Alternate Interior Angles: Definition, Theorem & More
Alternate interior angle is the angles formed when a transversal intersects two coplanar lines. They lie on the inner side of the parallel lines but the opposite sides of the transversal. The transversal crosses through the two lines which are Coplanar at separate points. These angles represent whether the two given lines are parallel to each other or not. If these angles are equal to each other then the lines crossed by the transversal are parallel.
An angle is formed when two rays, a line with one endpoint, meet at one point called a vertex. The angle is formed by the distance between the two rays. Angles in geometry are often referred to using the angle symbol so angle A would be written as angle A or When a line (called a transversal) intersects a pair of lines, AIAs are formed on opposite sides of the transversal. If the pair of lines are parallel then the alternate interior angles are equal to each other.
Alternate Interior Angles
A transversal line is a line that crosses or passes through two other lines. Sometimes, the two other lines are parallel, and the transversal passes through both lines at the same angle. The two other lines don’t necessarily have to be parallel in order for a transversal to cross them.
A straight angle, also called a flat angle, is formed by a straight line. The measure of this angle is 180 degrees. A straight angle can also be formed by two or more angles that sum to 180 degrees. Here, angle 1 + angle 2 = 180.
Parallel lines are two lines on a two-dimensional plane that never meet or cross. When a transversal passes through parallel lines, there are special properties about the angles that are formed that do not occur when the lines are not parallel. Notice the arrows on lines m and n towards the left. These arrows indicate that lines m and n are parallel.
AIAs are formed when a transversal passes through two lines. The angles that are formed on opposite sides of the transversal and inside the two lines are AIAs. Notice the pairs of blue and pink angles.
Alternate Interior Angles Theorem/Proof
The theorem states that if a transversal crosses the set of parallel lines, the alternate interior angles are congruent.
To prove: ∠4 = ∠5 and ∠3 = ∠6
Proof: Suppose a and b are two parallel lines and l is the transversal which intersects a and b at point P and Q. See the figure.
From the properties of the parallel line, we know if a transversal cuts any two parallel lines, the corresponding angles and vertically opposite angles are equal to each other. Therefore,
∠2 = ∠5 (i))(Corresponding angles)
∠2 = ∠4 (ii)(Vertically opposite angles)
From eq.(i) and (ii), we get
∠4 = ∠5 (Alternate interior angles)
∠3 = ∠6
Hence, it is proved.
Alternate Interior Angles Properties
- These angles are congruent.
- Some of the angles formed on the same side of the transversal which are inside the two parallel lines is always equal to 180°.
- In the case of non – parallel lines, alternate interior angles don’t have any specific properties.