Parallelograms Geometry

in this video we're going to focus on parallelograms we're going to go over some properties associated with parallelograms and talk about how to solve some problems regarding it so let's start with picture so let's say this is so the first thing you need to know is that opposite sides are parallel so is parallel to dc and we could also say that bc is parallel to now what else do we know regarding parallelograms well for one thing opposite sides are congruent so we can say that is congruent to dc and also bc is congruent to ad now the next thing that we need to know is that opposite angles are congruent so angle and angle are congruent to each other and angle and angle are also congruent to each other so let's say if angle is 100 degrees that means is 100 degrees now what's the value of and another rule we need to know is that consecutive angles are supplementary so that means and have to add up to 180. and has to add up to 180. so and they're equivalent to each other and they both must be 80. 100 plus 80 is 180 and so consecutive angles are supplementary now the next thing that you need to know is the diagonals the diagonals bisect each other so what does that mean ac bisex bd and also bd bisex ac so let's say if ac is 20 units long and let's call this that means ae will be 10 and ec will be 10. it's bisected into two congruent parts so is the midpoint of and so let's say if was 12 that means is six is six and so it's important to understand that these two are congruent to each other and so ae is one half of ac and ae is congruent to ec so we can write that ae is congruent to ec and ae is basically one half of ac so those are some other things that you want to keep in mind when dealing with parallelograms now let's work on an example problem so let's say if we have parallelogram abcd now let's say that angle is nine minus two and angle is ten plus thirty with this information calculate the measure of angle feel free to pause the video if you want to try now if you recall consecutive angles are supplementary so this means that angle plus angle must add to 180 degrees so angle is nine minus two angle is ten plus thirty so this has to add up to one eighty now let's combine like terms nine plus 10x is 19x negative 2 plus 30 that's 28 so now we need to subtract both sides by so 180 minus 28 that's 152. now let's divide both sides by 19. so 152 divided by 19 is 8. so now that we have the value of we can calculate the measure of angle so angle is nine minus two so let's plug in the value that we have so let's replace with eight and this will give us the answer nine times eight is seventy two and seventy two minus two is 70. and so that's the measure of angle it's 70 degrees and so that's it for this problem here's another example so here we have parallelogram and let's say once again this is and let's say that angle is squared plus 20 and angle is 7x plus 50. your task is to calculate the measure of angle go ahead and do this so one rule that you need to remember for parallelograms is that opposite angles are congruent so this means that angle is equal to angle so is squared plus 20 and is 7 plus 50. so let's go ahead and calculate the value of so how should we do this the best thing is to take everything on the right side and move it to the left side so it's going to be squared minus 7x plus 20 minus 50. now 20 minus 50 is negative 30. so this is what we now have so what two numbers we need to factor this quadratic equation by the way what two numbers multiply to negative 30 but add to the middle coefficient negative seven we need to find two numbers that multiply to negative 30 and add to negative seven this will help us to factor this trinomial so we have two and fifteen three and ten five and six notice that three and ten differs by seven and to make it negative thirty one of them have to be negative so it makes sense if we make it positive three and negative ten three times negative ten is negative 30 but 3 plus negative 10 adds up to negative 7. so to factor it's going to be plus 3 times minus 10. now what we need to do is set both factors equal to zero and so we could see that will equal three mean negative three if we subtract both sides by three and if we add 10 to both sides will equal positive 10. so it looks like we may have two possible answers if we plug in negative 3 both angle and angle will be the same and they will both have positive measure so for example the measure of angle is going to be squared plus 20 and so that's negative 3 squared plus 20 which is 9 plus 20 so that's 29 degrees and angle will have the same value it's going to be 7 times negative 3 plus 50. 7 times negative 3 is negative 21 plus 50 that's 29 degrees so one possible answer for angle is 29. let's see what the other possible answer i'm is going to be so let's say if is 10. so angle is going to be 10 squared plus 20. 10 squared is 100 100 plus 20 that's 120 degrees and if we check it by calculating angle it's going to be 7 times 10 plus 50 so that's 70 plus 50 which is also 120. so angle can be 29 or 120 degrees so now we can find the possible answers for angle keep in mind that and are supplementary so the measure of plus the measure of has to add up to 180 so let's calculate if is 29 so that's going to be 180 minus 29 so one possible answer for angle is 151 degrees now the other possible answer is 180 minus 120 so it could be 60. so there's two possible answers for this angle it could be 60 degrees or 151. now let's work out another example so this is going to be and as usual and this time we're going to have some diagonals and let's call this point so in this problem is going to be 4x and ec we're going to say it's 2x squared minus minus 3. so knowing this what is the measure what is the length of ac go ahead and work on this problem now if you recall the diagonals bisect each other so what that means is that ae is equal in length to ec these two sides