in this video we're going to focus on solving problems associated with triangles so let's start with the one in front of us what is the value of x in the figure below so we have the measure of angle a and b but we need to calculate the measure of angle c which is represented by x now what you need to know is that the three measures of a triangle the three angles have to add up to 180 angle a is 60 degrees angle b is 70 degrees angle c is equivalent to x and so if we combine like terms 60 plus 70 that's 130. now in order to calculate the value of x we need to subtract both sides by 130 and so these will cancel and x is 180 minus 130 which means it's equal to 50 degrees and so that's the answer for number one number two what is the measure of angle b and the figure below so go ahead and try this problem now we know that all three angles of a triangle has to add up to 180. angle a is 4x minus 2 angle b is equal to 10x and angle c is eight x plus six so these do we have to add up twenty so let's begin by combining like terms four x plus ten x is fourteen x and if we add 8x to that that's going to be 22x and then we can also combine negative 2 and positive 6 so that's plus 4. so let's subtract both sides by four 180 minus four is 176. next we need to divide both sides by 22. 176 divided by 22 is 8. so now we have the value of x which means we can now calculate the measure of angle b so angle b is equal to 10x so this is going to be 10 times 8 we've got to plug this in which is 80. so the measure of angle b is 80 degrees this is the answer now what about this problem how can we determine the measure of angle a in this picture so what we have is an isosceles triangle notice that these two sides are congruent so therefore the opposite angles to those sides must be congruent as well so our goal is to calculate the measure of angle a so let's call this angle x now this angle is equal to angle a so that's x as well so we know that angle a plus b plus angle c has to add up to 180 for a triangle so x is a and angle b is 50 and angle c is x so x plus x is 2x so 2x plus 50 is equal to 180. now let's subtract both sides by 50. so 180 minus 50 is 130 and now we just got to divide both sides by 2 and so 130 divided by 2 is 65. so therefore the measure of angle a is 65 degrees and as you can see this problem wasn't too bad it was very straightforward number four what is the measure of angle five and the figure below so go ahead and calculate the measure of this angle if you want to pause the video feel free to try so notice that these two they form a linear pair so that means that they add up to 180 so angle two is 180 minus 120. which means that this angle is 60 degrees now notice that these two angles also form a linear pair so angle 3 is 180 minus 100 which is 80. so now we can calculate angle 4 because these three angles are the interior angles of a triangle so they have to add up to 180. so we could say that angle 4 is 180 minus the other two interior angles so minus 80 minus 60. 180 minus 80 is 100 and 100 minus 60 that's 40. so that's the value of angle 4 it's 40 degrees so now we can calculate the measure of angle 5. because these two form a linear pair they add up to 180. so angle five is going to be 180 minus 40. so it's equal to 140 degrees and so that's the answer the measures of the three interior angles of a triangle are in the ratio two three four what is the measure of the largest interior angle of this triangle so what do you think we need to do in this problem well first let's start with a picture so let's call this angle a b and c now the best way to approach this is to say that a is 2x b is 3x and c is 4x so the three angles are now in the ratio 2 3 4. so now we can solve it so we know that angle a plus angle b plus angle c has to sum up to 180 so 2x plus 3x plus 4x is equal to 180. now 2 plus 3 plus 4 is nine so all we need to do is divide both sides by nine and eighteen divided by nine is two so one eighty divided by nine is twenty so now we have the value of x now our goal is to calculate the measure of the largest interior angle and clearly c is larger than a and b 4x is greater than 2x and 3x separately so angle c is 4x so that's going to be 4 times 20 and so this is going to be 80 degrees this is the answer right here a is 2x 2 times 20 is 40. b is 3x 3 times 20 is 60 and c is 80. so 40 plus 60 plus 80 adds up to 180 and this is the largest of the three angles number six what is the value of x in the figure below now to find the answer you can use the exterior angle theorem so the exterior angle of a triangle which in this case is x is the sum of the remote interior angles which are these two now there's three interior angles but these two are considered to be the remote interior angles so basically x is simply the sum of 40 and 55 so x is 95 now another way in which we can get that answer is by calculating the value of y so these three angles have to add up to 180 so we could say that y is 180 minus 40 minus 55 so 180 minus 40 is 140 and 140 minus 50 is 90 90 minus 5 this is going to be 85 so now notice that these two angles form a linear pair so y plus x is 180 so x is 180 minus y which is 180 minus 85 so 180 minus 80 is 100 and 100 minus 5 is 95 and so you have two ways in which you can get the same answer number seven calculate the value of x and y and the figures shown below so how can we do so well let's start with the figure on the left so m is the midpoint of segment a b as you can see a m and m b are congruent and also b n and n c are congruent so n is the midpoint of b c when you have basically a midline that touches the midpoints of two sides of a triangle and if it's parallel to this side then you could use this formula the length of the mid segment which is mn that's equal to one half of this segment ac so in this case x which is basically the length of segment mn that's going to be one half of ac which is 30. so x in this example is 15. now what about y intuitively you know that y has to be 40. you can still apply the mid-segment theorem here so h k is one half of df or you could say that df is twice the value of h k so df is y h k is 20 and two times 20 is 40. so this is the answer for this problem number eight determine the values of x y and z in the figure below so what do you think we need to do here well let's find x first so notice that these three angles add up to 180 so therefore x is going to be 180 minus 90 minus 30. so 180 minus 90 is 90 and 90 minus 30 is 60. so that's the value of x so this is 60 degrees so now we can