in this video we're going to talk about lines and angles the first type of line that you need to be familiar with are parallel lines parallel lines do not intersect so let's call the first line a and the second line B so we could say that a is parallel to line B so these two lines are also parallel they share the same slope so if the first line has a slope of 3 over4 the second line has the same exact slope perpendicular lines are different whenever you have two perpendicular lines intersecting each other they will form a right angle so they meet at 90° let's call this line a and line B so the way you can describe it you could say line a is perpendicular to line B so this symbol represents that those two lines are perpendicular so let's say that line C has a slope of 3 over 4 what do you think the slope will be for line D to find the slope of the perpendicular line you need to flip the fraction so instead of 3 over four it's going to be 4 over three now you need to change a sign from positive to negative so it has a slope ne4 over three when you have a third line that intersects two parallel lines that third line is the transversal so let's say this is line A and B so if a is parallel to B then C is going to be the transversal we'll call this angle one 2 3 4 5 6 7 8 angles 3 4 5 and six are known as interior angles when you think of the word interior what do you think of interior means on the inside so those four angles three 4 5 6 they're in between or on the inside of the two parallel lines angles one 2 7 and 8 are known as exterior angles because they're outside of the two parallel lines now which angles are alternate interior angles need to be able to identify those so the fact that they on the inter interior has to be between 3 4 and 5 6 3 and six are alternate interior angles they're on the inside of the two parallel lines but they're opposite of the transversal three is on the left side of the transversal and six is on the right side of the transversal so that's why they're called alternate interior angles now alternate andr angles are congruent so angle 3 is equal to angle six they have the same exact measure assuming if a and b are parallel angle four and five are also alternate interior angles they're congruent now what about alternate exterior angles angle one and 8 are alternate exterior angles it turns out also angle two and seven are alternate exterior angles alternate exterior angles are congruent they're the same and if you're wondering why they're called alterate exterior angles well first the word exterior they're on the outside or on the exterior of the two parallel lines and they alternate with reference to the transversal so Ang one is on the left side of the transversal or line c angle a is on the right side so they're on Alternate sides and 2 and seven are alternate exterior angles now the next term that you need to be familiar with are correspondent angles angle 2 and angle six are corresponding angles corresponding angles are congruent angles one and five are correspondent angles angles four and 8 are corresponding and three and uh seven correspond to each other so angle four is congruent to angle 8 and angle three is congruent to angle seven so those are corresponding angles they're on the same position on the parallel lines in relation to the transversal so if you look at two and six they're both on the right side of the transversal and they're both above the two parallel lines if you look at uh 3 and seven they're both on the left side of the transversal and they're both below the parallel lines so they're in the same position on the parallel lines in this case they're both below it and they're both on the left side of the transversal now there's some other terms that you need to be familiar with as well the next term is consecutive interior angles so four and six are consecutive interior angles they're on the inside of the transversal and consecutive means then basically the next one after so for example consecutive integers 2 3 4 5 6 three is consecutive to two because it comes right after so if you look at this angle the next one is six so those are consecutive interior angles consecutive interior angles they add up to 180 so basically they're supplementary so angle 4 plus angle 6 adds up to 180 angle three and angle five adds up to 180 so they're supplementary now the last term you need to be familiar with are vertical angles so whenever you have two lines that intersect each other they will form vertical angles let's say if this angle is 50 this angle is also 50 vertical angles are congruent one and four are vertical angles so one and four are congruent two and three are vertical angles seven and six are vertical angles so they equal each other and five and eight are congruent to each other so those are vertical angles so anytime you have a transversal or a third line that intersects two parallel lines you're going to have a lot of congruent angles so just make sure you're familiar with those terms now the next term that you need to be familiar with are complimentary angles so let's say that line a and line B are perpendicular to each other so let me just go ahead and extend those two lines so that means that this angle between A and B is 90° but let's draw a line let's say this angle is 30 and let's call this line C and let's call this uh vertex V so angle C VB is 