When two parallel lines are “cut” by a transversal, some special properties arise.  We will begin by stating these properties, and then we can use these properties to solve some problems.

PROPERTY 1:  When two parallel lines are cut by a transversal, then corresponding angles are congruent.

In the diagram below, angles 1 and 5 are corresponding, and so they are equal.  Similarly with angles 4 and 8, etc.

PROPERTY 2:  When two parallel lines are cut by a transversal, then adjacent angles are supplementary.  That is, when two parallel lines are cut by a transversal, then the sum of adjacent angles is \(180^\circ \).

In the diagram above, this property tells us that angles 1 and 2 sum to \(180^\circ \).  Similarly with angles 5 and 6.

EXAMPLE:  Solve for \(x\).

SOLUTION:  From property 2, we know that the 2 angles \(x + 75\), and \(A\) (so called by me) are supplementary.  That is,

\(x + 75 + A = 180\)

But from property 1, we know that \(A = x + 125\), since those two angles are corresponding.  Then \(x + 75 + x + 125 = 180\), so that \(2x + 200 = 180\). Then \(2x =  - 20\), and \(x =  - 10\).

EXAMPLE:  Solve for \(x\).

SOLUTION:  Since, by property 1, we know that corresponding angles are congruent, we know that

\(12x + 3 = 11x + 9\)

\(x = 6\)

Below you can download some free math worksheets and practice.