We are going to have a look at some angle relationships that are created when we have a pair of parallel lines (AB and CD) and a transversal (EF) that crosses them.

There are many angles in the above diagram. Lets review some pairs of angles that have special names.

X-Rule: Vertically opposite angles are equal.
U or C-Rule: Cointerior angles of parallel lines are supplementary
Z-Rule: Alternate angles of parallel lines are equal
F-Rule: Corresponding angles of parallel lines are equal.

Of course, these angle relationships won't look like X's, U's, Z's and F's in all diagrams because the diagram may be rotated and consist of more lines.

Angles in Parallel Lines

Vertically opposite angles are equal. (X rule).

Cointerior angles of parallel lines are supplementary. (U or C rule).

Alternate angles of parallel lines are equal. (Z rule).

Corresponding angles of parallel lines are equal. (F rule).

Worked Examples

Question 1

In the diagram, $AB$AB is a straight line.

Solve for $x$x. Give reasons.

Question 2

Consider the adjacent figure.

Find the size of angle $u$u.
Find the size of angle $v$v.

Question 3

Calculate $x$x giving reasons for your answer.

Question 4

Find the value of $x$x in the following figure.

Solving problems

When solving angle problems in geometry one of the most important components is the reasoning (or rules) you use to solve the problem. You will mostly be required in geometry problems to not only complete the mathematics associated with calculating angle or side lengths but also to state the reasons you have used. Read through each of these rules and see if you can describe why and draw a picture to represent it.

Angles in parallel lines

Cointerior angles in parallel lines are supplementary (U, C)
Corresponding angles on parallel lines are equal (F)
Alternate angles on parallel lines are equal (Z)
Vertically opposite angles are equal (X)

Geometry Rules with Angles

Exterior angle of a triangle is equal to the sum of the two opposite interior angles
Angle sum of an n-sided polygon is (n-2)[x]180
Sum of exterior angles of a polygon is 360°
Angle sum of a quadrilateral is 360°
Angles at a point sum to 360°
Vertically opposite angles are equal
Adjacent angles on a straight line are supplementary
Adjacent angles forming a right angle are complementary