hey welcome back this is Tammy and I do math for coffee and today we are reviewing topics that may show up on a first semester geometry final exam I picked 20 topics and you can jump around using the time stamps that are down in the description and I've put the topic in each time stamp but we're going to go through these in order and we are starting right now for this one this is the segment Edition postulate and you're supposed to find the length of segment PQ and you have this diagram and what we know is that the length of P to R is 19 long so that means what you'll do is you add 2x minus 2 2x minus 7 and set up an equation set it equal to 19. now we solve they're going to do some combining of like terms the two x's are going to go together and the two constants are going to go together add 9 to both sides and we end up with 4X equals 28 divide both sides by 4 and x equals 7 and if this is a multiple choice question you're looking at something like this that will be one of the two choices but it will be wrong and the reason is is because they wanted you to find the length of p q so you need to substitute that back in 2 times 7 minus 2 that will be the length of PQ so that's going to be 14 minus 2 so p q is 12. this problem is about angle addition postulate so the given information is at angle arq is X Plus 45 and angle SRA is X Plus 86 and the entire thing which is angle SRQ is 109. so when you add those two little angles together you are going to set that equal to 109. okay watch go through the algebra here real quick all right so now we know that x equals negative 11. now does that make any sense well it might because we're not solving for an angle here we're just solving for x substitute the negative 11 in for X in both of the smaller angles and see what we get there it is 75 and 34 is 109 we were supposed to get that the negative 11 for X works this is kind of a vocabulary question because we want to check and see if you know what these words mean complementary linear pair vertical or adjacent here we have two angles A and B that are right next to each other and on the other side of them you see a right angle well that's 90 degrees that means that angle A and B together must also add up to 90 degrees so these two are complementary now if they were a linear pair they would add up to 180 and if they were vertical they would be across from each other and they would be equal in size adjacent means they're just sitting right next to each other and it is true that A and B are adjacent they are right next to each other but the better answer is complementary this question is testing to see if you understand central angles or that an entire circle has a degree measure of 360 Degrees we can set this up 227 plus the missing angle B 2 plus 69 equals 360 combining like terms we end up with B plus 296 equals 360 subtract 296 from both sides so B must be 64 degrees this question is about complementary angles or even just angle addition postulate again because that little red square means they add up to 90 degrees so 3x plus 2 plus the 40 equals 90 and then we're going to solve the algebra and we end up with x equals 16. and to check your answer put 16 in for x or 3x plus 2 is and see if that all works out and it adds up to 50 and so 50 plus 40 equals 90. PW is an angle bisector that angle bisectors split angles exactly in half you end up with two smaller angles that are equal in size so find X if the measure of angle 1 is 17x and the measure of angle 2 is 18x minus 1. well even though those algebra expressions are different we know that the size of these angles has to be the same so 17x equals 18x minus 1. we subtract 18x from both sides and we end up with negative one x equals negative one dividing both sides by negative 1 we end up with x equals one I'll leave you to do the check this question is about vocabulary around the idea of two parallel lines in a transversal so they're asking you what is X and Y well X and Y in this diagram they're not the same size and they're also on the same side and they're also in between so they are clearly same side interior same type of problem but now you're going to be doing a little bit of math enough find the value of x that makes lines u and v parallel well in order for these two lines to be parallel these two are supposed to be supplementary they're going to add up to 180. so X plus 15 plus 70 equals 180. now instead of combining like terms here I I wanted to do it a little differently to show you that doing it a little differently will also work so I subtracted 70 from both sides right away partly because it's easier to do with Mental Math 180 minus 70 is 110. so now I have 100 X Plus 115 equals 110 subtract the 115 and I get x equal to negative 5. I'm always a little nervous about getting negative answers in Geometry so let's check that one we know if you add negative 5 to 115 you get 110. now that 110 plus the 70 that's down below it yes that does add up to 180. now we have to do the same thing again we're going to find the value of x that makes the two lines parallel but now we have a different pair 9x minus 4 and 7x plus 18. those are called alternate exterior angles even if you forget the name of the pairs if you remember the math relationships you'll be fine with a problem like this so we know that those two are going to be equal to each other almost all these angle pairs are equal except for the same side interior alternate exterior angles are congruent 9x minus 4 equals 7x plus 18 subtract 7x from both sides we get 2x minus 4 equals 18 add the 4 to x equals 22 x equals 11. let's do a quick check to make sure that's going to work out that angle is 95 and this angle is 95. yes classify each triangle by its angles and sides all three sides are different lengths that means that this is a scalene again this is a vocabulary problem if you look at the multiple choice scalene is a pretty safe option so we know it's scaling now we just need to figure out what it is based on the angles all the angles inside this triangle are less than 90 degrees that makes it an acute scaling the answer is B this problem uses the Triangle Sum Theorem which says that all three angles inside of a triangle have to add up to 180 degrees so that's what I'm going to do X Plus 87 plus 31 equals 180 combining those numbers I get X Plus 118 equals 180. I subtract that from both sides and I get x equals 62 degrees this problem uses the exterior angle theorem now this is the fastest way to do it but there's more than one way to do this problem the exterior angle theorem says that the two angles that are farthest away when you add them together are going to equal the angle that's on the outside so 61 plus 