Alternate Interior Angles Theorem
Alternate interior angles are the angles on the inner side of the two parallel lines but on the other side of the transversal.
When a transversal intersects two parallel lines, the resulting pairs of alternative interior angles are congruent.
Proof of Alternate Interior Angles Theorem
Given: Lines AB and CD are parallel.
To prove: ∠4 = ∠6 and ∠3 = ∠5
Proof: Suppose that a transversal, Y, crosses the parallel lines AB and CD. One special property of the parallel lines is that certain angles line up when a transversal passes across them.
In other words, comparable angles and vertically opposed angles become equal when a transversal joins parallel lines.
Taking this into consideration:
∠1 = ∠6 [Corresponding angles], and
∠1 = ∠4 [Vertically opposite angles]
Therefore, by comparing these relationships:
∠4 = ∠6 [Alternate interior angles]
Similarly,
∠3 = ∠5
As a result, it has been demonstrated that some pairings of angles on parallel lines cut by a transversal, such as ∠4 and ∠6, and ∠3 and ∠5, are truly equal!
Read More about Alternate Interior Angles.
Alternate Angles
Alternate Angles are a concept in geometry that arise when two lines are crossed by another line (known as the transversal). There are two types of alternate angles i.e., Alternate Interior Angles and Alternate Exterior Angles. Alternate angles are formed when a line intersects two or more lines, and if these lines are parallel, then alternate angles are equal.
In this article, we will explore the concept of alternate angles, including Alternate Interior Angles and Alternate Exterior Angles. We will also delve into the theorems related to these angles and provide proofs for them.
Table of Content
- What are Alternate Angles?
- Properties of Alternate Angles
- Types of Alternate Angles
- Alternate Interior Angles Theorem
- Alternate Exterior Angles Theorem