Lesson Video: Introduction to Inequalities in Triangles Mathematics

Video Transcript

In this video, we’ll 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 𝐶𝐷 is longer than the segment 𝐴𝐵. And it’s 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 𝐸𝐷𝐹 is greater than the measure of angle 𝐴𝐵𝐶.

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 𝑥 that satisfy the inequality 𝑥 plus one is less than two, we subtract one from both sides to find the solution set 𝑥 is less than one so that any value of 𝑥 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 𝐷𝐵𝐶 and 𝐶𝐵𝐴 is equal to the measure of angle 𝐷𝐵𝐴 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 𝑎, 𝑏, 𝑐, and 𝑑.

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

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

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

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

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

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

Show that the nonright angles in triangle 𝐴𝐵𝐶 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 𝐶𝐴𝐵 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 𝐴𝐵𝐶 and 𝐴𝐶𝐵 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.

Let’s 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 𝐴𝐶𝐷 what the measure of angle 𝐴𝐷𝐸. The measure of angle 𝐴𝐶𝐷 what the measure of angle 𝐴𝐵𝐶. The measure of angle 𝐴𝐷𝐶 what the measure of angle 𝐴𝐶𝐵. And the measure of angle 𝐴𝐷𝐸 what the measure of angle 𝐶𝐴𝐷.

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 𝐴𝐶𝐵 in triangle 𝐴𝐵𝐶 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 𝐴𝐶𝐵 is equal to 139 degrees, and making a note of this. Now, the measure of angle 𝐴𝐶𝐷 combines with the measure of angle 𝐴𝐶𝐵 to make a straight angle. And using our second fact, their sum must be 180 degrees. Substituting 139 degrees for the measure of angle 𝐴𝐶𝐵 and then subtracting this from both sides, we have the measure of angle 𝐴𝐶𝐷 equals 180 degrees minus 139 degrees, which is 41 degrees.

Now that we have two of the internal angles of triangle 𝐴𝐶𝐷, we can find the other again using our first fact. The measures of angles 𝐴𝐷𝐶, 𝐴𝐶𝐷, and 𝐶𝐴𝐷 must sum to 180 degrees. Hence, we have that the measure of angle 𝐴𝐷𝐶 equals 180 degrees minus 41 degrees minus 71 degrees. And that’s 68 degrees. Finally, we see that the measures of angles 𝐴𝐷𝐶 and 𝐴𝐷𝐸 combine to make a straight angle, so their measures sum to 180 degrees. With the measure of angle 𝐴𝐷𝐶 equal to 68 degrees, we find that the measure of angle 𝐴𝐷𝐸 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 𝐴𝐶𝐷 equals 41 degrees and the measure of angle 𝐴𝐷𝐸 equals 112 degrees. Hence, the measure of angle 𝐴𝐶𝐷 is less than the measure of angle 𝐴𝐷𝐸. And we can fill in our first blank with the symbol for less than.

Next, again, the measure of angle 𝐴𝐶𝐷 equals 41 degrees, and that of angle 𝐴𝐵𝐶 equals 17 degrees. Hence, the measure of angle 𝐴𝐶𝐷 is greater than the measure of angle 𝐴𝐵𝐶. Third, we have the measures of angles 𝐴𝐷𝐶 and 𝐴𝐶𝐵 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 𝐴𝐷𝐸 at 112 degrees is greater than the measure of angle 𝐶𝐴𝐷, 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 𝐵 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, 𝑎 plus 𝑏 plus 𝑐 is equal to 𝑏 plus the measure of angle 𝐶𝐵𝐷. And subtracting 𝑏 from both sides gives 𝑎 plus 𝑐 equals the measure of angle 𝐶𝐵𝐷.

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

Let’s 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 we’re not given the measures of any of the angles. So to solve this, we’ll 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 𝐴𝐵𝐶, for example, 𝐶𝐵𝐷 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 𝐴𝐶𝐷. Hence, its measure is greater than those of the two nonadjacent interior angles in this triangle. And these are angles four and five at 𝐴 and 𝐶, 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, we’ve 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 𝐴𝐶𝐷 is an isosceles triangle and that the measure of angle one is 60 degrees. We can add any line we like from 𝐷 to a point 𝐵 opposite 𝐷 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 that’s 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 𝐷𝐵 bisects the angle at 𝐷 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 we’ve 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.

