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.