30 equals 91 and 91 is the correct answer there's also a couple ways to do this particular problem but the way I saw it was I saw two congruent triangles that means that the 7 and the 2x plus 25 are actually equal to each other so doing the algebra here I end up with negative 18 equals 2x so x equals negative 9. we have a couple of things going on here we have two isosceles triangles sitting next to each other but they also make a 90 degree angle up there in the corner but the only number they gave us is this 50 degree angle that's over on the outside and we are supposed to find x and x has something to do with angle two so the measure of angle two is X plus 114. I can't even use that right away I mean I really need to dig in here and start to figure out what all these angles are and you have to start with the 50 degrees the angle across from the 50 degrees is a vertical angle those are the same size so that's also 50 degrees now if you look at this triangle it is an isosceles triangle base angles and isosceles triangles are congruent so if I know that one of them is 50 the other one must be 50. okay so now I'm over here where the right angle is and I know that part of it is 50 degrees so the other part of it has to be 40 degrees because they have to add up to 90. now we're looking at the isosceles triangle that's here on the side so if that's 40 then this one down here on the bottom is also 40. well those two add up to be 80. and since in triangle has to add up to 180 degrees total that means that angle 2 as an actual angle measurement has to be 100 degrees 100 plus 40 plus 40 is 180. okay now we're cooking so the 100 has to equal X Plus 114 because they told us sitting angle two equals X Plus 114. let's do some algebra subtract 114 from both sides we end up with x equals negative 14. I'll leave you to do the check if you want to this is one of my favorite topics and that is using the converse of the Pythagorean theorem we know that the Pythagorean theorem c squared equals a squared plus b squared is to help you find sides in a right triangle but what the converse is telling us that if you find c squared a squared and B squared you can also figure out if it's a right triangle or an acute triangle or an obtuse triangle if c squared comes out to be less than a squared plus b squared it is an acute triangle if c squared comes out to be greater than a squared plus b squared you have an obtuse triangle the trick here is you have to figure out which side is longest because that's going to be your C grab the calculator and I checked that the square root of 7 is about 2.6 so if that side's about 2.6 the other side is 3 4 is actually the C so c squared is going to be 4 squared and a squared plus b squared is going to be the square root of 7 squared plus 3 squared well 4 squared is 16 and root 7 squared is just 7 3 squared is 9. and what we end up with is 16 equals 16. because they're equal we know that this is a right triangle let's get into some congruent triangles if you need some practice with proofs and congruent triangles check out the link above and I'll put it in the description too state if the two triangles are congruent if they are State how you know well the how you know is the theorem well these two angles are congruent to each other so we have an angle the other two angles are congruent to each other so there's another angle so we know we're going to be using two A's for sure it's going to knock out A and B between the two congruent angles these two triangles are sharing a side so it is really angle side angle which is none of these congruent triangles again but in in a different way state what additional information is required in order to know that the triangles are congruent for the reason given the reason is side angle side I look at the triangles as given to me and I see that I have an angle that's congruent in a side that's congruent and I know that they want side angle side so it has to be in that order that a the angle has to be between the congruent sides side LM has to be congruent to side u t if side angle side is going to be true go look at our choices and it is d l m is congruent to UT we have a quadrilateral here one of the things we know is that the sum of all the angles in a four-sided figure has to equal 360 degrees it's very similar to what we know about triangles where all the angles have to add up to 180 so to make my job a little easier I'm going to add 64 and 108 that comes out to be 172. and I know that 15x and 22x is 37x 5 minus 2 is 3. so I have 37x plus 3 plus 172 has to add up to 360. I only did that that way just to save some space you could write them all out and then start combining like terms there's nothing wrong with that putting together the 3 and the 172 we get 37x 175 equals 360. subtract 175 from both sides 37x equals 185. we have to divide that by 37. let me grab a calculator and the answer equals five now they didn't ask us to solve for x this is one of those questions you'll need to be careful about because after all that work you just want to be done but they actually asked us to find the measure of angle V and V is 22x minus 2. so I got to substitute in that 5 do a little math and we get 108 degrees find the measurement indicated in each parallelogram this one feels like it's the same and it also feels like they didn't give you enough information so there's something you're supposed to know about parallelograms these behave a little nicer than just regular four-sided figures in a parallelogram if you add any two consecutive angles they have to add up to 180 because the whole thing is going to be 360. so I'm going to add these two together the 5x plus 3 and the 18x minus 7 and I get 23x minus 4 equals 180. 23x then equals 184 we divide by 23 and we get x equal 8 and like the last problem X is not what they were looking for you have to find it but you have to go a little farther because they want to know the measure of angle J so we're going to substitute that 8 into 18x minus 7 and we get 137 degrees again we're working with parallelograms and in parallelograms the diagonals actually bisect each other they won't be the same length length because one's going to be longer and one might be shorter but cut each other in half so we're told that DG equals 24 and that GB equals 2x plus 24 and when we mark this up we know that d g and GB are actually equal to each other so 2x plus 24 equals 24 you subtract 24 from both sides you get 2X equals 0. so X must equal zero it can happen It's Tricky and it's going to make you second-guess yourself but all the math is correct so that is the correct answer to review some more for your geometry final click into this playlist next