Proving angles are congruent

so we have an interesting looking diagram here let's say we know a few things about this diagram let's say we know let's say we know that that line m'kay so line MK is parallel to line and J and J so this line is parallel to this line this is line MK this is line and J now given that and all the other information on this diagram I'm hoping to prove that the measure of angle this angle right over here l MK l MK is equal to is equal to the measure of this angle right over here and this angle is L we could call it angle L and J is equal to the measure of angle L and J or another way of writing this the measure of LM K is B is B and the measure of L and K is a so we want to prove that B is equal to a using all of this information that we know and like always I encourage you to try this on your own before I walk through it all right so let's walk through it so the first thing you might see is okay look I have this I have a triangle formed up here and actually let me let me sketch out this triangle triangle M L K and what do we know about angle the measure of the angles of the interior angles of a triangle well the measures of the interior angles of a triangle are going to add up to 180 degrees so we know we know that B which is the angle the measure of this angle plus the measure of this angle plus C plus C plus this the measure of this right angle so it's gonna be plus 90 so let me write it plus 90 plus 90 degrees is going to be equal to if you these are this is this is sum of the measures of the interior angles of this triangle well for any triangle that's going to sum up to 180 degrees and so if we subtract 90 degrees from both sides we're going to get we're going to get B plus C B plus C is equal to is equal to 180 degrees minus 90 degrees is going to be it's going to be 90 degrees or if we wanted to write B in terms of or if we want to solve explicitly for B we could subtract C from both sides and we could write B B is equal to is equal to 90 degrees minus C 90 degrees 90 degrees minus C alright so that's interesting so that's that's one way of expressing B now can we express a in a similar way and once again at any point you get inspired I encourage you to do that well if we look carefully we see that we have a triangle N and it's hard to read we have a triangle let me see this I'll do it in purple we have triangle N and L J this really big triangle it's really most of the diagram nlj and what's interesting about n LJ is that this is another right triangle having a right angle here C is one of the measures of one of the interior angles and then a is a measure of the other interior angle so we can write something very similar we can write we can write a so that measure plus C plus C plus 90 degrees plus 90 degrees is going to be equal to is going to be equal to 180 degrees and actually the last time around I wanna I should write the C in that purple color it's very close to the magenta color but the C is in that purple color alright so what can we do here well we can do the exact same process to solve for a if we subtract 90 from both sides and we subtract C from both sides what do we get we get a a is equal to so once again if you subtract 90 from both sides you're gonna get 90 on the right hand side 90 degrees and if you subtract C from both sides you're gonna get 90 degrees minus C now this is interesting be is equal to 90 degrees my see and a is equal to 90 degrees minus C so 90 degrees minus C is equal to a it's also equal to B or we can now say that a we can now say that a must be equal to B that the measure of angle LMK which is B is equal to the measure of angle Ln J which is equal to a