In this video, we’re going to look at some key rules for working with angles and then apply them to calculate the missing angles in some different figures.
Okay the first rule that we’re going to look at is called the supplementary or linear pair theorem, and it tells us that two angles are supplementary if they form a linear pair, which is just another way of saying if they sit on a straight line together.
So you can see an example of this on the diagram on the right side of the screen. Angles 𝐴 and 𝐵 there sit on this straight line together. Now of course because they sit on a straight line, this means that the measure of these two angles must add up to a hundred and eighty degrees cause the key angle fact is that angles on a straight line add to one eighty.
So supplementary angles add up to one hundred and eighty degrees. The second rule is this; it’s called the complement theorem. Two angles are complementary if their noncommon sides form a right angle.
So noncommon sides are the sides that they don’t share, so the ones that I’ve highlighted in orange in the diagram here. Now of course the right angle is ninety degrees, so we have another fact about complementary angles, which is that the sum of their measures must be equal to ninety. So that’s our key fact for complementary angles.
The final fact considered here is about vertically opposite angles, and vertically opposite angles are defined as the nonadjacent angles, so not next to each other, formed by a pair of intersecting lines, so the ones that I’ve marked 𝐴 and 𝐵 on the diagram here. I could equally have marked the other pair of angles as a vertically opposite pair.
Now the key fact about vertically opposite angles is that they are the same as each other or congruent, so the measure of angle 𝐴 is equal to the measure of angle 𝐵. So these are the three main facts that we’re going to use in addition to the fact that angles in a triangle sum to one hundred and eighty degrees, and we’ll see how to apply these facts to some different problems.
So this is our first problem. We’re given a diagram and we’re asked to calculate the measure of angle 𝐴𝐶𝐵, so that means the angle formed when I move from 𝐴 to 𝐶 to 𝐵, so it’s this angle here.
So let’s think about how to approach this problem. And looking at the diagram, we can identify some different types of angles. If we look first of all at this angle here, now that angle you should recognize as vertically opposite the angle of a hundred and five degrees, because they’re formed by a pair of intersecting lines. So recalling our key fact about vertically opposite angles, they are equal to each other.
So we have that the measure of this angle 𝐶𝐴𝐵 is one hundred and five degrees. Next let’s look at the other angle at the base of the triangle, so this angle here which is angle 𝐴𝐵𝐶.
Now you’ll notice that this angle is supplementary with the angle of a hundred and forty-two degrees cause they’re a linear pair of angles. So remember our key fact was that supplementary angles add up to a hundred and eighty degrees.
So this tells that the measure of this angle 𝐴𝐵𝐶 plus a hundred and forty-two must be equal to one hundred and eighty. By subtracting a hundred and forty-two from both sides of this equation.
I can then see that this angle here is equal to thirty-eight degrees. So now I’ve worked out two of the angles inside this triangle and I’ve just labeled them both on the diagram. So finally, I want to work out the angle I was asked for, angle 𝐴𝐶𝐵, and to use this, I’m gonna recall the fact that the interior angles in a triangle add up to one hundred and eighty.
So I have that the measure of this angle 𝐴𝐶𝐵 plus the other two angles, a hundred and five and thirty-eight, must be equal to one hundred and eighty, and this gives me the equation that I can solve to work out this final angle.
So if I subtract a hundred and five and thirty-eight from a hundred and eighty, I have that the measure of angle 𝐴𝐶𝐵 is thirty-seven degrees.
So we used a number of different facts within this one question. We used the fact that vertically opposite angles are congruent to each other first of all, then we used the fact that supplementary angles add to one hundred and eighty degrees, and finally we used the rule that the interior angles in a triangle also add to one hundred and eighty degrees.
If you are asked within the question to give reasons in your answer, then you would need to drop down these worded reasons in your working out. So I would include those terms vertically opposite, supplementary, and interior angles in a triangle as part of my method.
Okay the next question, we’re given a diagram and we’re asked to find the value of 𝑥 which is used in the labels of two of the angles in this diagram. So from the diagram, we need to identify what type of angles we’ve been given and you’ll see that these angles are nonadjacent angles in intersecting lines; therefore they are vertically opposite angles.
So remember your key facts about vertically opposite angles; they are congruent to each other which means then that the algebraic expressions I have for these two angles are equal to each other. What I can do then is form an equation by setting these two expressions equal to each other.
