Prove that If two parallel lines are cut by a transversal then, any two angles are either congruent or supplementary?

Question

Autumn Armstrong · Accepted Answer

See the proof below

(1) Angles #/_a# and #/_b# are supplementary by definition of supplementary angles.

(2) Angles #/_b# and #/_c# are congruent as alternate interior.

(3) From (1) and (2) #=> /_a# and #/_b# are supplementary.

(4) Angles #/_a# and #/_d# are congruent as alternate interior.

(5) Considering any other angle in this group of 8 angles formed by two parallel and transversal, we (a) use the fact that it is vertical and, consequently, congruent to one of the angles analyzed above and (b) use the property of being congruent or supplemental proved above.

Avery Allen · Answer

undefined To prove that if two parallel lines are cut by a transversal, then any two angles are either congruent or supplementary, we can use the properties of parallel lines and transversals.

Given two parallel lines cut by a transversal, we have the following diagram:

```
        |
    ____|____
        |
```

Let's label the angles as follows:

```
        |
    ____|____
    a   |  b
        |
```

Now, we have several angle pairs formed by the transversal cutting the parallel lines:

1. Corresponding angles: angles on the same side of the transversal and on the same side of the parallel lines. These angles are congruent.
   In the given diagram, angles a and b are corresponding angles.

2. Alternate interior angles: angles on opposite sides of the transversal and inside the parallel lines. These angles are congruent.
   In the given diagram, angles a and b are alternate interior angles.

3. Alternate exterior angles: angles on opposite sides of the transversal and outside the parallel lines. These angles are congruent.
   In the given diagram, angles a and b are alternate exterior angles.

4. Consecutive interior angles: angles on the same side of the transversal and inside the parallel lines. These angles are supplementary.
   In the given diagram, angles a and b are consecutive interior angles.

Since any two angles formed by the transversal cutting the parallel lines fall into one of these categories, they are either congruent or supplementary. Therefore, the statement is proven.