what's up everyone welcome to this lesson where we're going to explore the different triangle congruence theorems so let's start out by asking the question what does congruence even mean we like to think of it as this equal sign with the squiggly line over it let's just take a few moments to really conceptualize what congruence means especially in terms of figures so let's look at a running shoe if I can somehow duplicate this running she would make an exact replica I would say that these two shoes are congruent because they are exactly the same size and shape now even if I change the color of one of the shoes or I rotate it around a certain number of degrees it's still congruent to the original sneaker because it's the same size and shape and we want to extend this kind of thinking now to comparing triangles to see whether or not they are congruent if their corresponding sides and angles have the same measure then we can say that the triangles are congruent and this applies even when the figures don't have the same orientation and visually it's not so easy to tell whether or not these triangles are congruent and this is where our theorems are going to come into play so the goal of this lesson is to help you to understand why the theorems that we use prove congruence and how to know which theorem to use in a particular situation so here are the five triangle theorems with their abbreviations we have side side side side angle side angle side angle angle angle side and hypotenuse leg so now we're ready to go ahead and explore each theorem individually now the side side side triangle congruence theorem states that if in two triangles we have three pairs of corresponding sides that are congruent to each other that means that they have the same length then that's enough information to say that the triangles are congruent to each other which again just means that both triangles are exactly the same size and shape so in this instance we can use the side-side-side theorem to conclude that the triangles are congruent and that their corresponding sides and corresponding angles have the same measure and remember that this relationship is still true even if the figures have different orientation our next theorem is side-angle-side now for this triangle congruence theorem we start with a pair of corresponding congruent sides and two triangles we also have a pair of corresponding angles that are congruent and another set of corresponding sides that are congruent now see that the letter a is in between the two S's in the name of the theorem this corresponds with that congruent angle being in between the two congruent sides now when this is the case we have enough information to say that the two triangles are congruent by using the side-angle-side theorem our next congruence theorem is the angle side angle theorem when we have two triangles with a pair of corresponding congruent angles a pair of corresponding congruent sides and another pair of corresponding congruent angles we have enough information to prove congruence using angle side angle now it's important to notice that the congruent side is in between the two congruent angles and now we can visualize that when this is the case we have enough information to prove that the two triangles are congruent by angle side angle which means that all of their corresponding sides and corresponding angles have the same measure now our next theorem is very similar to the one that we just looked at which was angle side angle this one is called angle angle side now when in two triangles we have a pair of corresponding congruent angles along with a second pair of corresponding congruent angles as well as a pair of corresponding congruent sides now we don't want to confuse the last theorem angle side angle and this theorem of angle angle side because both of them involve two congruent angles and one congruent side the difference is that an angle angle side the congruent angles are consecutive that is that they are one after the other and then the congruent side is next to them and setup in between them now even though the order is different we still have enough information here to prove that these two triangles are congruent which means again that their corresponding sides and corresponding angles have the same measure now our final congruence theorem is kind of a special case it's called hypotenuse leg and it applies to right triangles only so let's start out with the key feature of any right triangle that is a right angle now we know that the hypotenuse of a right triangle is its longest side and is always opposite the right angle so this congruence theorem says that if the hypotenuse in both right triangles are congruent and if a leg which is one of the other sides are congruent that is they have a pair of corresponding congruent sides and that is enough to say that the two triangles are congruent again which means that the triangles corresponding sides and corresponding angles have the same measure so now that we've covered the five triangle congruence theorems you may be asking what about angle side side please don't be uptight okay so now let's go ahead and explore why angle side side would not work as a triangle congruence theorem so if we have a set of corresponding angles that are congruent and a set of corresponding sides that are congruent okay so we have two sets we have an angle followed by two consecutive congruent sides the issue is with that second congruent side since we don't know the angle in between the two congruent sides we don't exactly know its position so I could take that side and swing it over to a new location and still show that angle side side is still the case here no matter how I choose to draw that second triangle and I have the option of drawing in either way since that's an isosceles triangle so both of those sides are definitely congruent to each other now if I draw it this way as an acute triangle I see that I have angle side side however the base of each triangle clearly are not the same and these triangles clearly are not congruent thank you so much guys for checking us out please subscribe to our YouTube channel for more free animated math lessons updated every week