So I have two 𝑥 minus thirty is equal to 𝑥 plus ten. Now the question has become an algebraic problem. Can I solve this equation to work out the value of 𝑥? So the first thing I’m going to do is add thirty to both sides of this equation.
And this will give me two 𝑥 is equal to 𝑥 plus forty; then I’m going to subtract 𝑥 from both sides of this equation. And when I do so, this gives me 𝑥 is equal to forty, which is therefore the answer to this question.
So in this question, I had to identify the type of angles that I had and then use the facts I know about vertically opposite angles being congruent to form an equation which I could then solve to work out this unknown letter 𝑥.
And next question is about supplementary angles. It tells us that a pair of supplementary angles are in the ratio three to two, and we’re asked to find the size of the larger angle.
Now I’m gonna go about this question in two different ways and then you can pick whichever is your favorite. So I’ll divide my page in two first of all.
So my first method, I’m gonna think about it in terms of working with ratio. Now key fact remember supplementary angles add to one hundred and eighty degrees. So in total in this ratio, I have five parts cause the three and the two together make five parts, so those five equal parts must be equal to one hundred and eighty.
So starting off with that, if I don’t want to work out what one part of this ratio is, I need to divide by five. So one part is a hundred and eighty divided by five which is thirty-six. I must find the size of the larger angle, so the larger angle will be the one that has three parts of the ratio. So in order to work out three parts, I need to multiply this thirty-six by three.
So three parts thirty-six multiplied by three is a hundred and eight, and that tells me then that the size of the larger angle is one hundred and eight degrees. So that’s one way of answering this question, by thinking about it in terms of ratio working out. Another way would be to think about it as an algebraic problem. So if these angles are in the ratio three to two, then I could call them three 𝑥 and two 𝑥, for example.
And remembering that key fact about supplementary angles summing to one hundred and eighty, I can form an equation. I could write the equation three 𝑥 plus two 𝑥 is equal to one hundred and eighty.
There’s my equation. Now if I simplify the left-hand side three 𝑥 plus two 𝑥 is five 𝑥, so I have five 𝑥 is equal to a hundred and eighty. Then if I want to work out the value of 𝑥, I need to divide by five.
So I have 𝑥 is equal to thirty-six. Now remember the larger angle was the angle that I was calling three 𝑥, so to work it out, I need to multiply thirty-six by three. And this of course gives me that same answer of one hundred and eight degrees. So the logic behind the two methods is very similar. The algebraic method is perhaps more formal, but either of those would be a valid way of approaching this question.
So it’s sensible to do a quick check where you can. So perhaps if we work out the size of the smaller angle. Well in both cases we worked out that one part or 𝑥 is equal to thirty-six, so the smaller angle will be two lots of thirty-six, which is seventy-two. And if we just check the sum of our two angles, then we can confirm that they do in fact add up to a hundred and eighty. So just a quick check at the end can give you a little bit of confidence in your working.
Okay the final question, again we’re given a diagram and we’re asked to calculate the measure of angle 𝐶𝑂𝐴, so the angle formed when I move from 𝐶 to 𝑂 to 𝐴; it’s this angle marked in green here.
Now that angle is made up of two parts, and I can already see that part of it is forty-two degrees, but I need to work out what the remaining part is. So inspecting that diagram again, you’ll see that we have a pair of complementary angles because there’s a right angle marked when I move from 𝐷 to 𝑂 to 𝐵, which means that these triangles that I’ve just marked in orange well remember complementary angles sum to ninety degrees.
So I can find the measure of this angle 𝐶𝑂𝐵 by using that fact. So I can write down this equation; the measure of angle 𝐶𝑂𝐵 plus thirty-one is equal to ninety. Subtracting thirty-one from both sides of this equation tells me then that the measure of this angle 𝐶𝑂𝐵 is fifty-nine.
So there it is, marked on the diagram. Now I’ve got everything I need in order to calculate the measure of angle 𝐶𝑂𝐴. So it’s this fifty-nine that I’ve just calculated plus the forty-two that we already knew.
So the measure of this angle is one hundred and one degrees. So to summarize then, you need to remember these three key angle rules that supplementary angles sum to one hundred and eighty degrees, complementary angles sum to ninety degrees, and vertically opposite angles are congruent to each other.
You also need to remember the fact that interior angles in a triangle add to one hundred and eighty degrees. When answering a problem, you need to inspect the diagram carefully to see which types of angles you can identify and then use the relevant angle rules to answer the question.