Alternate Interior Angles

In Geometry, the term alternate interior angles are often explained as when two lines are cut by a third line, a transversal. And generally, the Alternate Interior Angles 

If two parallel lines are transected in-between by a third line, the angles which are inside the parallel lines and on alternate sides of the third line are called alternate interior angles.

So, What is an Angle?

• An angle is basically formed when two lines each having one endpoint known as rays, meet at one point known as the vertex.

• The distance between the two rays leads to the formation of angles. 

• The maximum angle is equal to 360 degrees.

• In geometry, angles are often referred to using the angle symbol so angle A would be written as  Angle A.

What are Alternate Interior Angles?

• When a transversal passes through two parallel lines, alternate interior angles are formed. 

• They are also known as ‘Z angles’ as they generally form a Z pattern.

• Alternate interior angles are the angles that are formed on opposite sides of the transversal and inside the two lines are alternate interior angles.

• The pair of blue and pink angles denotes alternate interior angles. These pairs are alternate interior angles.

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Note: Alternate interior angle generally forms a z-pattern. Notice that in the diagram the pair of alternate interior angles makes a Z.

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Alternate angles are the angles found in a Z shaped figure. 

In the figure given above, we can see that two parallel lines are intersected by a transversal. Therefore, the angles inside the parallel lines are the alternate angles and they will be equal.

i, e., ∠a = ∠d and ∠b = ∠c

What are the Properties of Alternate Interior Angles?

Alternate Interior Angles: Theorem and Proof 

Statement for Alternate Interior Angles: The Alternate interior angle theorem states that “ if a transversal crosses the set of parallel lines,  then the alternate interior angles are congruent”.

Given: Line a is parallel to line b.

To Prove: We need to prove that angle 4 = angle 5 and angle 3 = angle 6

Proof: Suppose line a and line b are two parallel lines and l is the transversal which intersects parallel lines a and b at point P and Q. See the figure given below.

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From the properties of the parallel line, we know that if a transversal cuts any two parallel lines, then the corresponding angles and vertically opposite angles are equal to each other. Therefore we can write that,

∠2 = ∠5 ……….. Equation (1) (As angle 2 and 5 are Corresponding angles)

∠2 = ∠4 ………..Equation (2) (As angle 2 and 4 are vertically opposite angles)

From equation (1) and (2), we get;

∠4 = ∠5 (As  angles 4 and 5 are Alternate interior angles)

Similarly we can say that,

∠3 = ∠6

Hence proved.

Antithesis of The Theorem 

Statement: The Antithesis of the alternate interior angle theorem states that if the alternate interior angles produced by the transversal line on two coplanar are congruent, then the two lines are parallel to each other.

Given: Angle 4 = Angle 5 and Angle 3 = Angle 6

To prove: We have to prove that a is parallel to b.

Proof: Since we know that ∠2 = ∠4 (As angle 2 and 4 are vertically opposite angles)

So, we can now write,

∠2 = ∠5, (As angle 2 and 5 are corresponding angles)

Therefore, we can say that a is parallel to b.

Questions to be Solved:

Question 1) Find the measure of the angles 8 and 1 if the measure of angle 5 is 45 degrees and that of angle 4 is 135 degrees.

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Solution) Let’s list down the given information,

Measure of angle 5 is 45 degrees and that of angle 4 is 135 degrees.

Since 45° and angle 1 are alternate interior angles, they are congruent.

So, angle 1 = 45°

Since 135° and  angle 4 are alternate interior angles, they are congruent.

So, Angle 8 = 135°

Additional Information

What is a Transversal Line?

• A line that crosses or passes through two other lines is known as a transversal line. 

• At times, the two other lines are parallel, and sometimes the transversal passes through both lines at the same angle.

• The two other lines don't have to be parallel in order for a transversal to cross them.

What is a Straight Angle?

• A straight line forms a straight angle.

• It is also known as a flat angle.

• This angle measures equal to 180 degrees.

•  A straight angle or a flat angle can also be formed by two or more angles which on being added gives 180 degrees. 

• Here, in the diagram given below angle 1 + angle 2 is equal to 180.

What are Parallel Lines?

• Two lines on a two-dimensional plane that never meet or cross are known as parallel lines.

• There are special properties about the angles that are formed when a transversal passes through parallel lines; they do not occur when the lines are not parallel.