Inequalities and Relationships Within a Triangle

A lot of information can be derived from even the simplest characteristics
of triangles.
In this section, we will learn about the
inequalities
and relationships within a triangle that reveal information
about triangle sides and angles. First, let’s take a look at two significant inequalities
that characterize triangles.

Inequalities of a Triangle

Recall that an inequality is a mathematical expression about the relative size or
order of two objects. In geometry,
we see the use of inequalities when we speak about the length of a triangle’s sides,
or the measure of a triangle’s angles. Let’s begin our study of the inequalities
of a triangle by looking at the Triangle Inequality Theorem.

Triangle Inequality Theorem

The sum of the lengths of two sides of a triangle must always be greater than the
length of the third side.

Let’s take a look at what this theorem means in terms of the triangle we have below.

A triangle with points A, B, and C

The Triangle Inequality Theorem yields three inequalities:

All of our inequalities are not satisfied in the diagram above. The original illustration
shows an open figure as a result of the shortness of segment HG. If we rotate segment
FG to FG’ so that the segment does connect to form a closed figure, we see that
FG’ is too long.

Now, we will look at an inequality that involves exterior angles.

Exterior Angle Inequality Theorem

The measure of an exterior angle of a triangle is greater than the measure of either
of its remote interior angles.

For this theorem, we only have two inequalities since we are just comparing an exterior
angle to the two remote interior angles of a triangle.

Let’s take a look at what this theorem means in terms of the illustration we have
below.

A figure with four points, and three angles highlighted

By the Exterior Angle Inequality Theorem, we have the following two pieces of information:

We will use this theorem again in a proof at the end of this section. Now, let’s
study some angle-side triangle relationships.

Relationships of a Triangle

The placement of a triangle’s sides and angles is very important. We have worked
with triangles extensively, but one important detail we have probably overlooked
is the relationship between a triangle’s sides and angles. These angle-side relationships
characterize all triangles, so it will be important to understand these relationships
in order to enrich our knowledge of triangles.

Angle-Side Relationships

If one side of a triangle is longer than another side, then the angle opposite the
longer side will have a greater degree measure than the angle opposite the shorter
side.

Converse also true: If one angle of a triangle has a greater degree measure than
another angle, then the side opposite the greater angle will be longer than the
side opposite the smaller angle.

In short, we just need to understand that the larger sides of a triangle lie opposite
of larger angles, and that the smaller sides of a triangle lie opposite of smaller
angles. Let’s look at the figures below to organize this concept pictorially.

A figure of a triangle with a line illustrating angle-side relationships

Since segment BC is the longest side, the angle opposite of this side, ?A, is has
the largest measure in ?ABC.

A triangle with a line illustrating that the angle opposite the longest side in a triangle is the largest angle

Our smallest angle, ?C, tells us that segment AB is the smallest side of ?ABC.

Now, we can work on some exercises to utilize our knowledge of the inequalities
and relationships within a triangle.

Exercise 1

In the figure below, what range of length is possible for the third side, x,
to be.

A triangle with a side of unknown length

Answer:

When considering the side lengths of a triangle, we want to use the Triangle Inequality
Theorem
. Recall, that this theorem requires us to compare the length of
one side of the triangle, with the sum of the other two sides. The sum of the two
sides should always be greater than the length of one side in order for the figure
to be a triangle. Let’s write our first inequality.

So, we know that x must be greater than 3. Let’s see
if our next inequality helps us narrow down the possible values of x.

This inequality has shown us that the value of x can be no more than
17. Let’s work out our final inequality.

This final inequality does not help us narrow down our options because we were already
aware of the fact that x had to be greater than 3. Moreover,
side lengths of triangles cannot be negative, so we can disregard this inequality.

Combining our first two inequalities yields

So, using the Triangle Inequality Theorem shows us that x must
have a length between 3 and 17.

Exercise 2

List the angles in order from least to greatest measure.

A triangle to be used in a practice exercise to list angles from smallest to largest

Answer:

For this exercise, we want to use the information we know about angle-side relationships.
Since all side lengths have been given to us, we just need to order them in order
from least to greatest, and then look at the angles opposite those sides.

In order from least to greatest, our sides are PQ, QR,
and RP. This means that the angles opposite those sides will be ordered
from least to greatest. So, in order from least to greatest angle measure, we have
?R, ?P, and then ?Q.

Exercise 3

Which side of the triangle below is the smallest?

A triangle to be used in a practice exercise to determine the smallest side

Answer:

In order to find out which side of the triangle is the smallest, we must first figure
out which angle of the triangle is the smallest (because the smallest side will
be opposite the smallest angle). So, we must use the Triangle Angle Sum Theorem
to figure out the measure of the missing angle.

Since ?V has the smallest measure, we know that the side opposite
this angle has the smallest length. The corresponding side is segment DE,
so DE is the shortest side of ?DEV.

Exercise 4

A large triangle divided into two smaller triangles, with several angles highlighted

Answer:

While it may not immediately be clear that there are two exterior angles given in
the diagram, we must notice them in order to establish a relationship between the
two triangles’ angles. The exterior angle we will focus on is ?JKM.

We have been given that ?KLM and ?KMJ are congruent,
which means that the measure of their angles is equal.

We also know that the measure of ?JKM Is greater than either of the
remote interior angles of ?KLM. Thus, we know that the measure of ?JKM
is greater than the measure of ?KLM.

We have already established equivalence between the measures of ?KLM and
?KMJ, so but substitution, we have that the measure of ?JKM
is greater than the measure of ?KMJ. The two-column geometric
proof
for our argument is shown below.

A two-column geometric proof involving triangle inequalities

Exercise 5 (Challenging)

Two triangles connected at point C to be used in a practice exercise

Answer:

This problem will require us to use several theorems and postulates
we have practiced in the past. Judging by the conclusion we want to arrive at, we
will most likely have to utilize the Triangle Inequality Theorem also.

We begin by noticing that segments AD and BE are parallel.
This fact allows us to say that ?A is congruent to ?E
by the Alternate Interior Angle Theorem (with segment AE as
the transversal touching the set of parallel lines).

We were also given that C is the midpoint of segment AE.
This tells us that AC and CE are equal in length because
midpoints mark the middle of a line segment.

Next, we can say that ?ACD and ?ECB are congruent since
they are vertical angles. In other words, they have the same angle measure.

By the ASA Postulate, we can say that ?ACD??ECB, since we have
two pairs of congruent angles and one pair of congruent sides.

Now, we turn our attention to ?ACD. The Triangle Inequality Theorem,
which states that the sum of the lengths of two sides of a triangle must be greater
than the length of the third side, helps us show that the sum of segments AC
and CD is greater than the length of AD.

We know that CD and CB are equal in length since they
are corresponding parts of congruent triangles, so we can substitute CB
in for CD to arrive at our conclusion. Because there is a lot of information
to follow, we have a new illustration of this problem below that shows congruent
sides and angles.

Two triangles connected at point C, with congruent angles and sides marked

Our two-column geometric proof is shown below. It is easier to follow than the proof
in paragraph form we have already provided.

A two-column geometric proof involving triangle inequalities

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