Alternate Interior Angles Theorem

Alternate Interior Angles Theorem
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If you hold a stick across the horizontal parallel bars of a window as depicted in the figure. How many angles will be formed between the horizontal bars and the stick?

parallel lines and angles

A lot of special angle-pairs are formed, which are congruent. Alternate interior angles are one such congruent pair.

In this mini-lesson, we will explore the alternate interior angles theorem. Let's learn more about it.

Lesson Plan

What Are Parallel Lines and Transversal line?

Two lines that never intersect, are equidistant, and are coplanar are called parallel lines. The symbol for "parallel to" is II. Here line \(m \) II \(n\).

If we have two lines (they don't have to be parallel) and have a third line that crosses them, then the crossing line is said to be a transversal.

parallel lines

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transversal

<Create a similar image, name


What Are Alternate Interior Angles?

Alternate angles are of two types:

  1. Alternate interior angles
  2. Alternate exterior angles

Alternate interior angles: Alternate interior angles are the angles formed when a transversal intersects two coplanar lines. They are on the inner side of the coplanar lines but are on the alternate opposite sides of the transversal.

Here, in the image shown below,

Alternate interior angles are:

1. \(\angle 3\) and \(\angle 6\)

2. \(\angle 4\) and \(\angle 5\)

alternate interior angles

Alternate exterior angles: Alternate exterior angles are formed on the exterior of the coplanar lines but on the alternate opposite sides of the transversal.

Here, in the image shown above,

Alternate exterior angles are:

1. \(\angle 1\) and \(\angle 8\)

2. \(\angle 2\) and \(\angle 7\)


Alternate Interior Angles Examples

Let us try to spot alternate interior angles in the given figure. Remember the lines do not have to be always parallel for alternate angles to be formed.

Alternate interior an alternate exterior angle

Alternate interior angles are:

1. \(\angle 3\) and \(\angle 6\)

2. \(\angle 4\) and \(\angle 5\)

Alternate exterior angles are:

1. \(\angle 1\) and \(\angle 8\)

2. \(\angle 2\) and \(\angle 7\)


How Do You Find Alternate Interior Angles?

Z-test For Finding Alternate Interior Angles
To find the alternate interior angles, make a zig-zag line including the parallel lines as shown in the diagram. Here, \(\text p\) and \(\text q\) are alternate interior angles. Similarly, \(\text a\) and \(\text d\) are also alternate interior angles.

how to find alternate interior angles z test


What Does the Alternate Interior Angles Theorem States?

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

In the figure given below, a set of parallel lines \( m\) and \( n\) are intersected by the transversal and the following pairs of alternate interior angles are formed:

\(\angle 1\) and \(\angle 2\)

\(\angle 3\) and \(\angle 4\)

alternate interior angles

Since the given lines \( m\) and \( n\) are parallel, therefore the alternate interior angles will be congruent.

\[\angle 1 = \angle 2\]

\[\angle 3 = \angle 4\]

  • Interior angles on the same side of the transversal are called 'consecutive interior angles' or co-interior angles in short. Co-interior angles are supplementary when the lines are parallel.

\[\angle 2 + \angle 3 = 180^{\circ} \]

\[\angle 6 + \angle 7 = 180^{\circ} \]

Proof alternate interior angles theorem

  • Corresponding angles are a pair of angles on the similar corners of each of two lines on the same side of the transversal line. Corresponding angles formed by two parallel lines and a transversal are equal.

Corresponding angles are:

1. \(\angle 2\) and \(\angle 4\)

2. \(\angle 1\) and \(\angle 3\)

3. \(\angle 5\) and \(\angle 7\)

4. \(\angle 6\) and \(\angle 8\)


Explore Alternate Interior Angles Congruence

You can explore the alternate interior angle congruence in this simulation. Drag the red point to change the angle between parallel lines and the transversal. 

 
Challenge your math skills
Challenging Question
  • Find the value of \( x\), if two congruent alternate interior angles are \((2 x + 19)^{\circ}\) and \((4x -16)^{\circ}\).

What Is the Alternate Interior Angles Theorem?

Theorem And Proof

Statement: The theorem states that  “if a transversal intersects parallel lines, the alternate interior angles are congruent."

