### Video Transcript

In this video, weβll learn how to
define the concept of inequality in general and the axioms of inequality and compare
between the measures of angles given in a triangle. Before we can begin discussing
inequalities in triangles though, we need to first recall what inequalities
represent and how we can manipulate them.

First, remember that the symbol
representing less than is a chevron open to the right. For example, using this, we can say
that three is less than five since three is of smaller value than five. A chevron open to the left
signifies greater than. So in this case, for example, we
can use this to say 10 is greater than five, since 10 is a larger value than
five.

We can use inequalities to
represent a comparison between any two numbers, which is particularly useful if the
numbers represent a quantity such as distance. We can see clearly that the line
segment πΆπ· is longer than the segment π΄π΅. And itβs much more succinct to
write this as an inequality than in words. This extends to any quantities
represented by numbers. For example, if we consider the two
angles shown, the one on the right is clearly larger than the one on the left.

And recalling that we describe the
measure of an angle by a number, not the angle itself, we can write this as the
measure of angle πΈπ·πΉ is greater than the measure of angle π΄π΅πΆ.

We can also manipulate inequalities
to find equivalent inequalities which can be useful in solving algebraic
expressions. For example, to find all the values
of π₯ that satisfy the inequality π₯ plus one is less than two, we subtract one from
both sides to find the solution set π₯ is less than one so that any value of π₯ less
than one satisfies the given inequality.

This can be useful in comparing
lengths, angle measures, or any other quantities. For example, considering the angles
shown, we see that the sum of the measures of angles π·π΅πΆ and πΆπ΅π΄ is equal to
the measure of angle π·π΅π΄ but that the two summed angles make up less than the
measure of a straight angle. That is, their sum is less than 180
degrees. And we can write this as shown.

To manipulate inequalities, we
apply the same operation to both sides using rules called the axioms of
inequalities. And these apply for any real
numbers π, π, π, and π.

The first axiom states that any two
real numbers, say, π and π, can be compared such that either π is less than π,
π is greater than π, or π is equal to π.

Next, we have the axiom of
transitivity. Inequalities and equalities are
transitive relations on the set of real numbers, which means that if π is greater
than π and π is greater than π, then π is greater than π.

Adding or subtracting a real
constant to or from both sides of an inequality gives an equivalent inequality. So, for example, if π is greater
than π, then π plus π is greater than π plus π and π minus π is greater than
π minus π.

Next, multiplying or dividing both
sides of an inequality by a positive constant gives an equivalent inequality. So, for example, if π is greater
than π and π is a positive constant, then ππ is greater than ππ and π over π
is greater than π over π.

And finally, we can combine the
ideas of transitivity with the other axioms so that, for example, if π is greater
than π and π is greater than π, then π plus π is greater than π plus π.

These axioms apply to any
inequality. So letβs see how we can apply them
to the interior angles of a triangle.

Show that the nonright angles in
triangle π΄π΅πΆ are acute angles.

We can begin by recalling that the
sum of the measures of the interior angles of a triangle is 180 degrees. Since angle πΆπ΄π΅ is a right
angle, its measure is 90 degrees. And if we then subtract 90 degrees
from both sides, we see that the sum of the measures of angles π΄π΅πΆ and π΄πΆπ΅
equals 90 degrees. We know we can always compare the
sizes of real numbers and also that the measures of both angles are positive. So neither angle can have a measure
greater than or equal to 90 degrees. Hence, both angles must have
measures less than 90 degrees.

Letβs see some examples of
comparing geometric quantities using the axioms of inequalities.

Look at the figure. Use the symbols for less than,
equals, or greater than to fill in the blanks in the following. The measure of angle π΄πΆπ·
what the measure of angle π΄π·πΈ. The measure of angle π΄πΆπ·
what the measure of angle π΄π΅πΆ. The measure of angle π΄π·πΆ
what the measure of angle π΄πΆπ΅. And the measure of angle π΄π·πΈ
what the measure of angle πΆπ΄π·.

To compare the given measures,
we begin by recalling that the measures of the internal angles of a triangle sum
to 180 degrees and that the measures of the angles that make up a straight line
also sum to 180 degrees. We can use the first of these
facts to find the measure of angle π΄πΆπ΅ in triangle π΄π΅πΆ by noting that its
sum with 17 and 24 degrees must equal 180 degrees.

Now, subtracting 17 and 24
degrees from both sides, we have the measure of angle π΄πΆπ΅ is equal to 139
degrees, and making a note of this. Now, the measure of angle
π΄πΆπ· combines with the measure of angle π΄πΆπ΅ to make a straight angle. And using our second fact,
their sum must be 180 degrees. Substituting 139 degrees for
the measure of angle π΄πΆπ΅ and then subtracting this from both sides, we have
the measure of angle π΄πΆπ· equals 180 degrees minus 139 degrees, which is 41
degrees.

