Video: Inequality in One Triangle: Angle–Side

In this video, we will learn how to identify the relationships between angles and side lengths in a triangle to deduce the inequality in one triangle.

17:20

Video Transcript

In this video, we will learn how to identify the relationships between angles and side lengths in a triangle to deduce the inequality in one triangle. To look at this, let’s consider a scalene triangle. Remember that a scalene triangle is a triangle where all three sides are different lengths. In this triangle, the pink side, side one, is the shortest; the yellow side, two, is next smallest; and the blue side, side three, is the longest. What angle-side inequality lets us do is if we know the order from largest to smallest or from smallest to largest of the side lengths, we can also know the order of largest to smallest or smallest to largest of angle measures. Let’s look at this more closely.

We’ll let the angle opposite side one be called angle one, the angle opposite side two be labeled angle two, and the angle opposite side three be labeled angle three. What we’re saying here is that if we compare side one and side two, since side two is longer than side one, the angle opposite side two will be larger than the angle opposite side one. We could write it out this way. In a triangle, if one side is longer than another side, then the angle opposite the shorter side has a smaller measure than the angle opposite the longer side. And inversely, if the measure of one angle is smaller than that of another angle in a triangle, then the length of the side opposite the smaller angle is shorter than the side opposite the greater angle.

We can say side three is greater than side two. Therefore, angle three will be greater than angle two. And since side two is greater than side one, angle two will be greater than angle one. Or we can say the inverse, which means we start with the order of the angle measures and we can infer the order of the side lengths. If angle three is greater than angle two which is greater than angle one, then side three must be greater than side two which must be greater than side one.

How then should we use this angle-side inequality? Let’s consider two cases. If the lengths of the sides of a triangle are known, then the angles can be ordered according to their measures. And the second case, if the measures of the angles of a triangle are known, then the sides can be ordered according to their lengths. Now that we know how to use the angle-side inequality, let’s look at some examples. In our first example, we’ll only be ordering the angles in a triangle.

Which inequality is satisfied by this figure? (A) The measure of angle 𝐵 is less than the measure of angle 𝐵𝐴𝐶, which is less than the measure of angle 𝐶. (B) The measure of angle 𝐷𝐴𝐶 is less than the measure of angle 𝐵, which is less than the measure of angle 𝐶. (C) The measure of angle 𝐵𝐴𝐶 is less than the measure of angle 𝐶, which is less than the measure of angle 𝐷𝐴𝐶. Or (D) the measure of angle 𝐷𝐴𝐶 is less than the measure of angle 𝐵, which is less than the measure of angle 𝐵𝐴𝐶. And finally (E) the measure of angle 𝐶 is less than the measure of angle 𝐵, which is less than the measure of angle 𝐵𝐴𝐶.

For us to be able to order these angles, we’ll need to find the measures of a few of the missing angles. Currently, we don’t know the measure of angle 𝐶 or the measure of angle 𝐵𝐴𝐶. We should see that angle 𝐵𝐴𝐶 and angle 𝐷𝐴𝐶 make a straight line. If these two angles together make a straight line, they are supplementary angles and they’ll add together to be 180 degrees. If we plug in the measure for angle 𝐷𝐴𝐶, which we know is 92 degrees, then the measure of angle 𝐵𝐴𝐶 plus 92 degrees equals 180 degrees. And if we subtract 92 degrees from both sides of this equation, then we find that the measure of angle 𝐵𝐴𝐶 equals 88 degrees. We can add that to our figure.

And from there, we recognize that we have triangle 𝐴𝐵𝐶. And in a triangle, the three angles must add up to 180 degrees. So we say the measure of angle 𝐵 plus the measure of angle 𝐵𝐴𝐶 plus the measure of angle 𝐶 must equal 180 degrees. Angle 𝐵 is 52 degrees, angle 𝐵𝐴𝐶 is 88 degrees, and we want to find angle 𝐶. If we add 52 plus 88, we get 140 degrees. To find the measure of angle 𝐶, we then need to subtract 140 degrees from both sides of our equation to show that the measure of angle 𝐶 is 40 degrees. And then we can add that back to our figure.

What we can do now is list the angles that we know in order from least to greatest. Our smallest angle is angle 𝐶, which measures 40 degrees, followed by the measure of angle 𝐵, which is 52 degrees, followed by the measure of angle 𝐵𝐴𝐶, which is 88 degrees. And the largest of the angles we see in this figure is angle 𝐷𝐴𝐶, which is 92 degrees. Using this compound inequality, we can see which of the answer choices is true. And only option (E) list the angles in correct order, which says the measure of angle 𝐶 is less than the measure of angle 𝐵, which is less than the measure of angle 𝐵𝐴𝐶.

