### Video Transcript

In a triangle π΄π΅πΆ, if the
measure of angle π΄ equals one-half times the measure of angle π΅ equals two times
the measure of angle πΆ, what type of angle is the interior angle at vertex π΅?

Letβs begin this question by seeing
if we can work out any of the angles in the triangle using this relationship between
the angle measures. The best way to approach this would
be to take one of the angles and define it with a letter. So, for example, we could define
the measure of angle π΄ as π₯. Then, we know that the measure of
angle π΄ is half the measure of angle π΅. So the measure of angle π΅ must be
two π₯. If we find this hard to work out,
we can rewrite this part of the equation.

Since the measure of angle π΄ is
half the measure of angle π΅, then by multiplying by two, we have that two times the
measure of angle π΄ is equal to the measure of angle π΅. And since we defined the measure of
angle π΄ as π₯, then thatβs why we have that the measure of angle π΅ is two π₯.

Next, letβs write the measure of
angle πΆ in terms of π₯. This time, we can use the
relationship between the measures of angles π΄ and πΆ, which is that the measure of
angle π΄ equals two times the measure of angle πΆ. To find the measure of angle πΆ, we
would need to divide both sides of the equation by two. So one-half times the measure of
angle π΄ is equal to the measure of angle πΆ. And as we defined the measure of
angle π΄ as π₯, then we can write that the measure of angle πΆ is one-half π₯. And if we wished at this point, we
could sketch a diagram of triangle π΄π΅πΆ, something like this. And now we can consider if itβs
possible to find a value for each of these angle measures.

We can recall the property that the
interior angle measures in a triangle sum to 180 degrees. We can write that the three angle
measures of one-half π₯, π₯, and two π₯ sum to 180 degrees. And simplifying the left-hand side,
we have three and a half π₯, or seven over two π₯, which equals 180 degrees. We can then multiply each side by
two and divide by seven, either in one step or two steps, to find that π₯ equals 360
degrees divided by seven. We can keep this as a fraction. But sometimes it helps us to get an
understanding of the size of the value if we see it as a decimal. And 360 degrees over seven is
approximately 51.4 degrees to one decimal place.

We were asked to work out the type
of angle at vertex π΅. And so we first need to find the
value of this angle in degrees. We defined this angle to be two
π₯. And although the level of accuracy
doesnβt really matter here, itβs always best to substitute the most accurate value,
which is the fractional value. So we can write that the measure of
angle π΅ is equal to two times 360 over seven degrees. This can be simplified to 720 over
seven degrees, or 102.9 degrees approximated to one decimal place.

We can recall that one of the types
of angles is an obtuse angle, which is when the angle measure is greater than 90
degrees and less than 180 degrees. This is exactly what we have here
for the measure of angle π΅. We can therefore give the answer
that the type of angle at the interior vertex of π΅ is an obtuse angle. But of course in this type of
question, itβs always good to check our answers. In this case, finding the value of
all the angles in the triangle is a great way to do this.

We calculated that the measure of
angle π΄, which was π₯, is 360 over seven degrees, or approximately 51.4
degrees. Angle π΅ was two π₯, so 720 over
seven degrees, or approximately 102.9 degrees. Angle πΆ will be one-half π₯, which
is 180 over seven degrees, or approximately 25.7 degrees, to one decimal place. Adding up these values would indeed
give us the angle sum of the triangle, which is 180 degrees, although if we use the
decimal approximations, we may not always get a completely round value of 180
degrees simply due to the rounding errors.

As a final check of the values, we
can check that they do have the original relationship between the values that we
were given, that the measure of angle π΄ equals one-half times the measure of angle
π΅ equals two times the measure of angle πΆ. And these do, which confirms the
angle measure for π΅ and that the type of angle is an obtuse angle.