How can you determine when a triangle does not exist due to side lengths or angle measurements?
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’s boss is constructing a tortilla chip with side lengths \(0.5\) inch, \(1.0\) inch, and \(3.0\) inches. Mathalio pointed out to his boss that he did not agree with the measurements of the chips. Mathalio argued the chips would not work as the length of \(0.5\) inch and \(1.0\) inch do not total more than \(3.0\) inches. Mathlio’s boss argued that the chips will work because the sum of \(1.0\) inch and \(3.0\) inches add to more than \(0.5\) inches.
Determine who is correct and why.
Mathlio’s argument is correct. 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.
No.
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.
\(4.1\ \text{ft}\ +\ 2.7\ \text{ft}\ =\ 6.8\
\text{ft}\), which is greater than
\(1.3\ \text{ft}\).
\(4.1\ \text{ft}\ +\ 1.3\ \text{ft}\ =\ 5.4\
\text{ft}\), which is greater than
\(2.7\ \text{ft}\).
But
\(2.7\ \text{ft}\ +\ 1.3\ \text{ft}\ =\ 4.0\
\text{ft}\), which is NOT greater than
\(4.1\ \text{ft}\).
Yes.
The triangle sum theorem is satisfied because the sum
of the angles equals
\(180^\circ\).
The lengths of the sides satisfy the triangle
inequality theorem. The sum of the lengths of two
sides must be greater than the length of third side.
\(45\ \text{m}\ +\ 30\ \text{m}\ =\ 75\
\text{m}\), which is greater than
\(16\ \text{m}\).
\(45\ \text{m}\ +\ 16\ \text{m}\ =\ 61\
\text{m}\), which is greater than
\(30\ \text{m}\).
\(30\ \text{m}\ +\ 16\ \text{m}\ =\ 46\
\text{m}\), which is greater than
\(45\ \text{m}\).
Troy is correct.
The lengths of the sides of the garden posts 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.
\(10\ \text{feet}\ +\ 16\ \text{feet}\ =\ 6\ \text{feet}\), which is greater than \(2\ \text{feet}\) \(10\ \text{feet}\ +\ 2\ \text{feet}\ =\ 12\ \text{feet}\), which is greater than \(6\ \text{feet}\)
But
\(2\ \text{feet}\ +\ 6\ \text{feet}\ =\ 8\ \text{feet}\), which is NOT greater than \(10\ \text{feet}\)
Gigi is correct that \(48^\circ,\ 50^\circ,\ \text{and}\ 82^\circ\) can be the angles of a triangle. But her reasoning is incorrect. The angles can be a triangle because the triangle sum theorem is satisfied because the sum of the angles of the triangle equals \(180^\circ\).
\(48^\circ\ + 50^\circ\ + 82^\circ\ = 180^\circ\)
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