in this video we're going to talk about two column proofs as it relates to angles so let's draw a picture first okay that does not look like a straight line let's try that again so we're going to say this is angle one angle two and angle three so we're given that angle one is congruent to angle two our task is to prove that angle one is congruent to angle three so how can we do it well let's start with a two column proof in the first column we're going to write all of our statements and in the second column we're going to write the reasons for those statements so the first thing you want to write is what you're given that is angle one is congruent to angle two so that's statement one now what else can we say whenever you have two lines intersecting each other they form vertical angles so let's say if this angle is 60 this angle is 60 as well so notice that angle two and three are vertical angles so we can say angle two is congruent to angle three and the reason vertical angles are congruent now if angle one is congruent to angle two and if angle two is congruent to angle three we can make the statement that angle one is congruent to angle three and this is based on the transitive property so another example of the transitive property is if angle a is congruent to angle b and if angle b is congruent to angle c then we can say that angle a is congruent to angle c so that's the basic idea of the transitive property especially as it relates to angles so that's it for this problem now let's look at another example so let's call this angle 1 angle 2 angle 4 and angle 3. let's say that we're given that angle 1 is supplementary to angle 2. and also angle three is supplementary to angle four and also angle one is congruent to angle four your task is to prove that angle two is congruent to angle three so feel free to design a two column proof that proves the statement so if you want to pause the video now is the time to do it now the first thing we should do is write the statements that were given so number one we're given that angle one is supplementary to angle two so that's a given and then number two we're given that angle three is supplementary to angle four so that's given as well and also number three angle one is congruent to angle four so that's given so what can we do at this point now let's use numbers now we know that angle one and four congruent so let's say if angle one is a hundred degrees that means angle four is a hundred now notice that one and two form a linear pair that means that they add up to 180 they're supplementary so if angle 1 is 100 angle 2 has to be 80. and if angle 4 is 100 and 4 and 3 form a linear pair which means they're supplementary angle 3 has to be 80 as well so therefore we can make the statement that angle two is congruent to angle three and the reason is that supplements of congruent angles are congruent now let's see if we can make sense of that reason so let's focus on a sentence so what are the supplements that are talked about in that sentence the supplements are angle 2 and angle 3. angle 2 is supplementary to angle 1. angle 3 is supplementary to angle 4 so the supplements that is angle two and three of congruent angles the congruent angles are one and four because in statement three we said one and four congruent so supplements two and three of congruent angles which are one and four are congruent so if two and three are supplements of these congruent angles then two and three are congruent that's what that sentence is saying so hopefully that makes sense to you and that's how you can prove that angles two and three are congruent because they're supplementary to congruent angles let's work on another problem so this is going to be a b c and d so this is going to be angle one angle two three and four so given that angle bac is congruent to angle bca and that angle two is congruent to angle four go ahead and prove that angle one is congruent to angle three so you can try this out if you want let's start with a two column proof so what's the first thing that we should write so we should always write what we're given and that is angle bac is congruent to angle bca and that angle two is congruent to angle four so that's given now what else do we know the next thing that we need to do is make the statement that the measure of angle bac is equal to the measure of angle bca we can say that the measure of angle two is equal to the measure of angle four and the reason for this definition of congruent angles now let's move on to step three notice that angle b ac is the sum of angles one and two and also b c a is the sum of angles three and four so we could say that the measure of angle bac is equal to the sum of the measures of angle one and angle two we could also say that the measure of angle bca is the sum of the measure of angle three plus the measure of angle four as we highlighted earlier and so what reason can we use for this part this is going to be the angle addition postulate now number four since the measure of angle b ac is equal to the measure of angle bca we could say that the measure of angle one plus the measure of angle two is equal to the measure of angle three plus the measure of angle four because they equal to these two things which are equal to each other so the measure of angle one plus the measure of angle two is equal to the measure of angle three plus the measure of angle four and so this is based on the substitution property now notice that angle two is equal to angle 4. so therefore we could subtract these two angles from both sides because they're equal to each other so in step 5 we could say that angle 1 or the measure of angle 1 is equal to the measure of angle three based on the subtraction property and finally we can make our final statement that if the measure of angle one is equal to the measure of angle 3 then angle 1 has to be congruent to angle 3 based on the definition of congruent angles and so that's it that's how you can prove this statement so that's it for this video thanks for watching you