Alternate Interior Angles: Definition, Theorem & Examples

Alternate Interior Angles are the pair of angles formed when two coplanar lines are intersected by a transversal. These angles are formed on the inner corners of the intersections and lie on the opposite sides of the transversal. On the intersection of two parallel lines by a transversal, eight angles are formed. In these eight angles, the angles that lie on the inner side of the parallel lines but on the opposite sides of the transversal are called the alternate interior angles. Alternate Interior angles represent whether the two given lines are parallel to each other or not. If alternate interior angles are equal to each other, then, the lines intersected by the transversal are parallel.

Read More: NCERT Solutions for Class 9 Maths Lines and Angles

Key Terms: Alternate Interior Angles, Transversal, Parallel Lines, Angles, Co-Interior Angles, Corresponding Angles, Alternate Interior Angles Theorem

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What are Alternate Interior Angles?

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Alternate Interior Angles are formed when a transversal passes through two parallel or non-parallel lines. When two parallel lines are intersected by a transversal, the pair of angles that are formed on the inner side of the parallel lines, but on the opposite sides of the transversal are referred to as the alternate interior angles. Alternate Interior Angles are always equal. 

Given below is a figure in which two parallel lines are intersected by a transversal. Here AB and CD are the two parallel lines that are crossed by a transversal. 

Alternate Interior Angles

The pairs of alternate interior angles in the figure are:

• ∠4 and ∠6
• ∠3 and ∠5

As alternate interior angles are equal, so, 

• ∠4 = ∠6
• ∠3 = ∠5

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Properties of Alternate Interior Angles

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Here are a few important properties of alternate interior angles: 

• Alternate Interior angles are always congruent to each other.
• The sum of the angles that are formed on the same side of the transversal which are inside the two parallel lines is always equal to 180°.
• Alternate interior angles do not have any specific properties in the case of non-parallel lines.

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Alternate Interior Angles Theorem and Proof 

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According to the Alternate Interior Angles Theorem “If a transversal crosses the set of parallel lines, then the alternate interior angles are congruent to each other”.

Given: Line l is parallel to Line m.

To Prove: ∠4 = ∠5 and ∠3 = ∠6

Proof: Let line l and line m be two parallel lines and n is the transversal that intersects parallel lines l and m at points P and Q. The figure for the same is given as: 

From the properties of the parallel lines, we know that if a transversal cuts any two parallel lines, then the corresponding angles and vertically opposite angles are always equal to each other. Therefore, it can be said that

∠2 = ∠5 ……….. (1) (Corresponding angles)

∠2 = ∠4 ……….. (2) (Vertically Opposite Angles)

From equations (1) and (2), we get

∠4 = ∠5 

Similarly, we can say that,

∠3 = ∠6

Hence Proved.

Read More: Exterior Angle Theorem

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Converse of Alternate Interior Angles Theorem

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According to the Antithesis of the Alternate Interior Angle Theorem, “If the alternate interior angles produced by the transversal line on two coplanar are congruent, then the two lines are considered to be parallel to each other.”

Given: ∠4 = ∠5 and ∠3 = ∠6

To Prove: Line l is parallel to Line m (l || m)

Proof: Consider the figure given below: 

We know that, 

∠2 = ∠4 (Vertically Opposite Angles)

So, we can write that

∠2 = ∠5, (Corresponding Angles)

Therefore, we can say that Line l is parallel to Line m.

Read More: Lines and Angles MCQs

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How to Find the Alternate Interior Angles?

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The alternate interior angles theorem states that the alternate interior angles of two parallel lines are equal. We can find alternate interior angles using this theorem. 

Example: The following figure shows a map in which Street 1 runs perpendicular to Street 2 and Street 3, which are parallel. Another street named Street 4 makes an angle of 40° with Street 2. Find the measure of angle x.

Going with the alternate interior angles theorem, if the lines (parallel streets) are parallel, and Street 4 is considered to be the transversal, then x and 40° are the alternate interior angles. So, both these angles are equal. 

Therefore, x = 40°

Solved Examples on Alternate Interior Angles

Example 1: What will be the value of B and D in the given figure? (3 Marks)

Solution: In the given figure, 

45° and D are alternate interior angles, thus, both of them are congruent.

So, D = 45°

Similarly, 135° and B are also alternate interior angles, so, they are congruent.

So, B = 135°

Thus, the value of B and D in the given figure is 135° and 45° respectively.

Example 2: What will be the value of the missing angles A, C, and D in the following figure?

Solution: From the given figure, we can clearly see that

∠A, 110°, ∠C and ∠D are all alternate interior angles. Thus, 

∠C = 110°

We know that, 

∠C+∠D = 180° (Supplementary Angles)

∠D = 180° – ∠C = 180° – 110° = 70°

Now, ∠A = ∠D (Alternate Interior Angles)

∠A = 70°

Read More: Lines and Angles Important Questions

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Alternate Exterior Angles

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Alternate Exterior Angles are defined as those angles that have different vertices and lie on the alternate sides of the transversal and are exterior to the lines. In simpler terms, when a transversal intersects two parallel lines, the alternate exterior angles formed are always equal. 

In the given figure, ∠1 & ∠7 and ∠2 & ∠8 are the pairs of alternate exterior angles.

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Co-interior Angles

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Co-interior angles are two angles that lie on the same side of the transversal. They are the interior angles that sum up to 180 degrees. It can be said that the sum of two interior angles, which are on the same side of the transversal is supplementary. Co-interior angles are also known as consecutive interior angles or the same side interior angles.

• Co-interior angles have different vertices but they lie on the same side of the transversal.
• Co-interior angles lie between two parallel or non-parallel lines.
• They are non-adjacent angles.
• The sum of co-interior angles formed is always 180 degrees.

Co-interior Angles

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Things to Remember

• Alternate interior angles are the angles that are formed by a transversal intersecting two parallel or non-parallel lines. 
• They lie on the inner side of the two parallel lines but on the opposite sides of the transversal.
• Each pair of alternate interior angles is congruent, i.e, they are equal.
• The Alternate Interior Angle Theorem states that “ if a transversal intersects the set of parallel lines, the alternate interior angles are congruent”.
• The converse of the alternate interior angle theorem states that “if the alternate interior angles produced by the transversal are congruent, then the two lines are parallel.”