hey welcome to prealgebra lesson 10-3 draw triangles with given conditions in this lesson we'll be able to draw triangles when given information about their side lengths and angle measures so let's start with solve and discuss it cane has four pieces of wood available to build a triangle-shaped garden which pieces of wood can he use so we have all these different types of um different pieces of um wood right um we want to build a triangle shaped Garden which pieces of wood can he he use so the question is which pieces of wood can he not use like why would there be a piece we can't use right so in you need to think about this in order to be a triangle right it you need to have three sides and you need to have um yeah I mean that's very obvious but think about these two Woods the shortest Woods right the shortest pieces of wood um two and 3 feet could we use 5T you can you you can say yeah but think about it we have two feet here 3et here if we have 5et it's probably probably not going to be enough because they need to be flat 2 + 3 is 5 so this is not going to be a triangle it is going to be a straight line right so yes there are pieces of wood we cannot use at this point right so if two sides add up to be the third side we cannot use it in the two sides must be bigger than the the other side right so yes um other than two three and five um there could be yeah other possibilities like two three 4 is that okay yes the shorter sides are going to be what is that also okay two and three we have two and three right so 2 3 then this side could be shorter yeah so five may be a little bit too long yeah I said it the wrong way but five is going to be too long two three cannot meet the 5T piece right so in order to for them to meet it's it has to be straight but if it's shorter than 5T yes they can meet um yeah so the two sides if you add the sum of two sides must be smaller than the their sum so the other so wait the other the third piece must be smaller than the sum of the other two sides otherwise if it's bigger the two other the two sides cannot meet the third side like cannot meet the the end point the other end point of the third side right so yes um so you yeah you can do that but you can't do five ft with the two and3 feet two two and three ft right so yeah there we go this is um one combination there could be other combinations three four and five does that work the shorter pieces the shortest pieces three and four add up to seven and five is smaller than that so yes we have a triangle three four and five that works too but two three and five just do not work together okay yeah so 2 three and four 2 three and four could that be yeah 2 three and four Works 235 does not work and then 2 4 five what about 2 4 5 so 2 + 4 is 6 and five is smaller than that so yes two and four and five works yes and then 3 four five works too so the only thing that does not work would be two three and five yeah okay so they gave us a recommendation try all possible combinations of three pieces of wood and we did that the only combination that does not work would be two three and five let's look at focus on math practices are there any combinations of three pieces of wood that will not create a Le yes we just talked about that let's explain so yes we cannot use the pieces of wood that are 2 feet 3 ft and 5 ft to build a triangle because why because the two feet long p piece and 3 feet long piece so two feet let's make this simple two feet piece and three feet piece lie flat against the 5 ft piece okay so yeah there is definitely a rule about the size of a triangle right so you learn this again in Geometry there's a theorem about this um let's look at the actual lesson so the essential question for this lesson is how can you determine when it is possible to draw a triangle given certain conditions so we're going to talk about triangle in this lesson let's look at example one draw triangles with given side lengths students in wood shop class are measuring and cutting out a triangular base for a corner shelf with sides measuring 6 in 8 in and 10 in how can you determine if all the students will cut out the same triangle so we're going to see if um there are any possible ways that they could cut in different ways right if we have a combination of six 8 and 10 right so we could have six and eight have a right angle make a right angle and then the third side to be 10 units um and then we could we could say yeah using geometry software try different ways what if we have um six and 10 together six and 10 together but if they're writing this side must be longer than eight so that doesn't really work right so the only the only triangle that works would be where six units and eight units will make a right triangle and then the third side being 10 units and if you yeah if you draw them in different order you may get different orientation that means different um ways to look at but they're actually the same triangle so triangles with the same side lengths are the same shape and size so no matter how they're positioned they'll all have the same triangle okay so that's very important to note let's look at try how many unique triangles can be drawn with given side lengths of 8 in 10.3 in and 13 in we just figured out that any triangle with the same sides are the same shape and sides no matter how they're positioned so they're the same triangle so if we have all same sides that are exactly 8 and 10.3 and 13 in we would probably have one unique triangle so convince me when two sides of a triangle are switched why is it still considered the same triangle so what if we have this triangle and then that triangle like and then we have these two sides that are switched right so here we're switching eight and six six and eight was here and we're switching this side to be eight and then this side to be six this side is still 10 and if the sides are all together are they still the same triangle because they're the same sh shape and size of the sides are all the same right so let's write that down switching the sides can also be accomplished by flipping the original triangle turning it or a combination of both flipping and turning they're still the same size okay example two determine possible side lengths of triangles Steve gathers three pieces of wood from the scrap Pile in Wood Chop class can Steve make a triangle with these three wood pieces three four and eight so we just need to check the shorter sides actually three and four you add up to seven so the longest that they could make together would be seven right so even if they lie flat three four they would have seven alog together seven feet so there's no way we could have 8 ft um with this Thum and it's not going to be a triangle so you will see like missing pieces over here right so what can you conclude about the lengths that make a triangle possible so it cannot be exactly equal the sum cannot the sum of the two shorter sides cannot be equal to the longest side it has to be smaller than um the sum has to be bigger so the third side has to be smaller than the two shorter sides so this is another rule that we must remember let's move on to example three draw a triangle with a combination of given side lengths and angle measures can more than one triangle be drawn with the following conditions side lengths of 5 in and 6 in with which make an angle of 45° so let's try to try to draw that five inches and six inches we have exact sizes for those um sides and then they need to make 45 degrees then you'll have one side that could be exactly that right so actually there is no other way to have other triangles because our angle is fixed and then our sides are fixed and so the end points are fixed so no matter which way you draw it's still going to be the same triangle so only one triangle can be drawn with the given measures here let's look at Part B a