Let’s look at another example.

Which of the following inequalities is correct? (A) The measure of angle 𝐴𝐵𝐶 is less than the measure of angle 𝐵𝐴𝐶 is less than the measure of angle 𝐴𝐶𝐵. Or (B) the measure of angle 𝐴𝐵𝐶 is greater than that of angle 𝐵𝐴𝐶 which is greater than the measure of angle 𝐴𝐶𝐵. (C) The measure of angle 𝐵𝐴𝐶 is greater than that of 𝐴𝐵𝐶 which is greater than that of 𝐴𝐶𝐵. Or option (D), the measure of angle 𝐵𝐴𝐶 is less than the measure of angle 𝐴𝐵𝐶 is less than that of 𝐴𝐶𝐵. Or finally, the measure of angle 𝐴𝐶𝐵 is less than the measure of angle 𝐴𝐵𝐶 which is less than the measure of angle 𝐵𝐴𝐶.

We note that each of the five options involves the measures of the three angles 𝐴𝐵𝐶, 𝐵𝐴𝐶, and 𝐴𝐶𝐵 and that we’re given the measure of one of these. That’s angle 𝐴𝐶𝐵, which is 29 degrees. And so to answer the question, we need to find the measures of the two interior angles 𝐵𝐴𝐶 and 𝐴𝐵𝐶.

We can find the measure of angle 𝐵𝐴𝐶 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 𝐵𝐴𝐶 equals 180 degrees. Subtracting 109 degrees from both sides gives us the measure of angle 𝐵𝐴𝐶 equals 71 degrees. Next, we can find the measure of angle 𝐴𝐵𝐶 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 𝐴𝐵𝐶 plus 29 degrees equals 180 degrees.

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

In our final example, we’ll 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 𝐴𝐶𝐵 equals 62 degrees and the measure of angle 𝐴 equals 57 degrees, then the measure of angle 𝐴𝐵𝐷 what the measure of angle 𝐴𝐶𝐷.

We notice first that 𝐶𝐷 is equal to 𝐵𝐷, so the triangle 𝐵𝐶𝐷 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 𝐵𝐶𝐷 and 𝐶𝐵𝐷 are equal.

Now, we’re told that the measure of angle 𝐴𝐶𝐵 is 62 degrees and that the measure of the angle at 𝐴 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 𝐴 and those of angles 𝐴𝐵𝐶 and 𝐴𝐶𝐵 sum to 180 degrees.

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

Now, if we call the measure of our two congruent angles 𝑥. And making some space, we have 𝑥 plus the measure of angle 𝐴𝐶𝐷 equals 62 degrees. And 𝑥 plus the measure of angle 𝐴𝐵𝐷 equals 61 degrees. Now, if we add one to both sides of the second equation, we have that 𝑥 plus the measure of angle 𝐴𝐵𝐷 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 𝑥 plus the measure of angle 𝐴𝐶𝐷 equals 𝑥 plus the measure of angle 𝐴𝐵𝐷 plus one. Subtracting 𝑥 from both sides, we then have the measure of angle 𝐴𝐶𝐷 equals the measure of angle 𝐴𝐵𝐷 plus one. This means that the measure of angle 𝐴𝐵𝐷 is one degree smaller than that of angle 𝐴𝐶𝐷. Hence, the measure of angle 𝐴𝐵𝐷 is less than the measure of angle 𝐴𝐶𝐷. And the less than symbol completes the given statement.

Let’s finish by recapping some of the important points we’ve covered. The axioms of inequalities for real numbers 𝑎, 𝑏, 𝑐, and 𝑑 are such that, first, either 𝑎 is greater than 𝑏 or 𝑎 is less than 𝑏 or 𝑎 equals 𝑏. Next, the transitivity axiom, if 𝑎 is greater than 𝑏 and 𝑏 is greater than 𝑐, then 𝑎 is greater than 𝑐. Third, if 𝑎 is greater than 𝑏, then 𝑎 plus 𝑐 is greater than 𝑏 plus 𝑐 and 𝑎 minus 𝑐 is greater than 𝑏 minus 𝑐. If 𝑎 is greater than 𝑏 and 𝑐 is greater than zero, then 𝑎 multiplied by 𝑐 is greater than 𝑏 multiplied by 𝑐 and 𝑎 divided by 𝑐 is greater than 𝑏 divided by 𝑐.

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