Learn about Perpendicular Line Theorems In this lesson let’s learn about perpendicular line theorems and how to use them to calculate the shortest distance. Perpendicular line theorems are actually quite interesting we’ll do a few examples so the first one gives us a figure and it says name the shortest distance from point W to line XZ. This is point W we need to find the shortest difference between W and this line XZ and we need to write and solve than an equality for X. So key thing to remember is from any point the shortest distance to a line is figured out by drawing up a perpendicular line means this is 90 degrees. So the shortest distance from any point to a line is the line WY which is perpendicular to the original line XZ. So what is the shortest distance that is WY is the shortest distance which means WY is shorter than WX or WZ which means WY is shorter than WZ, right because WZ has to be longer than WY because this is the shortest distance. If WY is less than WZ we know WZ is 19 and WY is the length of WY is X+8 then this is the inequality, how do we solve for it subtract A from both sides, we X is less than 11. So if WY is the shortest distance and this is the link of WY the inequalities solution is X is less than 11. You get the shortest distance. Now let’s look at a couple of other examples. It says line CD which is the straight line here forms a linear pair of congruent angles on segments AB and EF. So this two are line segments AB and EF and CD forms a linear pair of congruent angles which means both these angles on 90 degrees. Right we need to prove that AB is parallel to EF, now the theorems states that if we have a line that intersects two lines and it forms perpendicular angles here and it forms perpendicular angles with the other line then these two lines are indeed parallel. If the is line intersects line AB with one set of angles in this case 90 degrees and the same measurement is the way then to sex line too which is line EF then these two lines are indeed parallel. So since this is 90, this is 90, and this is 90 and the intersection between CD and AB and EF are—CD intersects both these lines at 90 degrees. Since these two angles are equal that means AB is parallel to EF that is true and the reason it’s true is because angle CMB which is this angle is equal to angle C and F. Since they’re both equal to 90 degrees these two lines have to be parallel for these two to have the same angled measure. Let’s try one more. Here we need to compute X and Y. We can see these are two parallel lines, two parallel lines and it’s being intersected by AF, C and D are two parallel lines I intersected by AF so what does that mean if I can see this is 90 degrees right that means this is 90 degrees as well because AE is 180 degrees since C B intersects AE at a perpendicular angle that means angle CBA plus angles CBE which is these two angles that have marked in green equals 180 degrees. Since we’ve given that CBA is 90 degrees plus angle CBE equals 180 that means angles CBE equals 90 degrees, Right so this is 90 what do we know 6y equals 90 degrees. 6 x y is the measure of this angle but we know this angle is 90 degrees. So 6 x y equals 90 that’s divided by 6 on both sides what do I get y = 15. So I’ve already solved for y. Now that I know y equals 15, now let’s look at these two angles I’m going to raise what I’ve marked here and start all over. So what do I know these two are parallel lines since two are parallel lines if this is 90 degrees then this is also 90 degrees what does that mean that 6y equals 5x + 4y that means these two angles are equal, if these angles are equal 6 x 15 is 90 = 5x + 4 times 15 is 60, right so subtracting 60 degrees from both sides, 30 degrees equals 5 times x this gets cancelled, 30 degrees 5 times x let’s divide both sides by 5 to simplify and we get x=30/5 is 6. So x is 6 so what have we solved x=6 and y=15. And the reason we could solve this is because we know that if we have two parallel