Understanding Alternate Interior Angles

CEO Khai Intela
Have you ever wondered about the angles formed when a transversal intersects two coplanar lines? These angles are known as alternate interior angles, and they play a crucial role in determining whether the lines are...

Have you ever wondered about the angles formed when a transversal intersects two coplanar lines? These angles are known as alternate interior angles, and they play a crucial role in determining whether the lines are parallel or not. In this article, we will explore the definition, properties, theorems, and even solve some examples related to alternate interior angles.

What are Alternate Interior Angles?

Alternate interior angles are the angles formed inside two parallel lines when they are intersected by a transversal. These angles lie on the inner side of the parallel lines but on opposite sides of the transversal. In other words, they are like mirror images of each other. Take a look at the image below to see a visual representation of alternate interior angles.

Co-interior Angles Co-interior Angles

In the figure above, you can see two parallel lines intersected by a transversal. As a result, the alternate angles inside the parallel lines are equal:

∠A = ∠D ∠B = ∠C

Properties of Alternate Interior Angles

Here are some important properties of alternate interior angles:

  • These angles are congruent, meaning they have the same measure.
  • The sum of the angles formed on the same side of the transversal (inside the parallel lines) is always equal to 180°.
  • In the case of non-parallel lines, alternate interior angles do not have any specific properties.

Apart from alternate interior angles, there are several other types of angles formed when a transversal intersects parallel lines. Some examples include adjacent angles, corresponding angles, complementary angles, and supplementary angles. These angles have their own unique properties and relationships.

Theorem and Proof

Now, let's delve into a fundamental theorem concerning alternate interior angles:

Theorem: If a transversal crosses a set of parallel lines, the alternate interior angles are congruent.

Consider the following diagram to visualize the theorem:

Co-interior Angles Proof Co-interior Angles Proof

Here's a step-by-step proof of the theorem:

  1. Suppose a and d are two parallel lines, and l is the transversal that intersects a and d at points P and Q.
  2. Using the properties of parallel lines, we know that if a transversal cuts any two parallel lines, the corresponding angles and vertically opposite angles are equal to each other.
  3. Therefore, ∠2 = ∠5 (corresponding angles) and ∠2 = ∠4 (vertically opposite angles).
  4. From equations (i) and (ii), we can conclude that ∠4 = ∠5 (alternate interior angles).
  5. Similarly, ∠3 = ∠6.
  6. Thus, we have successfully proved the theorem.

Co-interior Angles

Apart from alternate interior angles, let's also explore the concept of co-interior angles or consecutive interior angles. These angles are on the same side of the transversal and are supplementary, meaning their sum is equal to 180 degrees. Co-interior angles resemble a "C" shape and are not equal to each other. Take a look at the image below for a visual representation.

Example 3

Co-interior Angle Theorem and Proof

The co-interior angle theorem states that if a transversal intersects two parallel lines, each pair of co-interior angles sums up to 180 degrees (they are supplementary). Here's a proof of the theorem:

Alternate Interior Angles

Let's consider the image above:

  • Angles 3 and 5 are co-interior angles, and angles 4 and 6 are co-interior angles.
  • To prove that ∠3 and ∠5 are supplementary, and ∠4 and ∠6 are supplementary, we start by using the definition of a linear pair.
  • ∠1 and ∠3 form a linear pair, and ∠2 and ∠4 form a linear pair.
  • By using the supplement postulate, ∠1 and ∠3 are supplementary (i.e., ∠1 + ∠3 = 180).
  • Similarly, ∠2 and ∠4 are supplementary (i.e., ∠2 + ∠4 = 180).
  • By using the corresponding angles theorem, we can write ∠1 ≅ ∠5 and ∠2 ≅ ∠6.
  • Therefore, by using the substitution property, we can conclude that ∠3 and ∠5 are supplementary, and ∠4 and ∠6 are supplementary.
  • Hence, the co-interior angle theorem (consecutive interior angle) is proved.

Examples

Let's solve a few examples to get a better understanding of alternate interior angles:

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

Alternate Interior Angles

Solution:

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

Similarly, 135° and B are alternate interior angles, so B = 135°.

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

Alternate Interior Angles

Solution:

As angles ∠A, 110°, ∠C, and ∠D are alternate interior angles, we can conclude that ∠C = 110°.

By the supplementary angles theorem, ∠C + ∠D = 180°.

Therefore, ∠D = 180° - ∠C = 180° - 110° = 70°.

Since ∠A = ∠D, we can say that ∠A = 70°.

Example 3: Find the value of x from the given figure.

Alternate Interior Angles

Solution:

We know that alternate interior angles are congruent.

Therefore, 4x - 19 = 3x + 16.

Simplifying the equation, we get:

4x - 3x = 19 + 16.

x = 35.

Video Lesson on Intersecting and Parallel Lines

If you want to enhance your understanding of intersecting and parallel lines, consider downloading the BYJU’S App. It provides personalized videos that can greatly improve your learning experience.

Remember, understanding alternate interior angles and their properties is crucial for geometry and other related mathematical concepts. So, make sure you grasp these concepts thoroughly to excel in your studies.

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