 So recall that two angles are complementary. If added together, they form a right angle. That is their angle measures together Form 90 degrees. So if you have the angle measure 40 degrees, its complement would then be 50 degrees Recall also that supplementary angles are those angles whose sums Turn out to be a flat angle or a half plane. In other words, a Two angles are supplementary if their angle sum is equal to 180 degrees So the supplement of 40 degrees is going to equal 140 degrees So notice that the complement of 40 is 50, but the supplement of 40 is 140 40 plus 50 adds up to be 90, which is why it's complementary and 40 plus 140 gives you 180 degrees, which is why they're supplementary Likewise, if you take the angle measure 110, which you'll notice this is an obtuse angle The complement of 110 degrees is actually negative 20 degrees You have to rotate clockwise 20 degrees to get 90 degrees 120 minus That's gonna be 110 minus 20 gives you 90 degrees. On the other hand the supplement of 110 degrees would be 70 degrees 70 degrees And so we can compute these things in general if you ever have an angle measure Theta and you want to compute its complement the complement will just be 90 degrees minus theta and the supplement would be 180 degrees minus theta That's all it is to computing complementary and supplementary angles the first angle 40 degrees Of course was an acute angle because it's less than 90 degrees If you have an acute angle, its complement will be acute its supplement will be obtuse So 140 is large and 90 degrees if on the other hand you have an obtuse angle its Component will actually be negative acute but negative and its supplement will in fact be a cute angle as well So consider the following angle diagrams, let's see if we can work through this a little bit So we have an angle diagram right here We have two angles one of them is three times one angle measured the degree measure is Three times some unknown quantity x the other is six times some other Quantity some unknown quantity x and we know that the union of the two forms a right angle We see that by the little square diagram right there So that would suggest that these two angles are Complementary angles so we know that if we take three x degrees plus six x degrees This is gonna add up to be 90 degrees so adding these together just by usual combined like terms We end up with nine x degrees is equal to 90 degrees. We can divide both sides by nine and We end up with x x degrees is going to equal 10 degrees in other words x equals 10 All right, and so that tells us that this angle measure is going to be 30 degrees and this angle measures gonna be 60 degrees So just using basic principles of algebra and knowledge of complementary angles We were able to fill in the unknown angles in this diagram Let's look at the second diagram right here in this situation We see that these angles are going to be supplementary the union of the two angles forms a half angle forms a half plane So this time one angle is going to be four times some unknown quantity The other is going to be six times an unknown quantity all being measured in degrees And so we see that four x plus six x is going to equal a hundred and eighty Okay, four x plus six x is ten x Which if we divide both sides by ten we see that x equals 18 Therefore the four x angle is going to be four times 18, which is going to be 72 degrees and Then six x that is six times 18 That's going to equal a hundred and eight degrees and therefore again using a little bit of algebra We all the fill in the missing information inside of these diagrams