 So first example is we want to prove that these two triangles are congruent first off Let's figure out which triangles we're talking about. So let's name them Now as you remember from other videos naming triangles in order to prove that they're congruent is an important thing It's important to keep the order correct So let's take a look at this triangle on the left. Maybe we could start at angle t here if we follow t And kind of head in the direction of I end of that C We could say triangle t I see well, what is angle t this angle up top here? What does that correspond with? Well in this other to this other triangle off to the right that angle is Made up of the shared side and also the segment with one Congruence mark so the angle that shares one congruence mark and fits the shared side is this angle So we'll begin naming our second triangle with C in Triangle t I see we follow t towards I End up at C in other words We go down the segment with one tick mark and follow the segment with two tick marks in the green triangle the one off To the right. We're gonna follow one tick mark Head towards these second congruence tick marks. So triangle C Is where we'll start head toward a end up at t can't spell apparently T so we want to prove that triangle C a t and T. I see our congruent How are we gonna do that? Well, let's take a look at those congruence marks again. I see I've got one pair of segments Let's call this I see I See and a T must be congruent because they both have these two pairs of congruence marks on them and likewise this segment T I is going to be congruent to segment AC and Then finally just from the way that these these two triangles are drawn. We've got a shared side up the middle and So what that means is I've got the green pair of sides I've got the blue shared side up the middle and I've got the red pair of shared sides So we're going to prove that these two triangles are congruent using side side side. So let's get to it So first off remember we are trying to prove that triangle TIC was congruent With triangle C a t We were given the two green sides are congruent So that's T I and C a and then also we have the two red sides are congruent So we've got what we have given we have what we're trying to prove now. Let's start filling in our three boxes In the previous slide we established that we are going to use the side side side congruence theorem for this proof so we can fill that in as the reason right now And so in order to prove that these two triangles are congruent. Well first off we should name them We wanted to prove that triangle TIC Was congruent to triangle C a t in order to do that We need to first prove that we've got a pair of sides another pair of sides And a final pair of sides that are congruent The reason we need to do that is in this Triangle congruence theorem we have the if part of the sentence the hypothesis and that must be satisfied in order for the then The conclusion to be true So which pair of sides do we have I? Know a t is congruent to I see and the reason that I know that to be true is that's given information Likewise the green segments segment ti and ca are congruent for the same reasons you've got given given information and Then lastly we need to talk about the shared side. So that shared side is kind of strange in triangle TIC in Triangle TIC I see that this segment is if we follow the order of the letters then the segment would be called TC however the corresponding segments in Triangle C a t Would be instead of TC. It would actually be C T and That that follows from the order of the letters in the names of the triangles So we have that shared side and we need to talk about the fact that that shared side gives us congruent segments So TC is shared by both triangles and as a result It must be congruent in both triangles and the statement that we use is called the reflexive property of equality And now we're done each of our different puzzle pieces has fallen into place We've proven what it is. We're trying to prove and now we're good to go Proof number one done