Given: Line p II line q

To prove: \(\angle 2 = \angle 7\) and \(\angle 3 = \angle 6\)

Proof alternate interior angles theorem

Proof: Suppose \(p\) and \( q\) are two parallel lines and t is the transversal that intersects \(p\) and \(q\).

We know that, if a transversal intersects any two parallel lines, the corresponding angles and vertically opposite angles are congruent.

   Therefore,

\(\angle 1 = \angle 3\) ………..(I)  [Corresponding angles]

\(\angle 1 = \angle 6\) ………..(II) [Vertically opposite angles]

From equations (I) and (II), we get-

\(\angle 3  = \angle 6\) ..................[Alternate interior angles]

Similarly,

\(\angle 2  = \angle 7\)

Hence, it is proved.


Alternate Interior Angles Theorem Converse

The converse of alternate interior angles theorem states that if two lines are intersected by a transversal forming congruent alternate interior angles, then the lines are parallel.

Thus according to the converse of alternate interior angles theorem, the below-given lines will be parallel if \(\angle D\) is \(40^{\circ}\) and \(\angle B\) is \(140^{\circ}\). This is because their corresponding alternate interior angles are of the measure \(40^{\circ}\) and \(140^{\circ}\).

Converse alternate interior angles theorem

In conclusion, the alternate interior angles theorem states that the alternate interior angles will be equal if the lines are parallel, whereas its converse states that lines will be parallel if the alternate interior angles are congruent.

 

important notes to remember
Important Notes
  • Interior angles are a pair of angles on the inner side of each of the two lines but alternate opposite sides of the transversal line.

  • Corresponding angles are a pair of angles on the similar corners of each of two lines on the same side of the transversal line. Corresponding angles formed by two parallel lines and a transversal are equal.

  • Co-interior angles are supplementary when the lines are parallel.

Solved Examples

Example 1

 

 

Cathy has been asked to find the pairs of alternate interior angles from the given diagram. Can you help her?

Solution

The alternate interior angles lie on "alternate" sides of the transversal, and they must be in the "interior" of the two parallel lines.

The alternate interior angles are:

\(\angle q\) and \(\angle z\)

\(\angle r\) and \(\angle a\)

Example 2

 

 

Julia asks her students if they can find the relationship between \(\angle q\) and \(\angle d\) if the given lines \(m\) and \(n\) are parallel. Can you tell how are \(\angle q\) and \(\angle d\) related to each other?

parallel lines

Solution

The given lines are parallel and are intersected by a transversal.

According to the alternate interior angles theorem, the two congruent alternate interior angles are produced. They are:

\(\angle q\) and \(\angle d\)

and

\(\angle c\) and \(\angle p\)

\(\therefore\) \(\angle q\) and \(\angle d\) are congruent alternate interior angles.

\(\therefore\) \(\angle q = \angle d\) 
Example 3

 

 

If the given two lines are parallel and are intersected by a transversal, what will be the measure of \(\angle X\) and \(\angle Y\)?

parallel lines

Solution

The given lines are parallel and according to the alternate interior angles theorem, the given angle of measure \(85^{\circ}\) and \(\angle X\)are congruent alternate interior angles.

Therefore, \(85^{\circ} = \angle X\)
Similarly, \(95^{\circ}\) and \(Y\) are congruent alternate interior angles.Therefore, \(95^{\circ} = \angle Y\)

\(\therefore\) \(\angle X = 85^{\circ}\)

    \(\angle Y = 95^{\circ}\)


Interactive Questions on Alternate Interior Angles Theorem

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 

 

 
 
 
 
 


Let's Summarize

The mini-lesson targeted the fascinating concept of the alternate interior angles theorem. We also learned about alternate interior angles theorem proof,  alternate interior angles theorem converse, and alternate interior angles theorem examples. The math journey around the alternate interior angles theorem starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

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Frequently Asked Questions

1. Are alternate interior angles congruent?

Yes, alternate interior angles are congruent.

alternate interior angles

2. How is the alternate interior angles theorem and the alternate exterior angles theorem alike?

The alternate interior angles theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles are congruent.

The alternate exterior angles theorem states that if two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.

Thus, the two theorems are alike with respect to the congruent angles produced.

  
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