In our next example, we’ll need to relate side-length inequality to angle inequality in a triangle.

Complete the following using less than, equal to, or greater than. If in triangle 𝐷𝐸𝐹, 𝐷𝐸 is greater than 𝐸𝐹, then the measure of angle 𝐹 is blank compared to the measure of angle 𝐷.

We need to think of a few things when solving this problem. We know that there is some triangle 𝐷𝐸𝐹 and 𝐷𝐸 is greater than 𝐸𝐹. So let’s label our sketch 𝐷𝐸𝐹 such that the side length we drew for 𝐷𝐸 is larger than the side length we drew for 𝐸𝐹. In this case, we’re trying to compare angle 𝐹 with angle 𝐷. And this is where we need to remember the angle-side inequality in a triangle, which tells us if a side is longer than another side in the triangle, the angle opposite the longer side length will be larger than the angle opposite the shorter side length. Since side 𝐷𝐸 is larger than side 𝐸𝐹, the measure of angle 𝐹 will be larger than the measure of angle 𝐷. And so we fill in the blank with the greater than symbol.

It’s important to say here that we were not given any information about the side length 𝐷𝐹. And that means we could not order side length 𝐷𝐹 or angle 𝐸. We would need more information to say anything about these values. Since we only know the inequality value of two of the line segments, we can only comment on the inequality of two of the angles.

In our next example, we’ll want to give an inequality value for side lengths of a triangle if we’re given some of the angle measures in the triangle.

From the figure below, determine the correct inequality from the following. (A) 𝐴𝐶 is less than 𝐶𝐵, (B) 𝐴𝐵 is greater than 𝐴𝐶, (C) 𝐴𝐵 is less than 𝐶𝐵, (D) 𝐴𝐵 is greater than 𝐶𝐵.

First, we need to look at our figure. We’re given one of the angles inside the triangle. But in order to do any kind of side length comparison, we’ll need to know the other two angles inside this triangle. And so we see that the segment 𝐶𝐵 is parallel to the ray 𝐴𝐷. And we remember that when parallel lines are cut by a transversal, we can say something about the angle relationships. Angle 𝐷𝐴𝐵 is an alternate interior angle to angle 𝐴𝐵𝐶. And when parallel lines are cut by a transversal, these two values will be the same, which means angle 𝐴𝐵𝐶 is also equal to 66 degrees.

And after that, we have two of the three angles inside the triangle. And we know that all three angles in the triangle must sum to 180 degrees. We can write that out like this and then plug in the values we know. Measure of angle 𝐶𝐴𝐵 is 52 degrees plus the measure of angle 𝐴𝐵𝐶, which is 66 degrees, plus the measure of angle 𝐵𝐴𝐶 must equal 180 degrees. 52 plus 66 is 118, so we bring everything else down from our equation. And to find the measure of angle 𝐵𝐶𝐴, we need to subtract 118 degrees from 180 degrees, which will give us 62 degrees.

And then we think about what we know of angle-side inequality in a triangle, which tells us if we know the angle values, we can order the side lengths. Side length 𝐶𝐵 is opposite the smallest angle, which will make side 𝐶𝐵 the smallest side length. Side length 𝐴𝐵 is opposite the middle angle value, which means 𝐶𝐵 is less than 𝐴𝐵. And our third side length 𝐴𝐶 is opposite the largest angle, so it’s the largest side length. And we can say that 𝐶𝐵 is less than 𝐴𝐵, which is less than 𝐴𝐶. This is an ordered list from least to greatest. We could also write it from greatest to least where 𝐴𝐶 is greater than 𝐴𝐵, which is greater than 𝐶𝐵. Using these two statements, we can see which of our answer choices is true. The only true option is 𝐴𝐵 is greater than 𝐶𝐵.

Let’s consider another example.

Use less than, equal to, or greater than to complete the statement. If 𝐴𝐵 equals 62, 𝐴𝐶 equals 63, and the measure of angle 𝐴𝑋𝑌 is equal to the measure of angle 𝐴𝑌𝑋, then compare line segment 𝑌𝐶 to line segment 𝑋𝐵.

