Welcome to a video on congruent and similar triangles the goals in this video were to determine congruent triangles, determine similar triangles and also to use Similar triangles to solve problems. Congruent triangles have the same size and the same shape. Therefore, the corresponding sides had the same length and the corresponding angles are congruent. So, the side is the same length as the side this side is same length as a side and side is the same length as this side and the corresponding angles are also congruent. Which means this angle equal to this angle this angle equals to this angle and this angle is equal in measure to this angle. Similar triangles have the exact same shape, but not necessarily the same size. So, the conditions for similarity are number one corresponding angles must have the same measure so again this angles is Congruent to this angle those are congruent and those are congruent. But, corresponding sides are proportional. Which means, the ratios at the corresponding sides are equal. So, what that's telling us is, let's say this is eight inches long and this is four inches long the ratio between these two would be to two to one Therefore, if this was ten inches long this would have to be five inches and this was twelve inches long this would have to be six inches the ratio stays consistent with corresponding sides. Here's a nice visual three Similar triangles as I animate one vertex the large triangle the other two triangles remain similar. Meaning, they are the same shape but not necessarily the same size. again these three triangles are Similar triangles So hopefully that gives you feel of what similarity means Throughout various other conditions that can be used to determine if triangles are similar Number one, two triangles are similar if they have three pairs of congruent angles sometimes called Angle-Angle-Angle or AAA but actually Angle-Angle is enough. So, for example here we have two triangles. Where, two pairs of corresponding angles are congruent However, since we know that the sum of these angles must equal 180 degrees we could also just conclude that these two angles with equal seventy degrees and if we have three pairs of congruent angles this guarantees that these two triangles are Similar. Number two, Two triangles are Similar if all three pairs of corresponding sides are proportional. So, what that means is if we take one pair corresponding sides For example, this side nine corresponds to this side six nine to six simplifies to the ratio of three to two and if all the other corresponding sides maintain this ratio they would also be proportional. and therefore, we would have two Similar triangles we can quickly see that fifteen to ten simplifies to three to two and eighteen to twelve simplifies to three to two so we have three pairs of corresponding sides are proportional. Therefore, these two triangles are Similar and lastly, two triangles are Similar, If two pare corresponding sides are proportional and included angles are congruent. So, what that means is, if we know this side was fifteen, and this side was ten this side was eighteen, this side was twelve. We know, these corresponding sides are proportional and because the angle formed by these two sides are Congruent this guarantees similarity. And, this is called Side-Angle-Side or SAS. So, take a look at couple of problems that demonstrate those ideas The two triangles are similar. Determine the measure of angle A and angle B what's first? Write the given information. We know, angle D is thirty-five. degrees and we know angle F is forty-three degrees and we know since they are Similar, the corresponding angles are congruent so angle C is also forty-three degrees an angle A is thirty-five degrees and the question they want us to determine is angle A which we just found but they also want angle B the angle here we know the sum of these angle so we can do is take the total sum 180 subtract thirty-five degrees give us 145 degrees and then subtract forty-three degrees we have 102 degrees so the measure of angle B equals 102 degree okay. On this problem we have similar triangles again, we want to determine the length of sides DE We will call this X and EF which, we'll call Y. and now lets go ahead and mark the given information CB equals fifteen AC equals nine AB equals ten and DF equals six. Okay, since we know, these triangles are Similar, these sides are proportional so we can setup proportions to solve for the unknowns we know the ratio of these two sides would be nine to six in that must be the same as a ratio of ten to x and the same thing for side Y. we can setup the proportion that would say nine to six must be equal to fifteen to y and now I can solve this apportioned to find the length of the missing sides. So, performing cross products here we would have nine x equals sixty. dividing both sides by nine X equals twenty-third which is equal to six and two third. To solve for y, we would have nine y equal six times fifty which equals ninety dividing by nine on both sides we have Y equals ten so we were able to solve for the missing sides because we knew the triangles were Similar. Which means, their corresponding sides were proportional. Now lets take a look at another type of application so for example here we can see that this building is casting a shadow onto the grass and so is this person standing over here but the idea here is that the shadow forms a triangle and these two triangles are Similar even as time passes the length of that shadow is going to increase but the result is to similar triangles so what this means is that the sides are these two triangle will be proportional. So, we could have proportioned that will compare corresponding sides have these two similar triangles let's take a look at the problem like this We have a tree that casts a shadow forty-five feet long, and at the same time the shadow cast by a vertical three-foot stick is five feet long. so here we have a three foot stick which cast a shadow five feet long and again the shadows forty-five feet so we have our too Similar triangles that will allow us to find the height of this tree without ever having to climb it because we know these size of proportion so X is to three as forty-five is to five so now I can solve for the unknown by performing cross-product five x must equal 135 dividing by five X equals twenty-seven feet so without ever having to climb this tree we were able to use Similar triangles to determine that the high of this tree is twenty-seven feet I hope you found this video helpful. Thank you for watching