Same Side Interior Angles Supplementary

Alternating Interior Angles

Alternate interior angles are the angles formed on the opposite sides of the transversal. In other words, when two parallel lines are intersected past a transversal, 8 angles are formed. Among these, the angles that lie on the inner side of the parallel lines but on the opposite sides of the transversal are known as the alternating interior angles.

i.	What are Alternating Interior Angles?
2.	Converse of the Alternate Interior Angles Theorem
3.	How to Find the Alternate Interior Angles?
4.	FAQs on Alternate Interior Angles

What are Alternate Interior Angles?

When ii parallel lines are crossed by a transversal, the pair of angles formed on the inner side of the parallel lines, just on the opposite sides of the transversal are called alternate interior angles. These angles are always equal. This can besides exist understood in another style. The alternate interior angles can show whether the given lines are parallel or not. If these angles are equal, then the given lines which are crossed past a transversal are said to be parallel.

Notice the post-obit figure to encounter the alternate interior angles. Here AB and CD are ii parallel lines crossed by a transversal.

Alternate Interior Angles

Past the alternate interior angles theorem, the pairs of alternate interior angles in the above figure are:

∠four and ∠6
∠three and ∠v

Permit us briefly discuss what are alternate exterior angles and how they are different from alternating interior angles.

Alternating Exterior Angles

Alternate exterior angles are those angles that accept unlike vertices, lie on the alternate sides of the transversal, and are exterior to the lines. When a transversal intersects 2 parallel lines, the alternate exterior angles formed are e'er equal. In the same figure, ∠1 & ∠7 and ∠ii & ∠8 are the pairs of alternating outside angles.

Converse of the Alternate Interior Angles Theorem

According to the converse of the alternating interior angles theorem, if a transversal intersects two lines such that the alternate interior angles are equal, then the two lines are said to exist parallel.

Let us understand this with the help of the following figure which shows: ∠i = ∠five (respective angles), ∠3 = ∠5 (vertically opposite angles). Thus, ∠1 = ∠3. Similarly, nosotros can prove that ∠2 = ∠4. This proves that since the alternating interior angles in the two given lines are equal, these lines are parallel to each other.

Converse of the Alternate Interior Angles Theorem

How to Observe the Alternate Interior Angles?

According to the alternate interior angles theorem, the alternate interior angles of two parallel lines are equal. We utilize this fact to find alternating interior angles. Let us understand this with an example.

Instance: The post-obit figure shows a map in which the route named Sixth Avenue runs perpendicular to the1^st Street and the 2^nd Street, which are parallel. Another route named Maple Avenue makes an angle of 40° with the two^nd Street. Can you find the measure of bending x?

Finding angle x using alternate interior angles property

Solution:

Following the alternate interior angles theorem, if the two streets are parallel, and Maple Avenue is considered to exist the transversal, then x and 40° are the alternate interior angles. Hence, both the angles are equal. Therefore, 10 = twoscore°.

Important Notes

Each pair of alternate interior angles is equal.
Each pair of co-interior angles is supplementary.
Each pair of corresponding angles is equal.
Each pair of alternate exterior angles is equal.

Challenging Question

In the post-obit effigy, \(\mathrm{AB}\|\mathrm{CD}\| \mathrm{EF}\)
Practice question on alternate interior angles

Find the value of ten.

☛Topics Related to Alternate Interior Angles

Check out the listed beneath interesting articles to acquire more near alternate interior angles and the related topics.

Vertical Angles
Alternate Angles
Same Side Interior Angles
Interior Angles of Polygon Estimator

Alternate Interior Angles Examples

Example i: Find the measure of angle ten in the following effigy if the two lines are parallel and they are crossed past a transversal.

Solution:

By the alternating interior angles theorem, x and xx° are the alternating interior angles. Hence, they are equal. Therefore, ten = 20°.
Instance 2: In the following figure given to Mathew, MN || OP and ON || PQ. If ∠MNO = 55° then assist Mathew in finding ∠OPQ.

Solution:

We will extend the lines in the effigy to solve this.

Here, MN || OP and ON is a transversal. Thus, 55° and 10 are co-interior angles, hence, they are supplementary, i.e., 55° + x° = 180°, 10 = 125°. Again, ON || PQ and OP is a transversal. Thus, x and ∠OPQ are corresponding angles, hence, they are equal, i.eastward., ∠OPQ = x = 125°. Therefore, ∠OPQ = 125°.
Case 3: In the following figure, l || g and due south || t. Observe the value of x + y - z.

Solution:

If l || m and t is the transversal, y° and 70° are alternate interior angles. Hence, they are equal in measure out (by the alternate interior angle theorem), i.due east., y° = 70°. Once more, south || t and grand is a transversal, 10°, and lxx° are corresponding angles hence, they are equal, i.e., x° = seventy°. At present, let us assume that the angle that is side by side to x° is westward°.

Since x° and due west° course a linear pair, ten° + w° = 180°
70° + west° =180°
westward° = 110°
Now, w° and z° are corresponding angles, hence, they are equal, i.eastward., z° = w° =110°.
Now, let us substitute the values of the angles: x + y - z = 70° + 70° - 110° = 30°.
Therefore, x + y - z = thirty°

become to slidego to slidego to slide

Nifty learning in high school using uncomplicated cues

Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and exist surprised by the outcomes.

Book a Gratis Trial Class

Practice Questions on Alternating Interior Angles

become to slidego to slide

FAQs on Alternate Interior Angles

What Are Alternating Interior Angles in Geometry?

In geometry, to define alternate interior angles we need to check the post-obit listed characteristics:

The alternate interior angles are equal in mensurate.
They lie on the alternate sides of the transversal.
They prevarication in between the interior of the two lines.
In other words, the angles that lie on the inner side of the parallel lines but on the opposite sides of the transversal.

How To Solve Alternating Interior Angles?

According to the alternating interior angles theorem, if ii parallel lines are crossed by a transversal, then the alternating interior angles are equal in measure out. Using this theorem, nosotros can notice the measure of alternate interior angle if nosotros know the measure of the respective alternating interior bending.

What is the Converse of the Alternate Interior Angles Theorem?

According to the converse of the alternate interior angles theorem, if a transversal intersects two lines such that the alternating interior angles are equal, then the two lines are said to be parallel.

What Is the Difference Betwixt Alternate Interior and Exterior Angles?

Alternate interior angles are those angles that take different vertices, they lie on the alternating sides of the transversal and are in between the interior of the 2 lines. Whereas alternating exterior angles are those angles that have unlike vertices, they lie on the alternate sides of the transversal, but they lie on the outer side of the 2 lines.

What Is the Departure Between Corresponding and Alternate Angles?

Corresponding angles are two angles that prevarication on the same side of the transversal in which one is interior and the other is exterior. The alternate angles are two angles that lie on the opposite sides of the transversal.

How To Solve Alternate Interior Angles?

According to the alternate interior angles theorem, if two parallel lines are crossed past a transversal, then the alternating interior angles are equal in mensurate. Using this theorem, nosotros can observe the measure of alternate interior angle if we know the measure of the corresponding alternate interior angle.

Are Alternating Interior Angles Congruent or Supplementary?

As per the alternate interior angles theorem, the alternating interior angles of two parallel lines are congruent.