30° what is angle a v c so you know they have to add up to 90 because angle a v b is 90 so 90 - 30 is 60 so therefore these two angles are complementary they add up to 90 so angle AV v c plus angle C VB has to add up to angle AVB so that's an example of complimentary angles they add up to 90 the next excuse EXC me the next one that you need to be familiar with are supplementary angles which we briefly talked about before supplementary angles add up to 180 so let's say if you have a straight line and there's another line that passes through it the total angle of a straight line will always be 180 a full circle is 360 so half a circle that's 180 so if this angle is 60 the other angle has to be 120 so let's call this a b c and d so angle AB D that's basically 120 plus DBC which is 60 is equal to basically the stength line ABC which is 180 so whenever you have a straight line all the angles that are part of that straight line has to add up to 180 so here's an example let's say This is 40 and this is 70 what is the missing angle and this is a straight line so 70 + 40 + x has to add up to 180 70 + 40 is 110 so to find the value of x we need to subtract 110 from both sides so X is 180 - 110 so X is also 70° so now you're familiar with the terms supplementary and complimentary so keep in mind if two angles are complimentary that means that they add up to 90° if two angles are supplementary they add up to 180° so now let's go over a few practice problems in uh solving uh certain things so I'm going to give you a few questions and I want you to find the value of x so in the first example what is what is the value of x now these two angles are congruent they're vertical angles so therefore X is simply 70 in the second example these two form a linear pair they're supplementary because they form a straight line when you add it together so therefore 80 + x is equal to 180 so X is going to be 180 minus 80 so X is 100° so some of these problems are not going to be very difficult so let's say if this is 30 and this is uh 2x + 5 and this is y find the value of X and Y actually let's say this is y + 8 feel free to pause the video and work on that problem so we know that 30 and y + 8 are vertical angles so therefore they're congruent so we could set 30 equal to y + 8 therefore Y is going to be 30 minus 8 so Y is 22 now what is the value of x now these two they form a linear pair so they're supplementary so 30 + 2x + 5 has to add up to 180 so now we could find the value of x 30 + 5 that's 35 so if we subtract both sides by 35 2x is 180 - 35 which is 145 and then we need to divide by two so half of 140 is 70 half of 5 is 2.5 so half of 145 is going to be 72.5 so that's the value of x let's say this angle is 35 what is the value of x let's call this a b c and d so and let's say that angle ABC is the right angle what is the value of angle DBC or basically what is the value of x so because this is a right angle it represents 90° therefore angle ABD which is 35 must add up to well angle ABD plus angle DBC which is X they have to add up to angle ABC which is 90 so basically these two angles are complimentary so X is simply 90 minus 35 which is 55 here's another example it's a very similar problem but slightly different so let's say this is 7x and this is 11x and let's say angle ABC is once again a right angle what is the value of angle AB D so feel free to pause the video and try this problem angle ABD which is represented by 7x plus angle DBC which is represented by 11 axis they have to add up to 90 so we can say that 7x + 11x is 90 so now let's find the value of x so 7 + 11 is 18 and 90 / 18 is five so X is equal to 5 but our goal is to find a measure of angle ABD not simply the value of x angle ABD keep in mind it's 7x so our goal is to find the value of 7x so that's 7 * 5 which turns out to be 35° if we wanted to find the angle of angle DBC that's 11x 11 * 5 is 55 now let's say if we have a straight line and let's say say this is a b c and d and let's say that angle DBC is 7 x - 6 and angle ABD is 13x + 26 go ahead and find the angle DBC find its measure feel free to pause the video so we said ABC is a straight line that means that it has an angle of 180 so angle a b d plus angle DBC has to add up to angle ABC ABD and DBC they're basically adjacent angles so this is ABD and its angle is 13x + 26 angle DBC is 7X - 6 so those two have to add up to ABC which is basically a straight line and the angle of a straight line is always 180 so therefore these two angles are supplementary since they add up to0 so now our goal is to find the value of x 13x + 7 x adds up to 20x and then 26 - 6 that's 20 so 20x + 20 is equal to 180 so now let's subtract both sides by 20 180 minus 20 that's 160 and now let's divide both sides by 20 we can cancel a zero so this becomes 16 / 2 which is 8 so that's the value of x now that we have the value of x we could find angle DBC which is 7X - 6 so we got to take this answer plug it in into 7X - 6 so it's going to be 7 * 8 - 6 7 * 8 that's basically 56 and 56 - 6 is 50 so therefore angle DBC is 50° which means that ABD has to be 180 minus 50 or 130 since these two are supplementary let's call this line C and line D and let's say that c is parallel to D and we have a transversal which we'll call line a now if this angle let's say is 50 find all the other angles formed by this transversal so go ahead and find