, which implies that it cannot be constructed.
The triangle sum theorem states the sum of all three interior angles in a triangle must equal \(180^\circ\).
So, when given a triangle with angle measurements \(x, y, z,\) as seen here:
\(x\ +\ y\ +\ z\ =\ 180^\circ\)
Take a look at this example.
Mathalio complained that the last chip design was impossible to make, so his boss gave him three angle measurements instead. They were \(48^\circ,\ 79^\circ,\) and \(60^\circ\).
Explain how Mathalio can use the triangle sum theorem to determine that a triangle with these given angle measurements cannot be constructed.
The angle measurements \(48^\circ\ +\ 79^\circ\ +\ 60^\circ\ =\ 187^\circ\), which is not \(180^\circ\).
This means that a triangle with interior angle measurements \(48^\circ,\ 79^\circ,\ \text{and}\ 60^\circ\) cannot exist, which implies it cannot be constructed.
Let's Practice
Now, use your knowledge of the triangle inequality theorem and the triangle sum theorem to complete the activity below. Read each description and determine whether the given triangle can exist. Then, click the description to check your answer.