calculate the value of y notice that x and y they form a linear pair so we can say that x plus y adds up to 180 which means y is 180 minus x and in this problem x is equal to 60 degrees so 180 minus 60 is 120 and so this is the value of y so now we can calculate the value of z and also these two angles form a linear pair so if this is 90 this has to be 90 because 90 plus 90 adds up to 180. now the four angles of a quadrilateral a four-sided figure must add to 360. recall that the sum of all of the interior angles is 180 times n minus two so for a four-sided figure like a quadrilateral n is four and so four minus two is two so 180 times 2 is 360. so y plus z plus this angle which is 90 plus this angle that's 90 as well has to add up to 360. so our goal is to calculate z now y is 120 and 90 plus 90 is 180 and 120 plus 180 is 300 so z is going to be 360 minus 300 so z is equal to 60 degrees so in this example x and z both have the same measure they're both equal to 60 degrees now we could get that same answer using another technique so let me draw the original picture so these two were right angles this is 30 this is 60 and this is 60. so let's call this a b c and then d and e so notice that e d is parallel to a i mean not parallel but perpendicular to a b and also c b is perpendicular to a b so if you have two lines that are both perpendicular to a third line then these two lines e d and c b are parallel to each other now granted that may not always be true but in this case e d and c b they both lie in the same plane because sometimes you could have let's say a b's in the x direction e d could be in the y direction c b could be in the z direction in that case e d and c b are not parallel to each other but because e d and c b lie on the same plane and they're both perpendicular to a b then e d has to be parallel to c b if they lie on different planes then that's a whole different story so if we can see that eb i mean ed and cb are parallel to each other then that means that these two angles are congruent notice that they're corresponding angles and so if you saw that in the beginning you can easily determine that a z has to be the same as x they both have to equal 60 degrees number nine angle bac is 50 degrees and angle acb is 60 degrees a d and c e are altitudes determine the measure of angle afc so our goal is to calculate the value of this angle now this problem is a little bit harder than the other ones so feel free to pause the video and tackle this problem go ahead and try it so i'm going to focus on the large triangle first that is triangle abc now we're told that angle bac that's equal to 50 degrees and angle acb is equal to 60 degrees so the missing angle b has to be 180 minus 60 minus 50. 180 minus 60 is 120 and 120 minus 50 is 70. so angle b is 70 degrees which i'm going to put here now we're told that ad and c e are altitudes that means that they form right angles with the opposite sides so this is a right angle and this is a right angle as well so now i'm going to focus on triangle abd so i'm going to redraw this way so here's a this is b and this is d now this is a right angle and this is 70 degrees so the missing side must be 180 minus 90 minus 70. 180 minus 90 is 90 and 90 minus 70 will give us 20. so this angle is 20 degrees now let's focus on this triangle so that's triangle b e c so this is a right angle and this is 70. so this has to be 20 as well now notice that angle a is 50 and if this part is 20 this part has to be 30. angle c is 60 and this part is 20 so 60 minus 20 is 40. so now we can focus on these three angles of triangle afc so those three angles have to add up to 180 so angle f or rather afc that's going to be 180 minus 30 minus 40. so 180 minus 30 is 150 150 minus 40 is 110. so therefore angle afc is equal to 110 degrees and so that's it for this problem so this is going to be the last problem of this video what is the measure of angle a in the figure below and we're given angle a b and angle bcd now angle bcd which we can call it y for now it's the sum of the remote interior angles that is angle a and b so y has to equal angle a plus angle b so y which is bcd that's 12x minus four and angle a is four x plus two angle b is x squared plus one so let's take everything on the left side and move it to the right side so on the right side i have x squared plus four x now the 12x was positive on the left but it's going to be negative on the right side and i have plus 2 plus 1 the negative 4 it's negative on the left but it's going to be positive on the right so everything changes sign if you move it from left to right so now i can combine like terms 4x minus 12x is negative 8x and then 2 plus one is three plus four that's seven so i have x squared minus eight x plus seven so now i need to factor this expression so to factor it we need two numbers that multiply to seven but add to negative eight so the only numbers that multiply to seven are one and seven but these two add up to positive eight so we need to make it negative one and negative seven so to factor the expression it's going to be x minus one times x minus seven so let's set each factor equal to zero so in the first one we need to add 1 to both sides if we do x is equal to 1 and for the second one if we add 7 both sides x is equal to 7. now the measure of angle a is four x plus two and let's see which answer makes sense well technically both can work but if we plug in one angle a is going to be four times one plus two which is six so that's one possible answer the other possible answer it could be four times seven plus two which is thirty now 30 seems like a more reasonable answer but it can be six as well so if we calculate the other values let's say angle b that's x squared plus one and we said that x could be one or seven so if it's one it's going to be one squared plus one and so that's going to be two now if we choose seven it's going to be 7 squared plus 1 that's 49 plus 1 which is 50. now if this is 30 and this is 50 that means that that's 80 180 minus 80 this is 100 which means this has to be 80 because they form a linear pair so these numbers they seem more reasonable a and b could be six and two in that case c would be 180 minus six minus two so c would be 172 and then bcd when i mean cmn acb by the way bcd would be 180 minus 172 it could be eight so those answers are possible but based on the way the figure is drawn it doesn't look like it's drawn to scale so for this problem the measure of angle a can be 30 or 6 degrees because mathematically they both can work you