Alternate Interior Angles

One of the most basic concepts in geometry is the relationship between angles formed by two coplanar lines, which are intersected by a transversal. These two coplanar lines can either be parallel or non-parallel lines. Alternate interior angles have applications not only in solving many geometric problems but also in real-life applications. Hence, solving alternate interior angles is of great importance and useful in solving many geometric problems and proofs.

1.0Definition of Alternate Interior Angles

Alternate interior angles are those pairs of angles which are located on opposite sides of a transversal and, as the name suggests, found in the interior or inside of the two lines intersected. These are angles formed when a transversal intersects two parallel lines or sometimes non-parallel lines, and they are found inside the parallel lines.

For Example: In the figure, m and n are two parallel lines with interior angles named 1, 2, 3, and 4. Hence, alternate interior angles will be:

∠1 = ∠2 

∠3 = ∠4 

2.0Properties of Alternate Interior Angles

1. Congruence for Parallel Lines: If two parallel lines are cut by a transversal, alternate interior angles are equal. This is the fundamental property that helps prove lines are parallel.
2. Formed by a Transversal: Alternate interior angles are formed when a transversal intersects two lines. The angles lie on opposite sides of the transversal and are inside the two lines.
3. Non-parallel lines do not promise equality: Alternate interior angles formed when the lines are not parallel do not equal each other, but the principle of alternate interior angles is not affected.
4. Can Be Used to Prove Parallelism: If a transversal cuts two lines and forms congruent alternate interior angles, it can be said that the two lines are parallel. 

3.0Theorems Related to Alternate Interior Angles

Alternate Interior Angles Theorem

Statement: If a transversal cuts two parallel lines, then every pair of alternate interior angles is congruent.

To Prove: ∠3 = ∠5 and ∠4 = ∠6

Given: AB∥CD 

Solution: Since AB∥CD, therefore,

∠2 = ∠4 (Vertically Opposite angles)...(1)

∠2 = ∠6 (Corresponding Angles) …..(2)

From Equations 1 and 2 

∠4 = ∠6 

Similarly, 

∠3 = ∠5

Hence, proved that If a transversal cuts two parallel lines, then every pair of alternate interior angles is congruent.

Converse of the Alternate Interior Angles Theorem

Statement: If two lines are crossed by a transversal and the pair of alternate interior angles are equal to each other, then these lines are parallel.

To Prove: AB ∥ CD

Given: ∠3 = ∠5 and ∠4 = ∠6

Solution: As, ∠2 = ∠4 and ∠6 = ∠8 (Vertically opposite angles) …..(1)

∠3 = ∠5 and ∠4 = ∠6 (Given) …..(2)

From Equations 1 and 2 

∠2 = ∠6

∠4 = ∠8 

Hence, AB ∥ CD by the converse of the corresponding angle theorem.

4.0Alternate Interior And Exterior Angles

Alternate interior angles and alternate exterior angles are two important concepts of geometry. While alternate interior angles lie on the inside of two coplanar lines by the transversal, alternate exterior angles are formed on the outside of these lines by the transversal. 

Just like alternate interior angles, Exterior angles are also equal to one another, given that the coplanar lines are parallel, as shown in the figure. Here, ∠1 = ∠4 and ∠2 = ∠3. 

5.0Common Mistakes And Misconceptions

• Confusing Alternate Interior Angles with Corresponding Angles: Corresponding angles always lie on the same side of the transversal, while alternate interior angles lie on opposite sides.
• Assuming All Interior Angles Are Equal: Lines have equal alternate interior angles when parallel; other interior angles may be unequal.
• Forgetting the Parallel Condition: Alternate interior angles are not necessarily equal unless the two lines are parallel.

6.0Solved Examples

Problem 1: Given that two parallel lines l₁∥l₂ are intersected by a transversal t, and form the following alternate interior angles: ∠1 = (4y + 15) and ∠2 = (6y − 5). Find the values of y, and then determine the value of ∠1.

Solution: Give that l₁ ∥ l₂, Hence, 

∠1 = ∠2

4y + 15 = 6y − 5

4y – 6y = –5 – 15

–2y = –20

y = 10 

Hence, ∠1 = 4y + 15 = 4(10) + 15

∠1 = 40 + 15 = 65°

Problem 2: Two parallel lines are cut by a transversal. One alternate interior angle is (3x - 10)° and the other is (2x + 20)°. Find x.

Solution: Give that 

3x - 10 = 2x + 20 (alternate interior angles)

3x – 2x = 20 + 10 

x = 30

Problem 3: Given that two coplanar lines are parallel, If one alternate interior angle measures 120°, what is the measure of its corresponding angle?

Solution: As the coplanar lines are parallel. Hence, 

The corresponding angle will also be 120° by the corresponding angle theorem.