Now that we have two of the
internal angles of triangle π΄πΆπ·, we can find the other again using our first
fact. The measures of angles π΄π·πΆ,
π΄πΆπ·, and πΆπ΄π· must sum to 180 degrees. Hence, we have that the measure
of angle π΄π·πΆ equals 180 degrees minus 41 degrees minus 71 degrees. And thatβs 68 degrees. Finally, we see that the
measures of angles π΄π·πΆ and π΄π·πΈ combine to make a straight angle, so their
measures sum to 180 degrees. With the measure of angle
π΄π·πΆ equal to 68 degrees, we find that the measure of angle π΄π·πΈ equals 112
degrees.

Now we have all the information
we need to fill in the blanks. So making some space, we have
the measure of angle π΄πΆπ· equals 41 degrees and the measure of angle π΄π·πΈ
equals 112 degrees. Hence, the measure of angle
π΄πΆπ· is less than the measure of angle π΄π·πΈ. And we can fill in our first
blank with the symbol for less than.

Next, again, the measure of
angle π΄πΆπ· equals 41 degrees, and that of angle π΄π΅πΆ equals 17 degrees. Hence, the measure of angle
π΄πΆπ· is greater than the measure of angle π΄π΅πΆ. Third, we have the measures of
angles π΄π·πΆ and π΄πΆπ΅ are 68 and 139 degrees, respectively. So our third symbol is less
than, since 68 is less than 139. And finally, the measure of
angle π΄π·πΈ at 112 degrees is greater than the measure of angle πΆπ΄π·, which
is 71 degrees.

From the figure then, we find that
the symbols required to fill in the blanks are less than, greater than, less than,
and greater than.

Using the same ideas as in this
example, we can demonstrate an interesting inequality that holds for triangles in
general. We know that the sum of the
measures of the interior angles of a triangle is 180 degrees. Similarly, we know that the
exterior and interior angles at π΅ make up a straight angle. So the sum of their measures is
also 180 degrees. And since the left-hand side of
both of these equations equals 180 degrees, we can equate them. Thus, π plus π plus π is equal
to π plus the measure of angle πΆπ΅π·. And subtracting π from both sides
gives π plus π equals the measure of angle πΆπ΅π·.

Now, since the measures of all
three of these angles are greater than zero, we can subtract either π or π to make
the left-hand side of our equation smaller. Hence, we have both π and π are
less than the measure of angle πΆπ΅π·. Stated more formally, this means
that the measure of any exterior angle of a triangle π΄π΅πΆ is greater than the
measure of either of the nonadjacent interior angles in the triangle.

Letβs see how we might apply this
result in an example.

In the given figure, which
angles must have smaller measures than the measure of angle one?

In this example, we need to
identify angles in the diagram whose measure is less than that of angle number
one. But weβre not given the
measures of any of the angles. So to solve this, weβll need to
use the diagram and the relationships between angles in triangles.

In particular, we recall that
the measure of any exterior angle of a triangle π΄π΅πΆ, for example, πΆπ΅π· in
our diagram, is greater than the measure of either of the nonadjacent interior
angles of the triangle. If we label all the vertices in
the given diagram as shown, we see that angle one is an exterior angle in the
triangle π΄πΆπ·. Hence, its measure is greater
than those of the two nonadjacent interior angles in this triangle. And these are angles four and
five at π΄ and πΆ, respectively.

Now we need to pay careful
attention to the wording of the question at this point. Note that the question asks
which angles must have smaller measures than that of angle one. So far, weβve demonstrated that
this is true of angles four and five. And we can show via an example
that the remaining angles may not necessarily in certain circumstances have a
measure smaller than that of angle one.

Suppose our triangle π΄πΆπ· is
an isosceles triangle and that the measure of angle one is 60 degrees. We can add any line we like
from π· to a point π΅ opposite π· to construct the missing angles. But since our triangle is
isosceles, we can use the fact that the angle bisector and perpendicular
bisector of the base are the same line. We see then that the measures
of angles six and seven in this scenario are both 90 degrees, which is greater
than 60 degrees. And thatβs the measure of angle
one. So these two angles can have
measures that are not smaller than that of angle one.

Next, we know that the line
π·π΅ bisects the angle at π· so that angles two and three have the same
measure. We see also that angles one,
two, and three combine to make a straight angle so that the sum of their
measures is 180 degrees.

Now, substituting 60 degrees
for the measure of angle one and recalling that angles two and three have the
same measure, we can solve this to find either of the measures of angle two or
three. Choosing angle two, we have its
measure equals 60 degrees. And of course, we know that
this is also the measure of angle three. So both of these are equal to
the measure of angle one. And weβve therefore shown that
angles two, three, six, and seven need not have smaller measures than angle
one. Hence, only angles four and
five must have measures smaller than that of angle one.

Letβs look at another example.

