here are the top 10 most important angle theorems that you have to know make sure you watch till the end where I give you a quick test to see how well you understand these theorems must know number one complementary and supplementary angles let's start with complimentary angles complimentary angles are two angles with a sum of 90° and if you were to put two complimentary angles together they would form a right angle if I were to draw two line segments that form a right angle and then break that angle up into two different angles let's call them angle a and angle B because these two angles combined together form a right angle I know the sum of those two angles is 90° which means those two angles are complimentary so let's do a quick example with complimentary angles in this example it says to calculate angle Theta if I zoom in on this diagram we can see that the 70° angle and Theta are complementary because together they combine to make a 90° angle so I could say thata plus 70 equal 90° so to calculate Theta I just have to find the difference between 90 and 70 which is 20° now I'll zoom out and we'll talk about supplementary angles supplementary angles are two angles with a sum of 180° when supplementary angles are combined they form a straight line so if I were to draw a line segment and break up one side of that line segment into two different angles let's call them angle a and angle B notice that when combining angle a and angle B together they form a straight line which means that angle a plus angle b equals 180° which means angle a and angle B are supplementary angles and let's do a quick example with supplementary angles in this example we need to calculate angle Theta again so let me zoom in and notice that angle Theta and 70° combined together form a straight line so they're supplementary that means that angle theta plus that 70° angle would have to equal 180° so to calculate Theta just find the difference between 180 and 70 which means the angle Theta would be 110° so I'll zoom out so you can see that whole section and let's continue on where we learn angle theorems that involve other shapes and also parallel line theorems must know number two sum of angles in a triangle and other polygons let's start with the sum of the angles in a triangle the sum of the three interior angles of any triangle will always be 180° so in this triangle that I'm drawing that has interior angles let's call them a b and c I know that the sum of those three Ang angles is 180° so let's do a quick example involving the interior angles of a triangle in this example I need to calculate angle Theta and if I zoom in I see that I know two of the interior angles of a triangle and I'm looking to calculate the third well I know all three interior angles of a triangle always add up to 180 so to calculate Theta I just have to subtract 70 and 75 from 180 and that would tell me that angle Theta is equal to 35° let me zoom out and now let's do the sum of the angles in any polygon to find the sum of the interior angles of any polygon you can just follow the formula which says do nus 2 time 180° where in this formula n stands for the number of sides that the polygon has so if I to draw a polygon with five sides and then I'll label the five angles of that polygon angles a b C D and E I know that I could figure out what the sum of those five angles would be by using this formula where I just Sub in five for n because this polygon has five sides so I would do 5 - 2 * 180° that's 3 * 180 which is 540° so the sum of the interior angles of this Pentagon would be 540° and now let's do an example where weal calate the measure of an interior angle inside of a four-sided polygon in this example we have to calculate angle Theta so let me zoom in on this polygon and because this polygon has four sides I know that the sum of the four interior angles would be equal to nus 2 * 180° so in this case the number of sides is four so 4 - 2 * 180° so the sum of the four angles would be equal to 2 * 180 which is 360° so to calculate angle Theta I would just have to subtract those three angles from 360 which would mean that angle Theta is 111° and let me actually leave you with a note about the exterior angles of a polygon the sum of the exterior angles of any polygon is always 360° so let me just zoom in here and give you a really quick diagram of what that would look like so if I had a triangle and if I were to extend the three sides of that triangle the exterior angles which I'll call A B and C would all add up to 360° and that would be true for the sum of the exterior angles of any polygon so I'll zoom out so you can see that whole page and let's continue moving on must know number three the isoceles triangle theorem this rule states that if two sides of a triangle are congruent then the angles opposite from those sides must also be congruent so I'll draw a triangle that has two sides of equal length I'll label the vertices of this triangle a b and c these tick marks indicate that the length of side a and the length of side CB are equal to each other because those two sides are equal in length based on the Isles triangle theorem I know that the angle is opposite from those two sides so angle a and angle C would be equal to each other so I could say that angle a equals angle C and then let's do a quick example in this example we have to find the measure of angle Theta because the measure of those two side lengths are equal to each other I know that the measure of the angles opposite from those two sides would have to equal meaning that this angle here must also be 35° and then the sum of the three interior angles of a triangle has to be 180 and then I could calculate Theta by subtracting those two angles from 180 and that would tell me that angle Theta is 110° must know number four the exterior angle theorem this rule Ru states that the exterior angle of a triangle is equal to the sum of the two opposite