# How do you prove corresponding angles are equal?

Dec 24, 2015

Suppose you have two parallel lines cut by a transversal.

Due to the straight angle (linear pair) theorem, we know that

{(mangle2+mangle3=180˚),(mangle5+mangle6=180˚):}

Thought the transitive property, we can say that

$m \angle 2 + m \angle 3 = m \angle 5 + m \angle 6 \textcolor{w h i t e}{\times \times}$ (1)

Though the alternate interior angles theorem, we know that

$m \angle 3 = m \angle 5$

Use substitution in (1):

$m \angle 2 + m \angle 3 = m \angle 3 + m \angle 6$

Subtract $m \angle 3$ from both sides of the equation

$m \angle 2 = m \angle 6$

$\therefore \angle 2 \cong \angle 6$

Thus $\angle 2$ and $\angle 6$ are corresponding angles and have proven to be congruent.