side length of 6 in with angles that that at each end measuring 40 and 60° so we can use um 6 in then 40 and 60 here we can maybe try to flip that maybe 60 could be here and 40 could be here but it's still going to be at meeting at the same point actually so this is just a reflection of this triangle but they're actually the same shape the same size so only one triangle so if you have a fixed um angle with two fixed sides so side angle side this is actually a theorem in Geometry okay SAS there's only one way to draw a triangle like that and also angle side angle so if you have a fixed side and fixed angles for two fixed angles for that one side then there is only one way to draw the triangle two okay so this is the side side side postulate theorem and then SAS theorem and S ASA theorem in order to remember them correctly but if you have um two sides and then a random angle like over here or over here there there are many other ways to draw that triangle this angle must be in between these two sides okay so this one pay a close attention to that angle right there angle must be in between those sides yeah let's look at TR it A and B write three side lengths that will form a triangle right three Sid lengths that will not form a triangle so give an example for each and then Part B can a triangle be drawn with a side length of 3 in and angles at each and measuring 90° and 89° explain so pause a video think about this and try to answer it yourself come back when you're ready for answers okay let's look at part A so give an example um yeah I wonder what your examples are yeah if you could share that in class or read it in your notes or if you're looking at this um from another country maybe write down in the comments I'll be interested to hear your thoughts but three side lengths that will form a triangle could be any combination where the two shorter sides are bigger than the third side okay so you know four if you've picked four maybe 4 cm 5 cm the sum of that would be 9 so it cannot be exactly N9 or bigger than nine the third side must be smaller than nine so seven or eight right so combination of that is a triangle and then combination let's do the same thing 4 cm 5 cm and then it fits exactly N9 or bigger than that it will not form a triangle okay Part B can a triangle be drawn with a side length of 3 in and angles at each end measuring 90 okay and 89 o so this is like almost 90° but it's like not so then it's like going to be super long and so the triangle is going to be really really long um but can it be possible yeah as long as the these two lines meet at some point which they will because this one's 90° and they're not parallel because this one's not 90° it's like 1° like it it's it's not 90° right so then it's going to meet that um side the other side so yes um the end of the triangle would probably just be 1° because the sum of triangle would be 180° uh for the angles right so the sum of the Interior like inside interior angles of a triangle is 180° and so the third side the third angle will will be 1° okay let's look at um example four draw a triangle with two given side lengths and a non-included angle measure can more than one triangle be drawn using two side lengths of six units and 9 units and 40° angle that is not formed by their intersection so draw a triangle ABC with side length six units N9 units and a non-included angle of 40° so yeah so included angle is this one included angle okay non-included angle is 40 here or 40 here okay we just chose this one um you can yeah so this angle could be any angle that angle could be any angle they just need to be um 9 unit and six units the sides the sides must be exactly 9 units and six units so yeah your triangle could look like this your triangle could look like that because this the other two angles don't really matter so more than one triangle can be made with these given measures which I shortly mentioned so this is why s is important this only works uh when you have an angle in between the sides es so there's only one triangle but then if you have just angle in other um non-included so side side and then you have an angle somewhere else uh it could be there could be two triangles so more than one so this one doesn't really work for one triangle let's look at example five draw a triangle with three given angle measures is there a unique triangle with angle measures of 30 60 and 90° so if you have all same angles right um what if cuz we have a fixed if we have fixed side measures then it's going to be one triangle right what about three same fixed angle measures as you can see here the angles can be the same but then the side lengths don't have to be fixed length so then the sizes are going to be different then they're going to be the same shape but they're not going to be the same size so then you can't say they're the same triangle so when we say there's a unique triangle is it going to be same shape and same size so for this one for the same angle yes they will have the same shape but they will not have have the same size so there is no unique triangle yeah so many different triangles so the only the only way you will have one unique triangle is the side side side and then side angle side where angle is included angle and then you have um what else did we look at side side angle side yeah SSS SAS is that the only thing we looked at okay sure so yeah just SSS and side angle side let's look at try it can more than one triangle be drawn with two side lengths of 6 in and a non-included angle of 60° we already know that two side lengths of 6 in and non-included angle of 60 so this needs to be 60 or that needs to be 60 but then you have two of the same sides so pause the video think about this and come back when you are ready for answers okay are we ready so this is a special case when we have two equal sides even though this is not an included angle what do we know because this is six and six and that's 60° the other two this one is an isoceles triangle even if we have this way this the angle must be the same so there's only one way to draw this that CU that must be 60 because the these two sides are equal so then if this angle is 60 and that angle is 60 the included angle must also be 60 because the total sum of the interior angles must be 180 does that make sense so then this is an equilateral triangle so if we have two of the same sides that are exactly the same side and then same side and the angle must be 60° then there is only one way right um yeah yeah so in this case they're cannot be more than one triangle so no if two sides have the the same so let's explain that in words if two sides have the same length then the triangle is isoceles that means you have two same length and has two equal BAS angles base angles just means these two angles on the bottom of 60° 60° um this means only one equilateral triangle with side lengths of 6 in can be drawn okay there so that is all about triangles let's look at Key concept to summarize our lesson you can analyze given conditions of side lengths and angle measures to determine whether one unique triangle more than one unique triangle or no triangle can be drawn so you can have have only one possible triangle given that um all three angles or two sides and a non-included angle wait this is more than one possible triangle so this is not unique if it's all if the angles are fixed and then side side a like there is an angle that is not included then more than one unique not not unique okay and then if there are sides that are fixed all same three sides that are fixed and then side and then angle side then you have angle side angle oh yeah we talked about angle side angle then this is unique triangle okay I hope that was very clear to you this was less than 10-3 draw triangles with given conditions we'll continue with the next lesson in the next video bye