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in this video were going to talk about the transitive property the substitution property and also vertical angles so heres the general idea of the transitive property if angles are congruent to the same angle then theyre congruent to each other so for instance lets say if angle 1 is congruent to angle 2 and if angle 3 is congruent to angle 2 then we can make the statement that angle 1 is congruent to angle 3 so now lets look at the sentence carefully if angles that is angle 1 and angle 3 so if lets say two angles are congruent to the same angle that is angle 1 and angle 3 are both congruent to angle 2 then the original two angles are congruent to each other the original two angles being 1 and 3 so 1 in 3 are congruent to each other and thats the basic idea of the transitive property the next property that you need to be familiar with is the substitution property so in a substitution property it just basically replaces something youre substituting a variable with another variable or an angle from that angle so lets say if a is equal to B and if B is equal to C well since a is equal to B I can replace B with a so Im substituting beat with a so I could say that a is equal to C so thats the substitution property but now you could use a transitive property to get the same result but the way in
which you go about doing it is a little bit different but the result is very similar so lets say that a is equal to B and C is equal to B then I can make the statement that a is equal to C so notice that I have lets say that these letters represent angles so angle a and angle C we have two angles that are congruent to the same angle the same angle is angle B a and C are both congruent to angle B so therefore I could say that the two angles angle a and angle C are congruent to each other so its really the way in which you use it the result is the same just remember if you want to use the substitution property youre replacing something youre substituting a variable with another variable if you want to use a transitive property youre basically linked in two equations to another equation now lets move on our discussion into vertical angles whenever two lines intersect they form two pairs of vertical angles lets call this angle one two three and four so the first pair of vertical angles are angle one and three I need to know that vertical angles are congruent so angle 1 is equal to angle 3 angles 2 and 4 are opposite to each other so those are vertical angles so angle 2 is congruent to angle 4 now it turns out that 2 & 3 are supplementary they form a linear pair so angle 2 plus angle
3 adds up to 180 and the same is true for 1 & 4 1 in 4 forms Alinea pair 1 in 2 forms Eleni pair and also 3 & 4 forms alumina pair but our focus is on vertical angles so what exactly are vertical angles there are two angles two angles are vertical angles if the race form in the sides of the other are opposite race so what does that mean well first need to know one whats an opposite rain to clean your rays to have a common endpoint and extend in different directions are opposite race so to illustrate that with draw a line and lets lets put some points on this line lets call this a B and C so ray ba which starts from B and points towards a is congruent or not congruent but its opposite to Ray BC BC starts with the common endpoint b and extends in the other direction so ba and BC are collinear rays because they exist on the same line but they share a common endpoint and they extend in opposite directions BC extends towards the right and the a extends towards the left so those are opposite race now in the case of vertical angles lets redraw so lets call this point a b c d and e so angle 1 and angle 3 are vertical angles angle 1 is formed from Ray CA which is opposite to Ray C II so Im just gonna put angle 1 and angle 3 now angle 1
is also formed two by Ray CB which is opposite to Ray CD so this is Ray CA thats opposite to see and Ray CD is opposite to C V so therefore the vertical angles are formed whenever you have raised forming the sides of those angles and it has to be opposite race so angle 1 and angle 3 are vertical angles now lets work on an example problem so consider the diagram lets call this angle 2 and angle 3 angle 2 is equal to x squared plus 6 angle 3 is equal to 7x plus 14 so given this information what is the measure of angle 2 now we know that vertical angles are congruent so therefore we could say that angle 2 is equal to angle 3 and value and that we could substitute angle 2 with x squared plus 6 unless you place angle 3 with 7x plus 14 our goal is to find the value of x unless we do that we could find the measure of angle 2 now what we have is in a quadratic equation so everything on the right side lets move it to the left so its going to be x squared minus 7x and we got to subtract 14 from both sides so 6 minus 14 is negative 8 and that we could factor this expression so what two numbers multiply it to negative 8 but add to negative 7 so this has to be negative 8 and 1 negative 8 plus 1 adds up to negative
its a factor its going to be X minus 8 times X plus 1 and thats equal to 0 so at this point we need to set each factor equal to zero so if we set X minus 8 equal to 0 X is equal to 8 and if we set X plus 1 equal to 0 X will be negative 1 now both answers may work out so lets try each one so lets start with X equals eight lets find the measure of angle two by plugging it into this expression so its eight squared plus 6 + 8 squared is 64 64 plus 6 is 70 so angle two is equal to 70 and it should be the same as angle three so if we plug it into seven X plus 14 we should get the same answer so seven times eight is fifty-six and 56 plus 14 is 70 so one possible answer is 70 degrees now lets find the other possible answer when X is equal to negative one lets make sure we get the same value so negative one squared plus six negative one squared thats negative one times negative one thats positive one and one plus 6 is 7 now angle three if we replace X with negative 1 7 times negative 1 is negative 7 and negative 7 plus 14 is positive 7 so the second possible answer that fits the equation is 7 degrees so angle 2 could be 70 degrees or 7 degrees both answers are acceptable in this example
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