Question Video: Determining the Type of Angle at a Vertex of a Triangle given a Relationship between the Angle Measures Mathematics

In a triangle π΄π΅πΆ, if πβ π΄ = 1/2 πβ π΅ = 2πβ πΆ, what type of angle is the interior angle at vertex π΅?

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Video Transcript

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

Letβs 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 thatβs why we have that the measure of angle π΅ is two π₯.

Next, letβs write the measure of angle πΆ in terms of π₯. This time, we can use the relationship between the measures of angles π΄ and πΆ, which is that the measure of angle π΄ equals two times the measure of angle πΆ. To find the measure of angle πΆ, 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 πΆ. And as we defined the measure of angle π΄ as π₯, then we can write that the measure of angle πΆ is one-half π₯. And if we wished at this point, we could sketch a diagram of triangle π΄π΅πΆ, something like this. And now we can consider if itβs 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 doesnβt really matter here, itβs 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, itβs 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 πΆ 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 πΆ. And these do, which confirms the angle measure for π΅ and that the type of angle is an obtuse angle.