Congruence and similarity | Worked example

- [Instructor] We're told in the figure above AB, so the length of that side right over there is equal to DE, so let me mark that also so AB is equal to DE, and angle A has the same measure as angle D, so angle A has the same measure as angle D. Which of the following statements, if true is not sufficient to show that triangles ABC and DEF are congruent? So we have to figure out which of these is not sufficient, and we only pick one of these, so pause this video and see if you can figure this out. Okay, so let's go through them one by one, and if any one of these is sufficient to show that these triangles are congruent then we would rule that out because we wanna figure out which of these choices which is not sufficient to show that the triangles are congruent. Okay, so choice A, AC is equal to DF, so this is saying that AC is equal to DF, so that's saying that we have a side, and then an angle and then another side that is congruent to a side, angle and another side. Now you might remember from geometry that side, angle, side which is a way of proving congruence, so this one actually would be sufficient to show that they're congruent, so we would rule it out. This is an interesting way that they asked this. Alright, this next one BC is equal to EF. BC is equal to EF, so BC is equal to EF right over there. Now is that sufficient for congruence, so this would be angle, side, side which is not a nice abbreviation, so we would say side, side, angle, and you might remember from geometry that this is not sufficient, and one way to think about this is that this angle up here, the measure of angle B, and the measure of angle E isn't necessarily fixed, and those aren't necessarily congruent, and so side BC or angle or side EF you could move them out. You can move them out at different angles, and so that would make DF different lengths, and so side, side, angle is not sufficient to show that the triangles are congruent, so I like this choice a lot, but we could look at the other ones, and see why they are sufficient to show that the triangles are congruent. Angle B has the same measure as angle E, so angle B has the same measure as angle E. Yep that is angle, side, angle. which is a way to show congruence and think about this, if this angle and this angle are converted to this angle and that angle and that side, and between them are the same, well, there's no way that you could move around the other two sides to get anything other than these triangles being congruent, so we'd rule that out. This is sufficient to show congruence. Angle C has the same measure as angle F. Angle C is the same measure as angle F. Yeah, because here, and I know this is getting confusing now, and actually let me erase some of this other stuff from some of the previous examples, so now we're saying that the measure of angle C is equal to the measure of angle F, and think about it this way, if you have two angles that are congruent, that means that the third angle is going to be congruent as well, so you know that they're similar, and then if one set of corresponding sides are congruent, and they're similar triangles well, then you know you're dealing with the congruent triangles, or another way to think about it is. This is angle, angle, side which is also a legitimate way of proving congruence, so this is sufficient, so we rule it out. Triangles ABC and DEF are equilateral triangles. Well, that also would be sufficient for showing congruence. Why is that? Because if they're equilateral triangles, then that means all of those sides have the same length. That side and that side all to have the same length, and so if all of the sides of the triangles are the same then you are going to be congruent, so since that is sufficient to show congruence, we'd rule that out.