angles 1 2 3 4 5 6 7 angle two is a vertical angle with respect to that angle so therefore they're congruent so this is 50 angle one is supplementary to angle 50 so they have to add up to 180 so 180 minus 50 is 130 angle three is a vertical angle with respect to 130 so angle three is also 130 now what about angle four notice that angle one and angle four are corresponding angles so they're congruent angle five is an alternate interior angle with respect to angle two alternate interrent angles are congruent so angle 5 is also 50° I think we call this angle six which is a vertical angle with respect to angle 5 so that's 50 and angle 7 is a vertical angle with respect to 130 so that's also 130 and so that's basically how you can solve that here's another example so let's say that these two lines are parallel we'll call it line a and line B and C is the transversal so if this is 70 go ahead and find all the other angles so this must be 70 as well and these two angles are supplementary 180 minus 70 is 110 so this is a vertical angle so these are congruent this angle corresponds to this angle so that's 110 which means this is 110 and this is 70 and that is 70 so you can quickly find all the angles for a problem like this now here's another problem let's say line a and line B are parallel to each other and there's going to be two transversals so this is one transversal we can call it line C and it intersects with another transversal which we'll call line D and let's say that this angle is 50° and let's say this angle is 40 go ahead and find all other angles that can be formed by this figure so this angle is 40 we know this has to be 40 they're vertical angles and therefore 180 minus 40 is 140 now this is 50 this must be 50 as well the vertical angles and these two are supplementary 180 - 50 is 130 which means this is 130 now these two angles are consecutive interior angles they add up to 180 which means this is 130 now these two angles are alternate interior angles so therefore this is 50 and this angle and that angle they're alternate exterior angles so they're congruent which means this must be 130 and so this angle on the inside is 50 now let's call this x what is the value of angle x what would you say now keep in mind this figure is not drawn to scale I just made this problem and put some numbers to it so if that is X you need to know that the three angles of a triangle must always add up to 180 so that's 40 and this is 50 and this is X 40 + 50 + x has to be 180 so if you take 180 and subtracted by 40 and then by 50 you should get let's see 180 - 40 is 140 140 - 50 is 90 therefore X is 90 which means this has to be 92 even though it's not drawn a scale so these are complimentary so basically all of these angles will be 90 they're right angles but they're not drawn a scale let's try another similar problem but one that hopefully is not 90 so this time we're going to make this angle 60 and this angle is going to be 110 so go ahead and find all other angles if that's 110 these two angles are supplementary which means this angle must be 70 and this has to be 70 as well which means this is 110 so now let's find a missing angle angle X so 60 + 70 + x has to add up to 180 60 and 70 is 130 and 180 minus 130 is 50 so therefore this angle is 50 now that we have all three angles of that triangle we could find everything else now those two are supplementary which means this must be 120 120 + 60 add 180 so that's 60 that's 120 now these two are correspondent angles so therefore this is 60 these two are alternate exterior angles so they're congruent this is 120 and these two are alternate interior angles so that's 60 and this is going to be 120 since these two are vertical angles now these two are vertical angles as well so this is 50 and then 180 minus 50 is 130 so this is 130 as well now these two are alternate interior angles so this must be 50 and those two are alternate inter angles so uh that's 130 and then these two are correspondent angles so corresponding angles are congruent and these two are alternate exterior angles which are also so congruent now notice that we have a quadrilateral in the middle a quadrilateral is a four-sided figure a three-sided figure like a triangle has an interior angle all the interior angles of a triangle adds up to 180 so any three-sided figure will always have the sum of their angles which is 180 if you have a four-sided figure like this quadrilateral highlighted in red the sum of those four angles will be 360 so if you add 60 and 50 that's 110 + 120 that's 230 + 130 230 + 130 that's going to be 360 so there's a rule where you could find the sum of the interior angles it's n - 2 * 180 so for a three sided figure like a triangle n is 3 3 - 2 is 1 180 minus or 180 * 1 is 180 now if you have a foursided figure like a quadrilateral n is 4 4 - 2 is 2 2 * 180 that's 360 so the sum of all the interior angles is 360 now let's say if you have a five-sided figure in this case a pentagon using the formula N - 2 * 180 this time n is going to be five since there are five sides 5 - 2 is 3 3 * 180 is 540 so the sum of all five angles let's say angle a b c d and e the sum of all five of these angles has to add up to 540 now now let's work on some word problems feel free to pause the video and try these problems yourself number one if angle a and angle B are complimentary what is the value of angle