First, let’s take the information we’re given and add it to the figure. We know that line segment 𝐴𝐵 equals 62 and line segment 𝐴𝐶 equals 63. We also see that the measure of angle 𝐴𝑋𝑌 is equal to the measure of angle 𝐴𝑌𝑋, which is already marked on our figure. But because angle 𝐴𝑋𝑌 is equal to angle 𝐴𝑌𝑋, we can say that the line segment 𝐴𝑋 will be equal in length to the line segment 𝐴𝑌, as in a triangle, when two angles have the same measure, the side lengths opposite those angles will also be equal.

If all of this is true, how do we compare line segment 𝑌𝐶 to line segment 𝑋𝐵? We know that 𝐴𝐶 is larger than 𝐴𝐵, but if the segments 𝐴𝑌 and 𝐴𝑋 are equal to each other, then the segment 𝑌𝐶 must be longer than the segment 𝑋𝐵. If this didn’t initially make sense, one way you could solve this is by just using some value for 𝐴𝑌 and 𝐴𝑋. If, for example, 𝐴𝑌 and 𝐴𝑋 both equal five, then 𝑌𝐶 would measure 58 but 𝑌𝐵 would measure 57. What if the equal segments measured 10? Then 𝑌𝐶 would have to equal 53 and 𝑋𝐵 would have to equal 52. Segment 𝑌𝐶 will always be greater than segment 𝑋𝐵.

And even though the question didn’t mention this, it’s also probably worth pointing out one more thing. Since line segment 𝐴𝐶 is greater than line segment 𝐴𝐵, the measure of angle 𝐵 will be greater than the measure of angle 𝐶. But this question was only asking us to compare 𝑌𝐶 and 𝑋𝐵, which we’ve done. 𝑌𝐶 is greater than 𝑋𝐵.

Let’s look at one more example of looking at inequality in a triangle.

Use less than, equal to, or greater than to fill in the blank to compare side length 𝐵𝐶 to side length 𝐴𝐶.

First, we can identify which side lengths we’re trying to compare. We want to compare side length 𝐵𝐶 to side length 𝐴𝐶. One strategy we can try to use is the angle-side inequality in a triangle. To do that, we’d need to compare the angles opposite the two side lengths we’re interested in. Now you might be thinking we don’t have any information about the angles. However, we can use some properties of triangles to find out some information about the angle measures. In a triangle, if two sides have the same length, then their opposite angles are the same, which means here angle 𝐴𝐵𝐹 will be equal in measure to angle 𝐹𝐴𝐵 and angle 𝐴𝐷𝐹 will be equal to angle 𝐷𝐴𝐹.

Both triangle 𝐴𝐵𝐹 and triangle 𝐴𝐷𝐹 are isosceles triangles, and they have two equal sides. This means we can say that segment 𝐴𝐷 is going to be equal in length to segment 𝐴𝐵 as these two triangles are congruent. It also means angle 𝐴𝐹𝐵 will be equal to angle 𝐴𝐹𝐷. This means we found that side length 𝐴𝐵 must be smaller than side length 𝐴𝐶. But how can we say something about side length 𝐵𝐶? For this, we’re going to think about the line segment 𝐴𝐹. We’ve seen that the line segment 𝐴𝐹 bisects the segment 𝐵𝐷. The line segment 𝐴𝐹 is a bisector of the isosceles triangle 𝐴𝐵𝐷. And when that happens, it is a perpendicular bisector, which means both of these angles must measure 90 degrees. And if that’s the case, in the smaller isosceles triangles, the smaller blue angles must be equal to each other and must therefore be equal 45 degrees each.

If all of these smaller angles measure 45 degrees, the big angle we were looking for, angle 𝐵𝐴𝐶, is a right angle. And since line segment 𝐵𝐶 is opposite that right angle, it’s the hypotenuse of the larger triangle 𝐴𝐵𝐶, and it is therefore the longest side length of this triangle. The order of the side lengths for the larger triangle 𝐴𝐵𝐶 must be 𝐴𝐵 is smaller than 𝐴𝐶, which is smaller than 𝐵𝐶. Since 𝐵𝐶 is the hypotenuse and is the longest side length in this triangle, we can say 𝐵𝐶 is greater than 𝐴𝐶.

To wrap up this video, let’s go over the key points. In a triangle, if one side is longer than another side, then the angle opposite the shorter side has a smaller measure than that opposite the longer side. And inversely, if the measure of one angle in a triangle is smaller than that of another angle, then the length of the side opposite the smaller angle is shorter than that of the side opposite the greater angle, which we can represent with this figure. Side three is greater than side two, which is greater than side one. Therefore, angle three will be greater than angle two, which will be greater than angle one.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.