 Can you prove that these two triangles are congruent? First off, let's go through and name the two triangles. So is it possible for us to prove that ABD and CBD are congruent? Well, these tick marks right here show me that AB and CB are congruent. Also, I've got a right angle right in there. And if that's a right angle, well, this angle and that angle, they're both right angles because we've got some perpendicular lines. Well, perpendicular lines make right angles and right angles are always congruent to each other. So that means angle ABD. So right now we have a pair of sides and a pair of angles. Then notice that this side length, BD, is a shared side. It's shared in both of the triangles. So that means in triangle ABD and CBD it's got to be the same in both of them. So yes, we can prove that these two triangles are congruent and we can prove it because of side angle and then that shared side. So we can prove that these two triangles are congruent using the side angle side there. What about these? Can we prove that these two triangles are congruent? First off, let's name them. So we have triangle DEF and DGF. Well, let's figure this out. We've got a pair of sides that are congruent that refers to these segments. So let's label those. We've also got a pair of angles that are congruent. Those maybe I'll just number them angle 1 and angle 2. So now we have a pair of sides and a pair of angles. We also have, just like in that previous example, we've got this shared side up the middle, DF. So that means that in both triangles DF will be congruent to itself. So we've got a pair of sides, a pair of angles and we've got that shared side. So it seems like this might follow side angle side. However, the angles, angle 1 and angle 2 are not included angles. And again, just a reminder of the definition of an included angle. The included angle must touch both sides that we know about or that we care about. In other words, angle 1, in order for us to use angle 1 as an included angle, did I say angle or side? In order for us to use angle 1 as an included angle, it would have to be created with this segment EF, but it's not. That red angle only is made up of this blue segment, the shared DF. And likewise, angle 2 doesn't touch segment GF. So these might be congruent, they might not be congruent. We can't prove that they're congruent. So they're not provable. Let's try another example. What about these triangles? Let's name them first. So we've got DEF and GHF. So what does it mean for two segments to bisect each other? Well bisect means cut in half. And so each of these segments is cutting the other one in half. I know EH is bisecting DG. So that means EH is doing the cutting and DG is being cut. In other words, DG is cut into two equal parts. And then also, since they bisect each other, DG bisects EH. In other words, EH is cut right in half. So I have two pairs of congruent segments that come from this fact. And then one last thing, whenever we have lines that intersect, we create these vertical angles. So I could talk about angle DFE and angle HFG. Vertical angles are always congruent. And so that means that those two angles are congruent. So DFE. So we have two pairs of sides, the reds, the greens, and a pair of angles. So it's possible that we can use side angle side. The only thing that's left to check is are these blue angles, are they the included angles? Well, how can we check? The way that we can check is the blue angle connected to both pairs of sides that we know about. In this case, they are. I see that blue angle touches both the red and the green. And as a result, that blue angle is made up of both of these sides that we know something about. So yes indeed, these triangles are indeed congruent, and they are congruent by SAS. How about these triangles? Can you prove that they're congruent? Well, I got one pair of sides. I got two pairs of sides. And I got the shared side up the middle, US. And so that means yes indeed, I can prove that these triangles are congruent by side, side, side. So here we have congruent triangles. Let's try another. All right, are these two triangles congruent? So it looks like from the drawing, we have that angle U and angle R are congruent. We know that T is the midpoint of RU. And RU refers to this segment from R to U. And so T is the midpoint, means that these segments are congruent in length. So we have a pair of angles. We have a pair of sides. We need one more pair of angles in order for these pairs of triangles to be congruent. We have intersecting lines, which means vertical angles are produced and vertical angles are always congruent. And so that means we have angle, side, angle. And so therefore we can prove that these two triangles are indeed congruent by ASA. Let's try one more example. Can we prove that these two triangles are congruent? Well, let's figure out the given information. We've got a given pair of angles, angle N and angle L are congruent. I got a pair of segments, pair of sides that are congruent, JK and MK. And the way that the triangles are drawn, we've got a pair of intersecting lines, sorry, intersecting line segments. And whenever line segments or lines intersect, we make vertical angles. And vertical angles, of course, are congruent. So we have two pairs of angles and a pair of sides. We have a choice between angle, side, angle or angle, angle, side. Which one do we pick? Well, the question is, is the side, is it included or not? And by included, I mean, is that side touching both of the angles? Well, JK is only connected to one pair of angles. It's kind of leaving angle N all by itself. Likewise, segment MK is connected to this middle angle, but it sort of leaves angle J by itself. So that means the sides are not included, but we can prove that the two triangles are congruent by angle, angle, side.