Let the lines AB and CD make angles θ 1 and θ 2 respectively with the positive direction of X-axis. Let the equation of two lines AB and CD be y = m 1x + c 1 and y = m 2x + c 2 respectively. And we are done.Angle between two lines Angle between the lines y = m 1 + c 1 and y = m 2 + c 2 and y = m 2x + c 2 Angle between the lines Slopes of these two lines are going to be the same. And so just like that, byĮstablishing similarity, we were able see the ratio of corresponding sides are congruent, which shows us that the Here, this is the slope of line, of line CD. Well, now, this is going to be change in y over change in x between point C and D. This second expression right over here, CE over DE, CE over DE. So this right over here is change in y, and this over here is change in x. To B, your change in x is AE, and your change in y is BE, or EB, however you want to refer to it. (m2-m1) // plugging x value in equation (4) > y c2 + m2 x: x (c1-c2) / (m2-m1) y c2 + m2 x // verify by plugging intersection point (x, y) // in orginal equations (1) and (2) to see if they intersect // otherwise x,y values will not be finite and will fail this check: if. You're going from A to B, it's change in y over change in x. // compute slope of line 2 (m2) and c2: double m2. I could write it like this, that is slope of, slope of A, slope of line AB. We could say that's the slope of line AB, slope of line connecting, connecting A to B. Now what is the ratio between BE and AE? The ratio between BE and AE. We can say the ratio of corresponding sidesĪre going to be the same. So once again, once we established these triangles are similar, And this just comes out of similar, the similarity of the triangles, CE to DE. So that's going to be the same as the ratio between CE and DE, and DE. That side over that side, well what is the corresponding side? The corresponding side to BE is side CE. Over here, the ratio of BE to AE, to AE, to AE, is going to be equal to, so The ratio of let's say the ratio of BE, the ratio ofīE, let me write this down, this is this side right If two lines are perpendicular and neither one is vertical, then. Triangles are a ratio of corresponding sidesĪre going to be the same. Theorem 104: If two lines have the same slope, then the lines are nonvertical parallel lines. So we could say triangle AEB, triangle AEB is similar, similar similar to triangle DEC, triangle DEC by, and we could sayīy angle, angle, angle, all the correspondingĪngles are congruent, so we are dealing with similar triangles. Use the geometric properties and theorems you have learned to solve for x in each. Measure as this blue angle, this magenta angle has the same measure as this magenta angle,Īnd then the other angles are right angles, theseĪre right triangles here. Also shown in the above figures: vertical angles are equal: m1 m2. Notice we have all three angles are the same in both of these triangles, well, they're not all the same, but all of the corresponding angles, I should say, are the same. So, because this thirdĪngle's just gonna be 180 minus these other two,Īnd so this third angle is just gonna be 180 Two angles in common, so if they have two angles in common, well, then their thirdĪngle has to be in common. Well, if you look at triangleĬED and triangle ABE, we see they already have Sometimes this is calledĪlternate interior angles of a transversal and parallel lines. And so we know that this angle, angle ABE is congruent to angle ECD. Now this angle on one side of this point B is going to also be congruent to that, because they are vertical angles. And so they're going toīe, they're going to have the same measure, they're

To this angle if we look at the blue transversal as it We also know some thingsĪbout corresponding angles for where our transversal Then that is a right angle right over there. First of all we know that angle CED is going to be congruent to angle AEB, because they're both right angles. So let's call that pointĪ, point B, point C, point D, and point E. So actually let me label some points here. That both of these lines, both of these yellow

Line angle properties to establish that this triangle and this triangle are similar and then use that to establish

And from this, I'm gonna figure out, I'm gonna use some parallel Perpendicular to each other, that these intersect at right angles. Green one is horizontal and the blue one is vertical. And then let me do a vertical transversal. I'm claiming that theseĪre parallel lines. Let me draw another line that is parallel to that. Do in this video is prove that parallel lines have the same slope.

Geometry the same line m1 and m2