Alternate Angles: Definition, Types, Theorem

Alternate angles are a special kind of angles in geometry and form the basics for understanding concepts relating to angles and parallel lines. Alternate angles are the angles formed on either side of the transversal which cuts through two or more parallel lines. These angles are always non adjacent. In this article, we are going to discuss the proper definition of Alternate Angles, types, Theorem, and some sample questions on alternate angles.

Key Terms: Alternate Angles, Transversal, Parallel lines, Interior Alternate Angles, Exterior Alternate Angles, Alternate Angle theorem

Also read: Isosceles Triangle Theorems

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Alternate Angles Definition

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Consider two coplanar parallel lines AB and CD as shown in the figure given above. If a line XY cuts these two parallel lines AB and CD at P and Q respectively, then the figure tends to form certain angles with respect to the lines. Whenever a transversal cuts two or more parallel lines, the alternate angles are always equal. 

 Hence in the given figure;

• ∠4=∠6 and ∠3=∠5
• ∠1=∠7 and ∠2=∠8

These are the pairs of alternate angles and are always equal.

Read More: Calculus Formula

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Types of Alternate Angles

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There are 2 types of alternate angles depending on the position of angles with respect to the transversal. They are:

1. Interior alternate angles: The pair of angles formed on the inner or interior sides of the two parallel lines when a transversal makes a cut through them. According to the figure given above, ∠3, ∠4, ∠5, and ∠6 form the interior alternate angles.
2. Exterior alternate angles: The pair of angles formed on the outer or exterior sides of the two parallel lines when a transversal makes a cut through them. According to the figure given above, ∠1, ∠2, ∠7, and ∠8 form the exterior alternate angles.

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Alternate Angles Theorem

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According to the alternate angles theorem, the alternate angles formed when a transversal cuts through two or more parallel lines, are always congruent.

To prove: When 2 or more parallel lines are cut by a transversal, the alternate angles formed are always congruent.

Proof: 

Consider two coplanar parallel lines AB and CD as shown in the figure given above. If a line XY cuts these two parallel lines AB and CD at P and Q respectively, then the figure tends to form certain angles with respect to the lines.

At the point of intersection of straight line AB, the following conditions will hold true:

 ∠3+ ∠4=180°_________(1)

∠1+∠4=180°__________(2) (Since AB and transversal are straight lines)

Now, from equations (1) and (2), we can conclude that ∠1=∠3.

Similarly,

At the point of intersection of straight line CD, the following conditions will hold true:

 ∠6+ ∠5=180°_________(3)

∠8+∠5=180°__________(4) (Since AB and transversal are straight lines)

Now, from equations (3) and (4), we can conclude that ∠6=∠8.

In the given figure above:

∠1=∠5 and ∠4=∠8 (Corresponding angles)

So now this implies that ∠3=∠5 and ∠4=∠6, where ∠3,∠4,∠5 and ∠6 are alternate angles.

Hence alternate angles are congruent OR alternate angles are equal.

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Things to remember

• Whenever a transversal cuts through two or more parallel lines, the alternate angles formed are always equal.
• The pair of angles formed on the inner or interior sides of the two parallel lines when a transversal makes a cut through them are known as interior alternate angles.
• The pair of angles formed on the outer or exterior sides of the two parallel lines when a transversal makes a cut through them are known as exterior alternate angles.

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