interior angles so if I were to draw a triangle and I'll label the interior angles angles a b and c and then if I extended one of the sides of this triangle this angle outside of the triangle which we call the exterior angle I could label angle D and then what you need to know is that this exterior angle D is equal to the sum of the two opposite interior angles so d would equal angle a plus angle B there would be two easy ways for me to prove to you why that is true first of all notice that angle D and angle C when combined together form a straight line meaning they're supplementary so they add to 180° but also I know that all three interior angles of a triangle when added together also has to be 180 notice adding two different things to C either D or a plus b when either of those are added to C we get 180 so that must mean that D and A plus b are equivalent to each other hence the exterior angle theorem another rationale would involve some parallel line theorems which I'll get to later in this video actually but I'll give you a brief preview of it now if I were to redraw a line parallel to this side starting here at angle D there's an alternate angle theorem or the Z pattern which would tell me that this angle and this angle are equal and there's a corresponding angle angle theorem which would tell me that this angle in here is exactly equal to this angle so that would equal angle a so you can see that angle A and B perfectly make up angle D now a quick example in this example I have to find the measure of angle Theta based on the exterior angle theorem I know that this exterior angle 110 would have to equal the sum of the two opposite interior angles and then I can solve for Theta by subtracting the 40 from 110 and that would tell me that theta equals 70° must know number five the vertical angle theorem this rule states that when two lines intersect the measure of the angles opposite from each other are equal so if I were to draw two lines that intersect I'm actually creating four angles I'll call those four angles angle a angle B angle C and angle D the vertical angle theorem tells me that the angles opposite from each other are equal in measure since angle a and angle B are opposite from each other I could say that the measure of those angles are equal and because C and D are opposite from each other the measure of those angles are also equal so a really quick easy example it asked me to find the measure of angle Theta well I notice I have two lines that are intersecting so the angles opposite from each other Theta and 61 must be equal to to each other so for that one we could just State Theta equal 61° must know number six alternate angles this is our first parallel line theorem if two parallel lines are cut by a transversal the alternate interior angles are equal and the alternate exterior angles are equal let me draw a diagram to show you what I mean let's start with the alternate interior angles I'll draw my two parallel lines and then I'll cut them with a transversal that means I'll draw a non-p parallel line which intersects both of those at two distinct points in this diagram I have two points of intersection the interior angles are the angles between those points of intersection so these four angles in here let me name those angles I'll name them a b c and d what this theorem tells me is that the alternate interior angles are equal at this point of intersection if I choose this interior angle a to find an angle equivalent to it I go to the other point of intersection and find the angle on the opposite side of the transversal so here angle C so angle A and C are equivalent to each other and you'll notice if I draw the letter Z it can make it easy to find these two angles that are equivalent to each other that's why this alternate angle theorem is sometimes called the Z pattern and then if I made a z going in the other direction the angles inside of that Z are B and D so those angles are also alternate interior angles so they're also equal to each other now not only are the alternate interior angles equal but so are the alternate exterior angles so let me draw two more parallel lines here and I'll cut them with the transversal as well here's my two points of intersection the exterior angles are these four angles outside here I'll label them e f g and H to find an angle equal to this exterior angle I go to the other point of intersection and pick the angle on the opposite side of the transversal so H so e and H are equal to each other and for that same reason f and g would also be equal to each other so let me make some room and let's do an example where we can practice finding this parallel line theorem in this diagram that I'll show you let's find the measure of the indicated angles so we need to find angle X and angle y i notied that X and 110° are alternate interior angles and I can draw that Z pattern so you can see that notice how there are both angles inside of that Z Drew so I can say that X is equal to 110° the next thing I notice is that angle X and Y are supplementary those two angles form a straight line so I know they add up to 180° and I know angle X is 110 which means angle Y Must Be 70° must know number seven co- interior angles this is our second parir line theorem this theorem states that interior angles on the same side of a transversal add up to 180 de so let me draw a diagram of this theorem I'll draw my two parallel lines and I'll cut them with the transversal and now I need to label the interior angles on the same side of the transversal so I'll go to the right side of the transversal and I'll label these two angles we'll call them A and B because they're the interior angles on the same side of the transversal I know they add up to 180 and this theorem is sometimes called the C pattern and if I draw letter C you can see that these two angles are inside of that letter C so that's why it's called a c pattern and now let me