Which of the following
inequalities is correct? (A) The measure of angle π΄π΅πΆ
is less than the measure of angle π΅π΄πΆ is less than the measure of angle
π΄πΆπ΅. Or (B) the measure of angle
π΄π΅πΆ is greater than that of angle π΅π΄πΆ which is greater than the measure of
angle π΄πΆπ΅. (C) The measure of angle π΅π΄πΆ
is greater than that of π΄π΅πΆ which is greater than that of π΄πΆπ΅. Or option (D), the measure of
angle π΅π΄πΆ is less than the measure of angle π΄π΅πΆ is less than that of
π΄πΆπ΅. Or finally, the measure of
angle π΄πΆπ΅ is less than the measure of angle π΄π΅πΆ which is less than the
measure of angle π΅π΄πΆ.

We note that each of the five
options involves the measures of the three angles π΄π΅πΆ, π΅π΄πΆ, and π΄πΆπ΅ and
that weβre given the measure of one of these. Thatβs angle π΄πΆπ΅, which is
29 degrees. And so to answer the question,
we need to find the measures of the two interior angles π΅π΄πΆ and π΄π΅πΆ.

We can find the measure of
angle π΅π΄πΆ by noting that the measures of the angles that make a straight line
sum to 180 degrees. And so we have 109 degrees plus
the measure of angle π΅π΄πΆ equals 180 degrees. Subtracting 109 degrees from
both sides gives us the measure of angle π΅π΄πΆ equals 71 degrees. Next, we can find the measure
of angle π΄π΅πΆ by recalling that the measure of the interior angles of a
triangle sum to 180 degrees. Thus, we have 71 degrees plus
the measure of angle π΄π΅πΆ plus 29 degrees equals 180 degrees.

Now, subtracting 71 and 29
degrees from both sides, we have the measure of angle π΄π΅πΆ equals 80
degrees. Since 80 is greater than 71,
which in turn is greater than 29, we see that the measure of angle π΄π΅πΆ is
greater than the measure of angle π΅π΄πΆ which is greater than the measure of
angle π΄πΆπ΅. And this corresponds to option
(B), so option (B) is correct.

In our final example, weβll use the
properties of isosceles triangles to compare the measures of two angles.

Use the less than, is equal to,
or greater than symbol to complete the statement. If the measure of angle π΄πΆπ΅
equals 62 degrees and the measure of angle π΄ equals 57 degrees, then the
measure of angle π΄π΅π· what the measure of angle π΄πΆπ·.

We notice first that πΆπ· is
equal to π΅π·, so the triangle π΅πΆπ· is an isosceles triangle. Then recalling that the angles
opposite the congruent sides of an isosceles triangle are congruent, we see that
the measures of angles π΅πΆπ· and πΆπ΅π· are equal.

Now, weβre told that the
measure of angle π΄πΆπ΅ is 62 degrees and that the measure of the angle at π΄ is
57 degrees. And recalling that the measures
of the interior angles of a triangle sum to 180 degrees, we have that the
measure of the angle at π΄ and those of angles π΄π΅πΆ and π΄πΆπ΅ sum to 180
degrees.

Substituting the given values
into our equation, we have 57 degrees plus the measure of angle π΄π΅πΆ plus 62
degrees equals 180 degrees. Now, subtracting 57 and 62
degrees from both sides, we find that the measure of angle π΄π΅πΆ equals 61
degrees. We can mark this on our diagram
as shown.

Now, if we call the measure of
our two congruent angles π₯. And making some space, we have
π₯ plus the measure of angle π΄πΆπ· equals 62 degrees. And π₯ plus the measure of
angle π΄π΅π· equals 61 degrees. Now, if we add one to both
sides of the second equation, we have that π₯ plus the measure of angle π΄π΅π·
plus one is equal to 62 degrees.

And now since both of our
right-hand sides equal 62, we can equate our left-hand sides to give π₯ plus the
measure of angle π΄πΆπ· equals π₯ plus the measure of angle π΄π΅π· plus one. Subtracting π₯ from both sides,
we then have the measure of angle π΄πΆπ· equals the measure of angle π΄π΅π· plus
one. This means that the measure of
angle π΄π΅π· is one degree smaller than that of angle π΄πΆπ·. Hence, the measure of angle
π΄π΅π· is less than the measure of angle π΄πΆπ·. And the less than symbol
completes the given statement.

Letβs finish by recapping some of
the important points weβve covered. The axioms of inequalities for real
numbers π, π, π, and π are such that, first, either π is greater than π or π
is less than π or π equals π. Next, the transitivity axiom, if π
is greater than π and π is greater than π, then π is greater than π. Third, if π is greater than π,
then π plus π is greater than π plus π and π minus π is greater than π minus
π. If π is greater than π and π is
greater than zero, then π multiplied by π is greater than π multiplied by π and
π divided by π is greater than π divided by π.

The transitivity axiom can be
combined with the others. So, for example, if π is greater
than π and π is greater than π, then π plus π is greater than π plus π. And finally, the measure of any
exterior angle of a triangle is greater than the measure of either nonadjacent
interior angle of the triangle.