B if angle a is 32 now if you recall whenever you hear the word complementary that means that the two angles add up to 90 so angle a plus angle b equals 90 now we're given the value of angle a is 32 so 32 plus angle B is 90 so to find angle B we need to subtract both sides by 32 so angle B is going to be 90 minus 32 90 - 30 is 60 and then 60 - 2 is 58 so that's going to be the value of angle B number two if angle a a and angle B are supplementary what is the value of angle a if angle B is 74° so keep in mind the we supplementary means that the two angles add up to 180° so angle a plus angle B is equal to 180° now this time we have the value of angle B which is 74° so let's go ahead and find the value of angle a so let's begin by subtracting both sides by 74 so angle a is going to be 180 minus 74 180 - 70 is 110 and 110 - 4 is 106 so this is going to be the answer number three angles R and S are complementary if angle R is 3x + 1 and angle s is 9x - 7 what is the measure of angle s so go ahead and take a minute and work on this problem so these two angles are complimentary so we know that r + S has to add up to 90 angle R is 3x + 1 and angle s s is 9x - 7 so now all we need to do is find the value of x and then we could find the measure of angle s 3x + 9x that's 12x 1 + -7 is -6 so 12x - 6 is 90 now let's go ahead and add 6 to both sides so 12x is equal to 96 and now let's divide both sides by 12 96 / 12 is 8 so X is equal to 8 now that we have the value of x we could find the measure of angle s angle s is 9x - 7 so all we got to do is take this value plug it into that expression so it's 9 * 8 - 7 9 * 8 is 72 and 72 - 7 is 65 so that's the measure of angle s it's 65° number four angles X and Y are supplementary if angle X is 11 x - 22 and angle Y is 5x + 10 what is the value of angle y so the angles are supplementary which means that they add up to 180 so x + y = 180 now angle X is 11 x - 22 let's not confuse these two letters with each other angle Y is 5x + 10 so now let's go ahead and find a value of x 11x + 5x adds up to 16x -22 + 10 adds up to -12 so now let's solve let's add 12 to both sides so 16x is equal to 192 now let's divide both sides by 16 192 / 16 that's about 12 so now we could find the measure of angle y angle Y is 5x + 10 so let's insert this value into that expression so it's going to be 5 * 12 + 10 5 * 12 is 60 60 + 10 is 70 so angle Y is 70° which means that angle X must be 110 degrees so let's say that these two lines line A and B are parallel to each other and we have a transversal line C and once again we're going to say that this is angle 1 2 3 4 5 6 7 8 so in this example let's say that angle two is equal to 5x and angle 3 is equal to 3x + 24 your goal is to find the value of x so feel free to pause the video and go ahead and try that problem so what is the relationship between angles 2 and three angles two and three are vertical angles and as we know vertical angles are congruent so we could say that angle two is equal to angle 3 now angle 2 is 5X angle 3 is 3x + 24 so now all we need to do is find the value of x so let's begin by subtracting both sides by 3x 5x - 3x is 2x so 2x is equal to 24 and 24 / 2 is 12 so that's the answer to the first problem X is equal to 12 now let's move on to our second example so let's say that angle 4 is equal to 5x + 5 and angle 6 is equal to 10 x - is 50 your task is to find the measure of angle six so go ahead and take a minute and do that the first thing we need to do is identify the relationship between angles four and six angles four and six are consecutive interior angles consecutive interior angles are supplementary they add up to 140 I mean not 140 but 180 so angle angle 4 plus angle 6 = 180 angle 4 is 5x + 5 angle 6 is 10 x - 50 so now all we need to do is find the value of x and then we can use that to find the measure of angle 6 5x + 10 x is 15x 5 +50 is- 45 so let's add 45 to both sides 180 + 40 is 220 and then if you add five to it this is going to be 225 225 ID 15 is 15 so X is equal to 15 so now we can find the measure of angle 6 angle 6 is 10 x - 50 so let's replace x with 15 so we're going to have 10 * 15 - 50 15 * 10 is 150 and 150 - 50 is 100 so 100 is the measure of angle six let's try one more problem let's say that angle one is equal to 9x + 16 and angle two I mean not two but angle 8 is 12x - 2 using this information go ahead and find a measure of angle one so what is the relationship between angles 1 and 8 1 and 8 are alternate exterior angles and they're congruent so because they're congruent angles 1 and 8 are equal to each other which means that 9x + 16 is equal to 12x - 2 so let's go ahead and find the value of x let's begin by subtracting both sides by 9x and at the same time let's add two to both sides 16 + 2 is 18 12x - 9x is 3x 18 / 3 is 6 so X is equal to 6 now that we have the value of x we could find a measure of angle one angle one is 9x + 16 so now let's replace x with six so angle one 1 is going to be 9 * 6 + 16 9 * 6 is 54 54 + 16 is 70 so the measure of angle 1 is 70° so that is it for this video so hopefully it gave you a good understanding of parallel lines perpendicular lines transversals and now you know how to identify corresponding angles alternate interior and exterior angles consecutive inter angles and basically everything else that we went over in this video so that's it and uh have a nice day