Complementary and Supplementary Angles

in this video we're going to talk about complementary and supplementary angles so what exactly is complementary angle complementary angles are angles that add up to 90. so consider this right triangle let's call it triangle abc now if angle is 30 then angle has to be 60. so notice that angle and angle are complementary to each other angle is the complement of angle any time you have two angles that add up to 90 the two angles are complementary to each other now the next term you need to be familiar with are supplementary angles supplementary angles add up to 180 as opposed to 90. so let's call this and angle abd let's say it's 120. so what is the value of angle cbd if you know abd and cbd are supplementary cbd has to be 60 because these two angles have to add up to 180. it turns out that they also form linear pair means we have straight line with two angles it forms the linear pair which also adds up to 180. now let's work on some practice problems let's start with this one find the supplement of 55 degree angle supplementary angles we know adds up to 180 so to find the other supplement it's going to be 180 minus 55. so if you want to subtract as mentally you could break up 55 into 50 and 5. now 180 minus 50 think of 18 minus 5 that's 13. so 180 minus 50 is 130 and 130 minus 5 is 125. so that's the supplement of 55 degree angle it's 125 degree angle number two what is the complement of 64 degree 26 minutes 37 seconds angle now to find the complement we need to subtract that angle from 90. so how can we do this if we're doing 90-64 that would be piece of cake but we have an angle in dms degrees minutes and seconds so how can we perform the operation well let's change 90 into dms value so first what we're going to do is we're going to borrow 1 from 90. so 90 is equivalent to 89 degrees and 1 degree is 60 minutes so 89 degrees in 60 minutes is basically equivalent to 90 degrees now we do have value in the seconds column so therefore we need to borrow minute from 60. so 89 degrees and 60 minutes is equivalent to 89 degrees 59 minutes and 60 seconds so now we could take this value and subtract it by that number minus sixty-four that's twenty-five fifty-nine minus twenty-six we know nine minus six is three five minus two is three and sixty minus thirty-seven that's 23. so the answer is 25 degrees 33 minutes and 23 seconds so that's the complement of this angle number three one of two complementary angles is 20 degrees larger than the other what is the measure of the two angles so let's say the two angles let's give it letter angle and angle if they're complementary to each other that means that they add up to 90 degrees now one of the angles is 20 degrees larger than the other one so that means is plus 20. so let's say if is 30 would be 50 if is 40 will be 60. so they differ by 20. so we have two equations and we have two variables how can we find the value of and so what i'm going to do is subtract both sides by in this equation if do so i'm going to have minus is equal to 20. so now have system of two equations and i'm going to solve it by the elimination method also known as the addition method so if we add the two equations and negative adds up to zero plus is two ninety plus twenty is one ten so to find the value of need to divide both sides by two hundred divided by two is fifty and ten divided by two is five so 110 divided by two is fifty-five so that's the value of angle now to find angle we know that these two angles have to add up to 90. so is going to be 90 minus 55. 90 minus 50 is 40. 40 minus 5 is 35. so angle is 35 so now we have the measure of the two angles so 55 plus 35 adds up to 90 and 55 is 20 units higher than 35 let's try this one number four two complementary angles exists in two to three ratio what is the measure of the larger angle so let's draw picture we can use the right triangle as an example just like we did in the beginning of this video and let's say angle is the right angle which means that angle and they have to be complementary to each other all three angles of triangle must always add to 180. so if angle is 90 and have to add up to 90 which means and are complements of each other now we could say that angle is 2x and angle is 3x and in that way they exist in 2 to 3 ratio so angle plus angle have to add up to 90. and angle is 2x angle is 3x and so 2x plus 3x is 5x so now we need to divide 90 by 5. now if you wish to perform the division mentally here's what you can do think of 90 as being 50 plus 40 and we need to divide each number by 5. 50 divided by 5 is 10 40 divided by 5 is 8. so 90 divided by 5 is 10 plus 8 or 18. so that's the value of but our goal is to find the measure of the larger angle which is angle and that's 3x so that's going to be 3 times 18 which if you don't know what 3 times 18 is you can break up 18 into being 10 plus 8. 3 times 10 is 30 3 times 8 is 24 and 30 plus 24 is 54. so that's 3 times 18. so angle is 54 degrees which means that angle is 2x 2 times 18 that's 36 and as you can see these two numbers 36 plus 54 adds up to 90. but the answer to the problem the measure of the larger angle is 54. number five one of two supplementary angles is 12 more than three times the value of the other calculate the measure of the smaller angle so let's say is the first angle and is the second one the fact that these two angles are supplementary means that they add up to 180. now one of the two angles is 12 more than three times the other so three times the other let's say if is the other one that will be three and then it's twelve more than that so three plus twelve our goal is to calculate the measure of the smaller angle so we're going to find the value of and and then we can see that is going to be larger angle is small angle so our goal is to find now in this problem i'm going to solve using substitution instead of elimination so i'm going to replace with 3x plus 12. so now we have the expression plus 3x plus 12 is equal to 180 and plus 3x is 4x next we need to subtract both sides by 12. 180 minus 12 is 168 and then we could divide both sides by four 16 divided by 4 is 4 and 8 divided by 4 is 2. so 168 divided by 4 is 42 and so that's the measure of the smaller angle to find the measure of the larger angle take this number and plug it into this expression or you could do 180 minus 42 which is going to be 138 so that's the measure of the larger angle but the answer to the problem is 42 which is what we wanted challenge problem number six the supplement of an angle is four times the complement of the angle find the measure of the supplement of the angle so feel free to pause the video and work on this problem now we need to realize is that there's three variables the first one is the measure of the angle the second one is the measure of the complement of the angle and the last one is the measure of the supplement of the angle so what we need to do is assign three variables to these three quantities let's say is the measure of the angle is for the complement is for the supplement so you can keep track of what's what now the first sentence says the supplement of an angle is four times the complement of the angle so we can say that is equal to four is four times the value of the complement now there are two other equations that we can write if you want to solve system of three variables you need three equations now the complement of angle is going to be 90 minus 8 because plus has to add up to 90. and the supplement and the angle have to add up to 180 so the supplement is going to be 180 minus the angle so now we have the three equations that we need so let's replace with 180 minus and let's replace with 90 minus so now we can have single equation that contains only one variable so now all we got to do is just some math let's distribute the 4 to 90 minus so 4 times 90 that's 360 because 4 times 9 is 36 and then we have 4 times negative which is negative 4a and now let's subtract both sides by 180 and simultaneously let's add 4a to both sides so on the left we have negative plus 4a which is 3a and 360 minus 180 that's 180. so next let's divide both sides by three so is 180 divided by three now 18 divided by three is six so 180 divided by three is 60. so the angle has measure of 60 degrees so i'm going to put that here is equal to 60. the complement is 90 minus 60 so the complement is 30. the supplement is 180 minus 60 which is 120. and we can see that our answers are correct because the supplement is indeed four times the value of the complement thirty times four is one twenty and so that's the end of this video hopefully this gives you good idea of how to solve geometry problems that are associated with complementary and supplementary angles you