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Alternate Interior Angles

Alternate interior angles are formed on the opposite sides of the line of transversal and inside the two lines. The line of transversal passes through the two lines who are coplanar at distinct points. The alternate interior angles will be on the opposite sides to this line and will always be inside the line. These angles can tell whether the two lines are parallel to each other or not. If these angles are equal then the two lines are parallel.

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Alternate Interior Angles Definition

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The angles which are on the opposite sides of the line of transversal and created between the two coplanar lines are known as alternate interior angles.
Alternate Interior Angles Diagram

  • $\angle$1 and $\angle$2 are alternate interior angles
  • $\angle$a and $\angle$b are alternate interior angles

Properties of Alternate Interior Angles

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Given below are some properties of alternate interior angles:
  • Alternate interior angles are congruent.
  • Sum of the angles formed on either side of the transversal which are inside the same parallel lines is 180$^o$.
  • Alternate interior angles formed in non - parallel lines has no specific properties.

The alternate interior angles will be congruent if the transversal will cut through a set of parallel lines.

Given: The parallel lines are "l" and "m" and line 't' is the transversal.

alternate  interior angles theorem

Proof:

Statement
Reasons
Step 1: $l$ and $m$ are parallel lines and $t$ is the transversal.
Given
Step 2: $\angle$ 2 and $\angle$ 6 are congruent ($\angle$ 2 = $\angle$ 6) Corresponding angles
Step 3: $\angle$ 4 and $\angle$ 6 are congruent ($\angle$ 4 = $\angle$ 6) Vertically opposite angles
Step 4: $\angle$ 4 and $\angle$ 6 are congurent ($\angle$ 4 = $\angle$ 6) From Step 2 and Step 3 (Using transitive property).

Converse of the Alternate Interior Angles Theorem


If the alternate interior angles formed by the transverse on two coplanar lines are congruent than the lines are parallel to each other.

Given: $\angle$4 = $\angle$6 and $\angle$3 = $\angle$5 (Using above figure)

To Prove: ${l}$ || ${m}$.

Proof: $\angle$2 = $\angle$4 and $\angle$5 and $\angle$7 are vertical opposite angles

$\angle$2 = $\angle$ 6 (Corresponding angles)

Thus, ${l}$ || ${m}$ are parallel.
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Alternate Interior Angles in Real Life

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In real life, we come across non coplanar lines in many places. Few such examples are given below.

A window pane and racket is a good example to understand different pairs of angles.

Alternate Interior Angles Examples

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Given below are some examples based on alternate interior angles.

Solved Examples

Question 1:

Find the value of angle a and b in the figure given below.

alternate interior angles examples


Solution:

As 76° and b are alternate interior angles, they are congruent.

So, b = 76°

As 104° and a are alternate interior angles, they are congruent.

So, a = 104°



Question 2:

Find the missing angles x , y and z in the following figure.

alternate interior angles  problem(2)


Solution:

As the pair 98°, x and z, y are alternate interior angles, they are equal in measure.

So, x = 98°

98° + y = 180°

y = 180° - 98°

= 82°

Since z and y are alternate interior angles, they are equal in measure

So, z = 82°



More topics in Alternate Interior Angles
Alternate Interior Angles Theorem
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