do an example where we try and identify that c pattern let's find the measure of angle Theta in this diagram since these are both the interior angles on the same side of this transversal I know they add up to 180° and let me draw that c pattern so you can clearly see that so I know that Theta plus 75 would have to equal 180° subtract the 75 over and I would figure out Theta must be 15° must know number eight corresponding angles this is our last parallel line theorem this rule states that angles which occupy the same relative position at each intersection of the transversal with the two parallel lines are equal so I'll draw the two parallel lines the transversal and show you what I mean by this at each of these points of intersection there are 1 2 3 4 angles let me label all of those angles and at each of those points of intersection the theorem tells me that the angles which occupy the same relative position are equal so if I go to both of those points of intersection and pick the top left angle those would be equal to each other so I know angle a will be equal to angle e and if I do the same thing but I pick the top right angle at both points of intersection those would be equal to each other so angle B is equal to angle F same thing with the bottom left angles D would equal H and the bottom right angles C would equal G now sometimes this rule is called the F pattern it's sometimes called this because if you're to draw the letter F it can help you find the corresponding angles that are equal the angles underneath the horizontal lines of this F C and G are equal to each other now let's try and example where we can use this relationship of corresponding angles let's find the measure of angle Theta in this diagram to find Theta I notied that Theta and 41° are equivalent because they're corresponding angles and I can see that more clearly if I draw the F pattern in that F pattern I know these angles inside of the f are equal to each other so Theta must equal 41° must know number nine angle subtended by an arc this is our first Circle theorem the rule states that inscribed angles subtended by the same Arc are equal so let me show you what that means I'll start by drawing my circle and then let me outline part of the circumference of this circle as my arc if I were to inscribe any two angles inside of the circle that are subtended by that Arc they will be equal so what does that mean to inscribe an angle subtended by this Arc that means pick any point on the circumference of this circle I'll pick right here and draw two line segments that intersect the ark at its end points so here and here the angle I've created is the angle up here at the circumference I'll call it angle y if I follow that same process but do it from another point on the circumference so over here as long as I draw segments that go through the end points of that same Arc that angle at the circumference which I'll call X this time is going to be equal to the other one y so angle X and angle y are equal to each other because they're subtended by this same Arc there's another interesting Circle theorem that's very related to this one that says that an angle inscribed in a semicircle actually equals 90° let me show you what that looks like so once again I'll draw a circle and since this theorem is about a semicircle let me draw a diameter line through the middle of the circle to cut it in half inscribing any angle in inside of this semicircle is going to make the angle equal to 90° so to inscribe an angle I pick a point on the circumference and then draw line segments to each of the end points of this chord that I've drawn of that Circle and this theorem tells me that this angle here is guaranteed to be 90° so let's try an example now let's calculate Theta in this diagram in this diagram I noticed that this angle and this angle are both subtended by the same Arc that Arc right there angle Theta is subtended by that Arc and 65 is subtended by that Arc so for that reason I know that Theta and 65 are equal to each other so I could say that theta equals 65° must know number 10 the angle at the center of a circle versus the angle at the circumference of a circle this rule states that the angle subtended by an arc at the center is double the angle subtended by the same Arc at any point on the circumference of the circle so let me draw that in a diagram in this circle if I draw an arc length I'm now going to create two angles that are subtended by that same Arc Length I'm going to create an angle at the center of the circle first so starting at the center of the circle I'll draw two line segments that go to the end points of the arc and now I'll inscribe another angle in the circle by choosing a point somewhere on this CC circumference of this circle I'll pick it over here and now I'm going to draw a line segment that goes from that point to each of the end points of that same Arc now what this theorem tells me is that the angle at the center of the circle is going to be double the angle that's at the circumference of the circle so if the angle at the circumference of the circle is we'll call it angle X the angle at the center of the circle would be double that I could call it 2X and now let's use that relationship to solve this example because I noticed that both of these angles are subtended by this same Arc I know that the angle at the center of the circle has to be double the angle that's at the circumference of the circle so 142 is equal to 2 * X and then to calculate X I could just cut 142 in half and I would figure out that X is equal to 71° now that you know the 10 most important angle theorems test yourself out see if you can find all of the missing indicated angles in these three questions leave your answer in the comments I'll make sure to like a comment that has all of the right answers for you to be able to check your answers against also let me know what top 10 do you want to see next J