Alternate Angles

Alternate Angles
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You must have studied or heard about various types of angles so far formed by a transversal falling on two lines. One such type of angle is alternate angles. 

In the simulation below, tick the box to visualize the alternate angles. You can drag the point of intersection to change the values of angles. 

In this mini-lesson, we will explore more about angles by learning about alternate angles including its definition, types of alternate angles, theorems on alternate angles with the help of interesting simulation, some solved examples and a few interactive questions for you to test your understanding.

Lesson Plan 


What Are Alternate Angles?

When two straight lines are cut by a transversal, then the angles formed on the opposite side of the transversal with respect to both the lines are called alternate angles. 

So, alternate angles are those angles that: 

  • have different vertices
  • lie on the alternate sides of the transversal 

examples of alternate angles

The pairs of alternate angles in the above figure are:

\(\angle 3\) and \(\angle 5\)

\(\angle 4\) and \(\angle 6\)

\(\angle 1\) and \(\angle 7\)

\(\angle 2\) and \(\angle 8\)


What Are the Types of Alternate Angles? 

There are two types of alternate angles- alternate interior angles and alternate exterior angles. 

Alternate angles that lie in the interior region of both the lines are called alternate interior angles. 

Alternate angles that lie in the exterior region of both the lines are called alternate exterior angles

Example

In the above image, two pairs of alternate interior angles are \(\angle 3\) and \(\angle 5\) and \(\angle 4\) and \(\angle 6\). 

                                two pairs of alternate exterior angles are \(\angle 2\) and \(\angle 8\) and \(\angle 1\) and \(\angle 7\).

 
important notes to remember
Important Notes
  1. Alternate Angles can be formed with two non-parallel lines also, but there is no relation between the angles so formed. 
  2. Alternate Interior Angles are popularly known as Z-angles, because we can easily identify a pair of alternate interior angles in a Z-shaped figure.

Alternate Angles Theorem

Now, let us learn about the two theorems related to Alternate Angles.

  1. Alternate Interior Angle Theorem
  2. Alternate Exterior Angle Theorem

Alternate Interior Angle Theorem

Suppose two parallel lines are intersected by a transversal, as shown below:

Alternate interior angles

What is the relation between any pair of alternate interior angles?

This relation is determined by the "Alternate Interior Angle Theorem."

When a transversal intersects two parallel lines, each pair of alternate interior angles are equal.

By alternate interior angle theorem converse, if a transversal intersects two lines such that a pair of interior angles are equal, then the two lines are parallel.

Alternate Interior Angle Theorem Proof

Refer to the figure above.

We have:

\(\angle1=\ \angle5\ \) (corresponding angles)

      \(\angle3=\ \angle5\ \) (vertically opposite angles)

Thus,

\(\angle1=\ \angle3\)

Similarly, we can prove that \(\angle2=\ \angle4\)

In the simulation given below, click on 1st pair and 2nd pair and carefully observe each pair of alternate interior angles.

Alternate Exterior Angle Theorem

If two lines are parallel, then the pair alternate exterior angles formed are congruent

alternate exterior angles pairs - two parallel lines and a transversal

In the figure above, we can observe that angles 1 and 2 is a pair of the alternate exterior angle.

We will now prove that they are congruent ( i.e. they have equal measure).

In the simulation below, click on any angle and carefully observe the pairs of alternate exterior angles.

Alternate Exterior Angle Theorem Proof

To prove this result, we will consider the vertically opposite angle of \(\angle1\)

Let's denote it by \(\angle3\)

alternate exterior angles step1 - two parallel lines and a transversal

Now, \(\angle1=\angle3\) as they are vertically opposite angles.

alternate exterior angles step2 - two parallel lines and a transversal

Since \(line\ a\ \parallel\ line\ b\)

\(\angle3=\angle2\ \)(corresponding angles axiom)

\(\therefore\ \angle1=\angle2\ \)(transitivity)


Solved Examples

Example 1

 

 

Two roads are running parallel to each other as shown below.

alternate angles example1 - two parallel roads and another road cutting them

If \(a=(2x)^o\) and \(b=(30-4x)^o\) , then what will be the value of \(x\)? 

Solution

To solve this problem, we will be using the alternate exterior angle theorem.

The pair of angles in this figure forms a pair of alternate exterior angles.

So, we have:

                 \((2x)^\circ= (30-4x)^o\)

\(2x+4x=30\)

        \(6x=30\)

        \(x=5\)

\(\therefore\ x=5\)
Example 2

 

 

In the given figure, find the value of \(p\)

alternate angles problem

Solution

In the given figure, \(\angle ADE\) and \(\angle FGJ\) forms a pair of alternate exterior angles. 

By using alternate exterior angle theorem, we have, \(\angle ADE = \angle FGJ\)

So,

 \(p^o+55^o= 135^o\)

                    \(p^o= 135^o-55^o\)

        \(p^o=80^o\)

\(\therefore\) The value of \(p\) is \(80^o\).
Example 3

 

 

Find \(x\), if line \(p\) \(\left |  \right | \) line \(q\)

problems on alternate exterior angles - two parallel lines and a transversal

Solution

Given that, p\(\left |  \right | \)q,

So by alternate exterior angle theorem we get,

\begin{align}(2x+26)^{\circ}\ &=(3x-33)^{\circ}\\\ 2x-3x&=-33-26\\-x&=-59\\\therefore x&=59\end{align}

\( \therefore\) \(x=59\)
Example 4

 

 

Sixth Avenue runs perpendicular to both 1st street and 2nd street, which are parallel.

However, Maple Avenue makes a \(40^\circ\) angle with 2nd street. 

What is the measure of angle \(x\)?

How to find alternate interior angles?

Solution

By the alternate interior angles definition, \(x\) and \(40^\circ\) form a pair of alternate interior angles.

Hence they are equal (by alternate interior angle theorem).

\(\therefore\) \(x=40^\circ\)
 
Challenge your math skills
Challenging Questions
  1. In the following figure, \(\mathrm{AB}\|\mathrm{CD}\| \mathrm{EF}\).
    Interior angles challenging question
    Find the value of \(x\)

Interactive Questions on Alternate Angles

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 Alternate Angles. The math journey around Alternate Angles 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.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions(FAQs)

1. What are alternate interior angles?

Alternate interior angles lie in the interior side of the parallel lines and are on the opposite side of the transversal.

2. What are alternate exterior angles?

Alternate exterior angles lie in the exterior side of the parallel lines and are on the opposite side of the transversal. 

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