are equivalent to each other so therefore we could set ae which is 4x equal to ec which is 2x squared minus minus 3. so we have another quadratic equation so let's begin by subtracting both sides by 4x so this is going to be 0 is equal to 2x squared negative 1x minus 4x is negative 5x and so we have this expression so how can we factor 2x squared minus 5x minus 3. so notice that we have trinomial where the leading coefficient is not 1. so what do we need to do in this case the first thing you need to do is you need to multiply the leading coefficient by the constant term so 2 times negative 3 is negative 6. next you need to find two numbers that multiply to negative six but add to the middle coefficient negative five this is going to be negative six and positive one negative six plus one adds up to negative five next replace the middle term with these two so we're going to replace negative 5x with negative 6x plus 1x because they add up to negative 5x after that factor by grouping so in the first two terms take out the gcf the greatest common factor in this case it's going to be 2x 2x squared divided by 2x is negative six divided by two is negative three now in the last two terms factor out the gcf if there is no gcf factor on one which means what's convenient on the inside is the same minus 3. if these two are the same that means you're on the right track so we're going to factor out minus 3 and then what goes inside the second parentheses is what we see out here 2x plus 1. and so that's how you could factor this particular trinomial now just like before let's set each factor equal to 0. so for this one let's add 3 to both sides so we can see that one possible answer is that is equal to 3. now here let's subtract both sides by negative 1. so two is equal to negative one and then we'll have to divide both sides by two so the other possible answer is that is negative one half now which value should we use can we use both of them like the last problem or just one of them now notice that if we plug in negative one-half into 4x it gives us negative value and you can't have negative length so therefore this answer doesn't make sense now if the negative value gave us positive 4x value or let's say if that was 4x squared then it would work but you can't have negative side length it just it is not practical like what would that mean so has to equal 3 in this example now ae is 4x so it's 4 times 3 that's 12. ec is 2x squared minus minus 3 or 2 times 3 squared minus 3 minus 3. three squared is nine and negative three minus three is negative six two times nine is eighteen eighteen minus six is twelve so we can see that these two are indeed the same now ac is the sum of ae and ec so ae is 12 ec is 12 therefore ac is 24 and this is the answer that we're looking for in this problem here's another math problem for you so we're going to use the same letters once again we're going to have diagonals in this picture and this is once again so let's say that be is 4x minus 6 and bd we're going to say it's 7x minus 8. so with this information calculate the length of go ahead and work on it so what's the relationship between and notice that be is one half of bd so let's say if be was seven that means ed would be seven but bd would be 14 7 is one half of 14 for example so be is 4x minus 6 bd is 7x minus 8. what i'm going to do is i'm going to multiply both sides by 2 because if distribute the one-half i'm going to have 7 over 2 and don't want any fractions so let's multiply this side by 2 and the left side by 2. two so one half times two is one now on the left side need to distribute the two to four minus six so two times four is eight and two times negative six is negative 12. on the right side because cancel the fraction no longer need parenthesis so can simply write 7x minus 8. now let's subtract both sides by 7x and let's add 12 to both sides so these two will cancel 8x minus 7x is 1x which is simply and negative 8 plus 12 is 4. so now we have the value of is equal to 4. so now we can calculate be so that's 4x minus 6 or 4 times 4 minus 6. 4 times 4 is 16 and 16 minus 6 is 10 so be is 10 which means ed is also 10. and so that's the answer ed is 10 units long now let's consider another problem so this one is going to be little bit different than the other ones so let's say that is equal to three plus four and let's say bc is equal to five plus three and cd is equal to 7x minus 12 and also ad is 2y plus 12. so with this information calculate the perimeter of the parallelogram go ahead and try it so keep in mind opposite sides are congruent now is three plus four and is five plus three okay that doesn't look like so let's fix that and cd is 7x minus 12 is 2y plus 12. so since opposite sides are congruent we know that is congruent or equals so is three plus four we can set that equal to seven minus twelve so let's subtract both sides by 3x and let's add 12 to both sides 4 plus 12 is 16 and 7x minus 3x is 4x so if we divide both sides by 4 16 divided by 4 is 4. so is equal to 4. now if is equal to 4 we can calculate the measure of side so is three plus four so that's gonna be three times four plus four and so this is twelve plus four which is sixteen so and they're both equal to sixteen since they're congruent to each other now we could also set bc equal to because they have the same left now bc is 5y plus three is two plus twelve so let's subtract both sides by two and by three five minus two is three twelve minus three is nine and 9 divided by 3 is 3. so now we can find the measure of bc so it's 5 plus 3 so that's 5 times 3 plus 3. 5 times 3 is 15 15 plus 3 is 18. so bc and ad they both equal 18. so now we can calculate the perimeter the perimeter is going to be 2 times 16. because we have two of them plus two times eighteen so two times sixteen is thirty-two two times eighteen is thirty-six thirty-two plus thirty-six is 68 so that's the perimeter of this particular quadrilateral you