Alternate Interior Angles - Definition, Theorems, Examples

An angle is made when two line segments or two rays meet at their endpoints. When a line segment or ray is drawn from a point on a line, a pair of linear angles is created. When two lines intersect each other, four angles are created. These contain linear pairs of angles and vertically opposite angles.

Moreover, when we have two parallel lines intersected by a non-parallel line, eight angles are created. This non-parallel line is known as a Transversal for the set of parallel lines. These eight angles may create different types of pairs depending on their positions and characteristics. For example, Corresponding angles, alternate interior angles, co-interior angles etc.

What Do We Mean By Pair Of Alternate Interior Angles?

As we can observe from the diagram below, a pair of parallel lines is being intersected by a transversal creating eight different angles, four on each of the parallel lines. Out of these four angles on each parallel line, two angles lie in the region between two parallel lines and hence, known as interior angles. Similarly, the other two lie on the opposite side of the other parallel lines and thus, known as exterior angles.

Now, on each parallel line, the two interior angles lie on two sides of the transversal. Now, let us consider one of the interior angles on one parallel line. Corresponding to this angle, when we move along the transversal line towards the other parallel line, we observe another interior angle which lies on the other side of the transversal on the second parallel line. As the other side correlates to the term alternate side. And, hence, the two angles together are known as a “Pair of Alternate Interior Angles”.

Interestingly, a pair of alternate interior angles possess a property that they are equal in magnitude. This becomes a very important result in solving geometric problems. One can easily find alternate interior angles by drawing a Z-shape. This is why they are also referred to as "Z angles" because they typically create a Z pattern.

In the above figure, you can see how a transversal intersects two parallel lines. The alternating angles between the parallel lines will therefore be equal.

The Characteristics Of Alternate Interior Angles

• These angles are congruent, which means that their measurements are equal.

• Every time an angle is generated on the same side of a transversal that is inside of two parallel lines, the sum of those angles is always 180 degrees, which means that they are supplementary. These are referred to as co-interior angles and are therefore supplementary to each other.

• Alternate interior angles for non-parallel lines don't have any special characteristics.

Alternate Exterior Angles

Alternate exterior angles are those that lie in the exterior region of the parallel lines, have different vertices, and lie on different sides of the transversal.

A transversal always results in alternate exterior angles that are equal when it crosses two parallel lines.

Alternate Interior Angles Theorem

According to the Alternate Interior Angles Theorem, when transversal cuts two parallel lines, the alternate interior angles created are congruent.

We are aware that the corresponding angles and vertically opposite angles are equal whenever a transversal crosses any two parallel lines.

Corresponding angles,

\angle2=\angle6

Vertically opposite angles,

\angle2=\angle3

The two equations above clearly show that,

\angle3=\angle6

Similarly,

\angle4=\angle5

Hence, the alternate interior angles are congruent, and the alternate interior angles theorem is proved.

Converse Of Alternate Interior Angles Theorem

According to the converse theorem, If the transversal line on two co-planar creates alternate interior angles that are congruent, then two lines are parallel.

Now as,

\angle3=\angle6

and \angle4=\angle5

And as angles 2 and 4 are equal, being vertically opposite angles. Thus, it can be said that,

\angle2=\angle5

These are corresponding angles.

The two lines are therefore parallel to one another.

Hence, the converse of the alternate interior angles theorem is proved.

Co-Interior Angles

The two angles that are on the same side of the transversal and between the parallel lines are known as co-interior angles. These are interior angles that sum up to 180 degrees. The two internal angles that are on the same side of the transversal are said to be supplementary i.e, their sum is 1800. The co-interior angles are not equal to one another and resemble a "C" shape. The consecutive interior angles or the interior angles on the same side are other names for the co-interior angle.

Properties Of Co-Interior Angles

• Despite having different vertices, co-interior angles have a common side to the transversal.

• When a transversal is drawn on two parallel lines, the sum of co-interior angles that are produced is always equal to 180 degrees.

Allied Angles

When the sum or difference of two angles equals 0°, 90°, or a multiple of 90°, they are said to be allied angles. As an illustration, we can claim that 120 degrees and 60 degrees are allied angles since their sum is 180 degrees, and 180 is a multiple of 90.