Alternate Angles

Last updated on 26 October 2025

Table of contents

Alternate Angles Definition
Types of Alternate Angles
1. Alternate Interior Angles
2. Alternate Exterior Angles
Alternate Angles Theorems
1. Alternate Interior Angle Theorem
2. Alternate Exterior Angle Theorem

Angles play a fundamental role in geometry, providing a framework for understanding the relationships between lines and shapes. One crucial concept within this realm is that of alternate angles. Alternate angles are a specific type of angle pair formed when a transversal intersects two parallel lines. In this article, we will delve into alternate angles, exploring their definition, properties, and applications.

Alternate Angles Definition

Alternate angles, also known as "Z-angles", are pairs of angles formed when a transversal intersects two parallel lines. Specifically, alternate angles are angles that are on opposite sides of the transversal and located at the intersection points on the parallel lines. Angles in the pair of alternate angles don't have the same vertices. Visually, they often resemble the letter "Z", which is why they are sometimes referred to as Z-angles.

Observe the above figure.

Lines $PQ$ and $RS$ are parallel.
Line $MN$ intersects the lines $PQ$ and $RS$ .
Angles $∠ a$ , $∠ b$ , $∠ c$ and $∠ d$ are formed at the intersection point of lines $PQ$ and $MN$ .
Angles $∠ w$ , $∠ x$ , $∠ y$ and $∠ z$ are formed at the intersection point of lines $RS$ and $MN$ .
The pairs of alternate angles in the above figure are:
- $∠ d$ and $∠ x$
- $∠ c$ and $∠ w$
- $∠ a$ and $∠ y$
- $∠ b$ and $∠ z$

Types of Alternate Angles

There are two types of alternate angles formed by a transversal intersecting two parallel lines: Alternate Interior Angles and Alternate Exterior Angles.

Alternate Interior Angles
: Alternate angles that lie in the interior region of the parallel lines are called Alternate Interior Angles.
Alternate Exterior Angles
: Alternate angles that lie in the exterior region of the parallel lines are called Alternate Exterior Angles.

In the above figure,

$∠ d$ , $∠ x$ and $∠ c$ , $∠ w$ are the pairs of alternate interior angles.
$∠ a$ , $∠ y$ and $∠ b$ , $∠ z$ are the pairs of alternate exterior angles.

Alternate Angles Theorems

Now, let's learn about the theorems related to Alternate Angles.

Alternate Interior Angle Theorem

This theorem states that, when a transversal intersects two parallel lines, the alternate interior angles are equal.

In the figure above, Lines

L_{1}

and

L_{2}

are parallel and line

L_{3}

is transversal.
Hence by Alternate Interior Angle Theorem,

∠ a = ∠ y

and

∠ b = ∠ x

Alternate Exterior Angle Theorem

This theorem states that, when two lines are parallel and are intersected by a transversal, then the alternate exterior angles are considered as congruent angles.

In the figure above, Lines

L_{1}

and

L_{2}

are parallel and line

L_{3}

is transversal.
Hence by Alternate Exterior Angle Theorem,

∠ p = ∠ m

and

∠ q = ∠ n

Alternate Angles

Alternate Angles Definition

Types of Alternate Angles

Alternate Interior Angles

Alternate Exterior Angles

Alternate Angles Theorems

Alternate Interior Angle Theorem

Alternate Exterior Angle Theorem

Triangle Similarity Theorems

Circle