Consecutive Interior Angles

Also known as co interior angle, consecutive angles are vertically opposite angles that are equal to each other. That being said, when two lines are cut by a transversal, the pair of angles on one side of the transversal and on the interior of the two lines are known as the consecutive interior angles. In the figure shown below, angles 3 and 5 are said to be consecutive interior angles.

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Alternate Interior Angles

Angles on the opposite side of the transversal are called alternate interior angles.

Example: angle 1 and angle 2 are alternate interior angles.

Using the concept of angles and their results we can prove that angles in a triangle add to 180°

Draw two parallel lines E and F

Since alternate interior angles are equal and pairs of angle 1 and angle 2 form alternate interior angles, therefore, angle 1 and angle 2 are equal. The same is for angle 3 and angle 4.

Since angle 1, angle 3 and angle 5 are around the straight line, therefore they add up to 180°

From this, we can conclude that angles in a triangle add to 180°

Since angle 1 = angle 2 and angle 3 = angle 4, therefore angle 5 + angle 4 + angle 2 = 180°

Also, angle 6 is the exterior angle of the triangle and the exterior angle of the triangle is equal to the sum of the two interior angles.

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Properties of Consecutive Interior Angles

You know the co interior angles definition. Now refer to the following key properties of consecutive interior angles:

• Consecutive interior angles are formed when two lines are cut by the transversal the pair of angles formed inside the lines on one side.

• Consecutive interior angles are called supplementary when the two lines are parallel, and the pair of angles add to 180°

• The angles are called “consecutive” because they follow each other consecutively on the same side of the line.

• The pair of angles which lie on the exterior of the transversal on the same side are called consecutive exterior angles.

• Angles formed inside the parallel lines are cut by the transversal on the same side are called co-interior angles. Co-interior angles are supplementary but they are not equal when the lines are parallel.

Consecutive Interior Angles Theorem- Proof

See the figure shown below.

Since we are already aware that the two lines are parallel, therefore we have:

∠1=∠5 (corresponding angles)

From the aforementioned two equations, we get

∠1+∠4=180⁰

In the same manner, we can show that

∠2+∠3=180⁰

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Converse of Consecutive Interior Angle Theorem 

Proof

Suppose that

∠ 1+∠4=180⁰→(1)

Because ∠5 and ∠4 forms a linear pair,

∠5+∠4 =180∘ →(2)

From the equations (1) and (2),

∠1=∠5

Therefore, a pair of corresponding angles are equivalent that can only take place if the two lines are parallel.

Thus, the converse of consecutive interior angle theorems is proven.

Solved Examples

Example:

Are the following lines naming l and m in the figure below parallel?

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Solution:

In the figure, 125o and 60o are the consecutive interior angles given that they are supplementary.

But

125 + 60 =185o

Hence, 125o and 60o are NOT supplementary angles.

Therefore, the given lines are NOT parallel as per the "Consecutive Interior Angle Theorem,”

Hence, l and m are NOT parallel.

Example:

Consider the given figure, in which L1 and L2 are the parallel lines. Calculate the value of ∠C?

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Solution:

Through C, draw a line parallel to L1 and L2, as given in the figure.

We have:

∠x = ∠β = 60 degrees (alternate interior angles)

∠y = 180o – 120 degrees (consecutive interior angles)

∠y = 60degrees

Therefore, we obtain

∠C = ∠x + ∠y = 120degrees

Hence, ∠C=120 degrees

Fun Facts

• The eight angles together form four pairs of corresponding angles. 

• Corresponding angles are in congruence to one another if the two lines are parallel.

• Vertical angles are formed by only two intersecting lines and they are non-adjunct angles.