Alternate Interior Angles

Alternate interior angles are 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.

Alternate Interior Angles Definition

The angles which are formed inside the two parallel lines, when intersected by a transversal, are equal to its alternate pairs. These angles are called alternate interior angles.

Alternate interior angles

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

i,e. ∠A = ∠D and ∠B = ∠C

Theorem and Proof

The theorem states that “ if a transversal crosses the set of parallel lines, the alternate interior angles are congruent”.

Given: a//b

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.

Alternate Interior Angles Theorem Proof

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]

Similarly,

∠3 = ∠6

Hence, it is proved.

Antithesis of Theorem

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: ∠4 = ∠5 and ∠3 = ∠6

To prove: a//b

Proof: Since ∠2 = ∠4 [Vertically opposite angles]

So, we can write,

∠2 = ∠5, which are corresponding angles.

Therefore, a is parallel to b.

Properties

  • These angles are congruent.
  • Sum 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.

Examples

Question 1: Find the value of B and D in the given figure.

Alternate Interior Angles Example

Solution:

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

So, D = 76°

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

So, B = 135°

Question 2: Find the missing angles A, C and D in the following figure.

Alternate Interior Angles Problem

Solution: As angles ∠A, 110°, ∠C and ∠D are all alternate interior angles, therefore;

∠C = 110°

By supplementary angles theorem, we know;

∠C+∠D = 180°

∠D = 180° – ∠C = 180° – 110° = 70°

Since,

∠A = ∠D

Therefore, ∠A = 70°.

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