Proving theorems about angle pairs formed by transversals.
1.) Use previously known theorems and definitions:
Vertical angles theorem, linear pair theorem, supplementary, definition of congruence, definitions of different angle pairs, transitivity, substitution
2.) Even with all the previous theorems and definitions, we still can’t prove any of the transversal theorems about making one assumption:
Corresponding angles postulate- If the lines are parallel, then corresponding angles are congruent.
Remember this cannot be proven! We just agree that it needs to be true and can’t be any other way.
3.) All of the angle pair theorems show what conclusion we can reach under the conditions that the two lines are parallel. That’s the only time angle pairs formed by transversals are congruent or supplementary.
Ex.) Prove: If the lines are parallel, the alternate interior angles are congruent.
Create a situation with two parallel lines. (line a//line b)
We want to say something about alternate interior angles, so we identify angle 1 angle 2 as alternate interior angles.
Now we need to think about prior and felt that we know is true, and that applies to our diagram.
Identify an angle that is corresponding with angle 1 and label it angle 3. Since the lines are parallel, we can make use of the corresponding postulate, which says that angle 1 is congruent to angle 3.
You will also notice that angle 2 and angle 3 are vertical angles, so by the vertical angles theorem we know that angle 2 is congruent to angle 3.
So we know that angle 1 is congruent to angle 3, and angle 2 is congruent to angle 3, based off of prior knowledge. Transitivity says that since angle 1 and angle 2 are both congruent to angle 3, they must be congruent to each other. So angle 1 is congruent to angle 2 by transitivity.
Since angle 1 and angle 2 are alternate interior angles, we have shown that when the lines are parallel, the alternate interior angles must be congruent.