Proving Parallel Lines With Two Column Proofs Geometry Practice Problems

in this lesson we're going to focus on proven parallel lines so let's say if we have two lines let's call this line and line and these two lines are cut by transversal line which we'll call for transversal and let's say this is angle 1 2 3 seven and eight now the first thing you need to know is that if angles three and six are congruent so let's write that if angle 3 and 6 are congruent then the two lines are parallel so is parallel to now angles three and six they're known as alternate interior angles notice that they are on alternate sides of the transversal angle three is on the left side of the transversal and six is on the right side so they're on alternate sides now notice that three and six are in the interior of the two parallel lines and they're on the inside of those lines they're in between them so therefore three and six are alternate interior angles and so if the alternate interior angles are congruent then the lines must be parallel so you can write it this way the converse of alternate interior angles if they're congruent then the two lines are parallel now this symbol means that line and line are parallel so this is the symbol for parallel lines and that's the symbol for perpendicular lines now the next thing we need to talk about are alternate exterior angles so angle 1 and angle 8 are alternate exterior angles if these two angles are congruent then line is parallel to line so you can write it this way if the alternate exterior angles are congruent then the lines will be parallel so this is the converse of the alternate exterior angle theorem so let's focus on one and eight there are alternate angles because they're on opposite sides of the transversal one is on the left eight is on the right and notice that they're outside of the two parallel lines they're not on the inside they're on the exterior of those parallel lines exterior means it's on the outside interior means it's on the inside and so if you can prove that two angles are alternate exterior angles and if they're congruent then you know that the lines are parallel the next term that you need to be familiar with is corresponding angles angles 2 and 6 are corresponding angles so if those two angles are congruent to each other then line will be parallel to line so you could describe it this way so if the corresponding angles are congruent then the lines will be parallel and forgot the word so you can add that there now there's one more term you need to be familiar with and that's same side interior angles so four and six they're on the same side of the transversal that is they're both on the right side of the transversal and if these two angles are supplementary so if angle four plus angle six it if they add up to 180 meaning that they're supplementary then the two lines that is and are parallel to each other so that's something else that you want to keep in mind so if you have alternate interior angles such as three and six or 4 and 5 those are alternate inter angles the two lines are parallel if you have alternate exterior angles like 1 and 8 or 2 and 7 then is parallel to if you have correspondent angles such as two and six or one and five that's corresponding or you could say three and seven is corresponding and four and eight they're corresponding angles then the lines are parallel let's work on an example problem so let's say if we have this picture and let's say that this is and and in this problem you're given the following information so first you're given that is congruent to dc that's the first given and also you're given that is congruent to bc so using that information prove that line or segment if you want to use that is parallel to dc so go ahead and try this problem feel free to pause the video if you want to work on it so let's make two column proof statements and reasons so what's the first statement that we can make it's always good to start with given so we know that is congruent to dc and so that statement is given to us and let's mark it on figure so this is and it's congruent to dc now in step two we know that ad is congruent to bc and that is also given to us so that's number one number two and this is it's congruent to bc now what's our next step where do we go from here now let's look ahead our goal is to prove that is parallel to dc and in order to do so we need to prove that these two are congruent to each other those are alternate interior angles so if you look dc and if we're trying to prove that those lines are parallel then we can see that db is the transversal which makes these two angles alternate interior angles and those angles will be congruent if these two triangles triangle and triangle if we could prove that they're congruent then we could prove that those two angles are congruent and therefore the two lines so that's mental outline of what we need to accomplish in this problem so in order to prove that the two triangles are congruent we need third side and that is db db is the common side between both triangles so we can say that db is congruent to itself based on the reflexive property now let's move on to number four so now we can make the statement that triangle is congruent to triangle now what's our next step now let's make sure we put reason for this the reason is side side side it's the sss postulate and we've used statements one two and three to prove that so now we could say that angle abd that's this angle is congruent to angle cdb which is that angle and the reason for that cpctc now we can make our final statement that is is parallel to dc and because these two angles are congruent they are alternate interior angles and if the alternate internal angles are congruent then we know that the lines are parallel so we can write this statement so whenever the alternate interior angles are congruent then the lines the two lines are parallel or we could say the converse of the alternate interior angle theorem so that's one way in which you can prove two lines of parallel and typical two-column proof thanks for watching you