Parallel Lines, Transversals, and Angles of a Triangle
i am having trouble with knowing the diffrences between parallel lines and skew lines. i have a problem that says, if two lines do not intersect, are they always parallel? can i prove the statement wrong? or is skew same as parallel?
i also have a problem with this one:
if two parallel lines, are cut by a transversal, then the corresponding angles are congruent. they tell me to point out which angles are corresponding. also same question just altrnate interior angles. which angles are they?
one last problem that i have in my homework is in triangles. they tell me to figure out the interior angles. okay, so i know that a triangle must add up to 180 degrees, so i basically figured that out. then they ask me to figure out exterior. how do i figure that out?
The Difference between Parallel and Skew
As you will see below, if two lines do not intersect, they are not always parallel. Skew lines and parallel lines are different. Though skew and parallel both imply non intersecting lines, parallel is the special case where the two lines are also in the same plane ...
I have attached two pictures below:
The first one is depicting two parallel lines AB and CD. Parallel lines are lines that are in the same plane but do not intersect. AB and CD are both in the plane ABCD and do not intersect therefore they are parallel.
The second picture is of skew lines TR and AC. Though they do not intersect, they are not in the same plane. It is not possible to draw a plane that could have both lines TR and AC in it. Lines that do not intersect and are not in a shared plane are skew lines.
Corresponding and Alternate Interior Angles are Congruent
I have attached two pictures . . .
The first of which shows which angles are corresponding when parallel lines have a transversal line. If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.
1 and 5,
2 and 6,
3 and 7,
4 and 8 are all corresponding angles; they are congruent.
The second picture shows which angles are alternate interior angles. If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.
3 and 6,
4 and 5 are both alternate interior angles; they are congruent.
Exterior Angles on a Triangle
To figure out the angles of the exterior of a triangle you must take into account two things:
1. The sum of the interior of the angles of a triangle is equal to 180 degrees.
2. A circle has 360 degrees in it.
Please take a look at the attached picture to follow along.
Angles A+B+C = 180
Angles A+D = B+F = C+E = 360
For example exterioir angle D = 360 - A
If you want the sum of the exterior angles of a triangle using the two equations from above :
(A+D) + (B+F) + (C+E) = 360 +360 +360
A + B + C + D + E + F = 1080 now we subtract angles A B & C from both sides
D + E + F = 1080 - 180
D + E + F = 900 degrees.
The sum of the exterior angles of a triangle is 900 degrees, this is always true for any triangle.