In Geometry, one of the special kind of angles is alternate angles. Alternate angles are the set of non-adjacent angles on either side of the transversal. In this article, let us discuss the proper definition of alternate angle, types, theorem, and an example in detail.

Table of Contents:

• Definition
• Types
• Theorem
• Example
• FAQs

Alternate Angle Definition

If a straight line intersects two or more parallel lines, then it is called a transversal line. When the coplanar lines are cut by a transversal, some angles are formed. Those angles are known as interior or exterior angles. Alternate angles are shaped by the two parallel lines crossed by a transversal.

Consider the given figure,

EF and GH are the two parallel lines.

RS is the transversal line that cuts EF at L and GH at M

If two parallel lines are cut by a transversal, then the alternate angles are equal.

Therefore, ∠3 = ∠ 5 and ∠4 = ∠6

∠2 = ∠8 and ∠1 = ∠7

Types of Alternate Angles

Based on the position of the angles, the alternate angles are classified into two types, namely

• Alternate Interior Angles – Alternate interior angles are the pair of angles on the inner side of the two parallel lines but on the opposite side of the transversal.

          From the above-given figure,

          ∠3, ∠4, ∠5, ∠6 are the alternate interior angles

• Alternate Exterior Angles – Alternate exterior angles are the pair of angles on the outer side of the two parallel lines but on the opposite side of the transversal.

          From the above-given figure,

          ∠1, ∠2, ∠7, ∠8 are the alternate exterior angles

Alternate Angles Theorem

Alternate angle theorem states that when two parallel lines are cut by a transversal, then the resulting alternate interior angles or alternate exterior angles are congruent.

To prove:

If two parallel lines are cut by a transversal, then the alternate interior angles are equal.

Proof:

Assume that PQ and RS are the two parallel lines cut by a transversal LM. W, X, Y, Z are the angles created by a transversal

At the intersection point on the straight lines PQ and LM,

∠W + ∠Z = 180° (PQ is the straight line)—-(1)

∠X + ∠Z = 180° (LM is the straight line)—-(2)

So, from (1) and (2), we get

∠W = ∠X

Again, at the intersection point on the straight lines RS and LM,

∠W + ∠Z = 180° (RS is the straight line)—-(3)

∠W + ∠Y = 180° (LM is the straight line)—-(4)

So, from (3) and (4), we get

∠Z = ∠Y

Therefore, it is concluded that the alternate interior angles are congruent.

Hence, proved.

Alternate Angles Example

Question:

From the given figure, find the angles Y, X, and Z.

Solution:

Given:

∠Y = 60°

From the alternate interior angle theorem, ∠Y = ∠Z.

Therefore, the value must be equal.

∠Z = 60°

Since RS is the straight line, ∠W + ∠Z = 180°

So, ∠W = 180° – ∠Z

Substitute the value ∠Z = 60°

∠W = 180° – 60°

∠W = 120°

Again, from the alternate interior angle theorem, ∠W = ∠X.

Therefore, ∠X = 120°

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