Consecutive Interior Angles

Introduction to Consecutive Interior Angles

Consecutive interior angles are formed on the inner sides of the transversal and are also known as co-interior angles or same-side interior angles. When a transversal crosses any two parallel lines, it forms many angles like alternate interior angles, corresponding angles, alternate exterior angles, consecutive interior angles.

Consecutive interior angles are defined as the pair of non-adjacent interior angles that lie on the same side of the transversal. The word 'consecutive' refers to things that appear next to each other. Consecutive interior angles are located next to each other on the internal side of a transversal. Observe the following figure and the properties of consecutive interior angles to identify them.

• Consecutive interior angles have different vertices.
• They lie between two lines.
• They are on the same side of the transversal.
• They share a common side.

What are Consecutive Interior Angles

When two lines are cut by a transversal, the pair of angles on one side of the transversal and inside the two lines are called the consecutive interior angles. If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary. The consecutive interior angles theorem states that when the two lines are parallel, then consecutive interior angles are supplementary to each other. Supplementary means that the two angles add up to 180 degrees. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.  

When a transversal crosses a pair of parallel lines, many pairs of angles are formed other than consecutive interior angles. They are corresponding angles, alternate interior angles, and alternate exterior angles.

The Consecutive Interior Angles Theorem states that if the two lines cut by the transversal are parallel, then the two consecutive interior angles are supplementary. Parallel means the two lines have the same slope (rate of change) and will never intersect. Two angles are supplementary when the sum of the two angles after being added together equals 180.

Consecutive Interior Angles 

There is only one situation in which two consecutive angles are congruent. In this situation, the two lines must be parallel and the transversal must be perpendicular to those two lines. Recall, perpendicular means the two lines create a 90 angle. If this case happens, every single angle in the diagram would be a right angle (90) and therefore all be congruent, including the consecutive interior angles. This is called the perpendicular transversal theorem.

Non-consecutive angles

The standard form of a line can be particularly helpful when solving a system of equations. For instance, when using the elimination method to solve a system of equations, we can easily align the variables using standard form. As applied to a polygon, a diagonal is a line segment joining two non-consecutive vertices. Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. A diagonal of a polygon is a line segment that connects any two nonconsecutive vertices.

When are Two Angles Consecutive?

Proof

If we draw to parallel lines and then draw a line transversal through them, we will get eight different angles. The eight angles will together form four pairs of corresponding angles. The eight angles include corresponding angles, alternate interior and exterior angles, vertically opposite angles, and co-interior angles.

A quick way to prove this theorem is to use the knowledge that two angles forming a line are also supplementary. Referencing the same diagram as before, two angles that create a perfect straight line are 2 and 4, and 6 and 8. Allowing angle 2 to be 120 means that angle 4 is 60. Since the two lines are parallel, angle 6 and angle 8 will hold the same measurements. Angle 6 = 120 and angle 8 = 60. Without any extra math, the two consecutive interior angles (angle 4 and angle 6) are already defined. Angle 4 = 60 and angle 6 = 120, therefore they are supplementary as 120+60 = 180. This process will work with any angle measurements used because the two angles will always create a straight line above and below the consecutive interior angles.

Since two consecutive interior angles are supplementary only if the two lines being crossed by the transversal are parallel, the reverse is also true. This means if two consecutive interior angles are supplementary, then the two lines forming them will be parallel.

Example

Are the two lines in the diagram parallel if angle 3 = 35 and angle 5 = 150?

Since 35 + 150 = 185, the two lines are not parallel. This does not mean that angle 3 and angle 5 are not consecutive interior angles, it just means that the lines forming the angles are not parallel.

In geometry, a consecutive interior angles is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, consecutive exterior angles, corresponding angles, and alternate angles. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

Proposition 1.27 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of alternate angles of a transversal are congruent then the two lines are parallel (non-intersecting).

It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of alternate angles of a transversal are congruent (Proposition 1.29 of Euclid's Elements).

Euclid's Proposition 27 states that if a transversal intersects two lines so that alternate interior angles are congruent, then the lines are parallel. Euclid proves this by contradiction: If the lines are not parallel then they must intersect and a triangle is formed.