Angle Relationships

Parallel Lines Cut by a Transversal

It is amazing how you can find the measures of the angles of intersecting lines given only one angle when the lines are parallel.

Vocabulary

  • transversal interior angles
  • exterior angles
  • corresponding angles
  • alternate interior
  • alternate exterior
  • same side interior
  • same side exterior

Transversal

A transversal is a line that intersects two or more lines. Notice below how line h is the transversal because it intersects lines m and n.

Lines m and n are mostly horizontal, but not parallel. Line h is diagonal and passes through both

Angles Formed

Also, notice the 8 angles that are formed when a transversal intersects two lines.

Line h creates 4 angles from its intersection with m and 4 angles with its intersection with n

Parallel Lines Cut by Transversal

Something unique happens when parallel lines are intersected by a transveral. Certain pair of angles are congruent while other pairs are supplementary. Below are a pair of parallel lines cut by a transversal. Can you identify which angles are congruent and which ones are supplementary?

Two horizontal lines are parallel, and a third lines crosses both at a slight angle

Your Turn

Why don't you try? Can you identify which angles are congruent to ∠a?

Angle a forms linear pairs with angle b and angle c, and is vertical to angle d. Angle e is a corresponding angle to angle a, and angle e forms linear pairs with angle f and angle g, and is vertical to angle h.

Video example.

Can you identify which angles are supplementary to ∠a?

Video example.

Congruent Angles

Write the information below in your notes for future reference.

Below are the pairs of angles that are congruent to each other when two parallel lines are intersected by a transversal.

Alternate interior angles

Showing angle c and angle f to be alternate interior angles. Showing angle d and angle e to be be alternate interior angles.

Alternate exterior angles

Showing angle a and angle h to be alternate exterior angles. Showing angle b and angle g to be alternate exterior angles.

Corresponding angles

Showing angle a and angle e to be corresponding angles. Showing angle b and angle f to be corresponding angles. Showing angle c and angle g to be corresponding angles. Showing angle d and angle h to be corresponding angles.

Supplementary Angles

Write the information below in your notes for future reference.

Below are the pairs of angles that are supplementary to each other when two parallel lines are intersected by a transversal.

Same side interior angles

Showing angle c and angle e to be same side interior angles. Showing angle d and angle f to be same side interior angles.

Same side exterior angles

Showing angle a and angle g to be same side exterior angles. Showing angle h and angle b to be same side exterior angles.

Video example.

Interactive Diagrams

Each link below allows you to move lines to see how this affects each pair of angles. Also, make sure you click "other pair" to the right of the diagram to see all pairs of each set.

Math Open Reference: Alternate Interior Angles

Math Open Reference: Alternate Exterior Angles

Math Open Reference: Corresponding Angles

Math Open Reference: Same Side Interior Angles

Math Open Reference: Same Side Exterior Angles

Explore more!

Alternate Interior Angles

Alternate Exterior Angles

Corresponding Angles

Same Side Interior Angles

Same Side Exterior Angles

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