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Alternate Interior Angles: Examples, Definition, Theorem

The angles created on the opposing sides of the transversal have a name which is alternate interior angles. In other words, eight angles are generated when two parallel lines are crossed by a transversal. The alternate interior angles are those that are on the inner side of the parallel lines but on the opposite sides of the transversal.

Alternate Interior Angles

When a transversal connects two coplanar lines, alternate interior angles are generated. They are on the inside of the parallel lines but on the outside of the transversal. The transversal cuts through two lines that are Coplanar at different locations. These angles indicate whether or not the two specified lines are parallel. If these angles are equivalent, the lines intersected by the transversal are parallel.

In this article, we are talking about these angles. So, please keep reading to know more.

Alternate Interior Angles are

When two parallel lines are intersected by a transversal, the pair of angles created on the inner side of the parallel lines but on opposing sides of the transversal is known as alternate interior angles. These angles are always the same size. This can also be interpreted another way. The different interior angles can demonstrate whether or not the provided lines are parallel. If these angles are equivalent, the specified lines that are intersected by a transversal are said to be parallel.

To examine the different interior angles, look at the image below. In this case, AB and CD are two parallel lines joined by a transversal.

Alternate Interior Angles

The pairs of alternate interior angles in the preceding picture, according to the alternate interior angles theorem, are:

Angle 4 and angle 6

Angle 3 and angle 5

Let us first define alternate outside angles and how they differ from alternate interior angles.

Alternate Interior Angles example

Example 1

Determine the values of B and D in the given graph.

Alternate Interior Angles

Solution:

45° and D are equivalent since they are opposite inner angles.

As a result, D = 45°.

Because 135° and B are inner angles that alternate, they are congruent.

As a result, B = 135°.

Example 2

Find the missing angles A, C, and D in the diagram below.

Alternate Interior Angles

Solution:

Because angles angle A, 110°, angle C, and angle D are all alternative internal angles,

Angle C = 110°

We know this because of the additional angles theorem.

Angle C + angle D = 180°

Angle D = 180 degrees – angle C = 180 degrees – 110 degrees = 70 degrees.

Example 3

Find the value of x in the figure below.

Alternate Interior Angles

Solution:

Alternate interior angles are known to be congruent.

As a result, 4x – 19 = 3x + 16

4 x – 3 x = 19 + 16

35 = x

Since,

Angle A = angle D

As a result, angle A = 70°.

Example 4

Determine the angle x in the accompanying diagram if the two lines are parallel and intersected by a transversal.

Solution:

The alternate interior angles are x and 20°, according to the alternate interior angles theorem. As a result, they are equal. As a result, x = 20°.

Example 5

l || m and s || t are shown in the diagram below. Determine the value of x + y – z.

Solution: If l || m is the transversal and t is the interior angle, y° and 70° are alternate interior angles. As a result, they are equivalent in measure (according to the alternative internal angle theorem), i.e., y° = 70°. Again, because s || t and m are transversals, x° and 70° are comparable angles and hence equal, i.e., x° = 70°. Let us now suppose that the angle close to x° is w°.

Because x° and w° are linear, x° + w° = 180° 70° + w° = 180° w° = 110°

Because w° and z° are matching angles, they are equal, i.e., z° = w° =110°.

Let us now swap the angles’ values: 70° + 70° – 110° = 30° x + y – z = 70° + 70° – 110° = 30°

As a result, x + y – z = 30°.

Example 6

In the illustration below, MN || OP and ON || PQ are presented to Mathew. If MNO = 55°, then assist Mathew in determining OPQ.

Solution:

To remedy this, we shall expand the lines in the illustration.

MN || OP and ON is a transversal in this case. As a result, because 55° and x are co-interior angles, they are supplementary, i.e., 55° + x° = 180°, x = 125°. ON || PQ and OP is a transversal once more. As a result, because x and OPQ are matching angles, they are equal, i.e., OPQ = x = 125°. As a result, OPQ = 125°.

Alternate Interior Angles theorem definition

According to the theorem, “if a transversal crosses the set of parallel lines, the alternate interior angles are congruent.”

Assumed: a/d

To demonstrate: angle 4 = angle 5 and angle 3 = angle 6

Assume that a and d are two parallel lines and that l is the transversal that crosses a and d at locations P and Q. Take a look at the diagram below.

We know from parallel line qualities that if a transversal intersects any two parallel lines, the corresponding angles and vertically opposite angles are identical. Therefore,

angle 2 = angle 5………..(i) [Corresponding angles]

angle 2 = angle 4………..(ii) [Vertically opposite angles]

We derive the following from eqs. I and (ii):

[Alternate interior angles] angle 4 = angle 5

Similarly,

angle 3 = angle 6

As a result, it is established.

Alternate Interior Angles theorem

The two lines are parallel if the alternate interior angles created by the transversal line on two coplanar are congruent.

Given: angle 4 Equals angle 5, and angle 3 = angle 6.

To demonstrate: a/b

Since angle 2 = angle 4 [vertically opposite angles],

As a result, we may write,

Angle 2 = angle 5, which corresponds to similar angles.

As a result, a is parallel to b.

Co-interior angles, also known as Consecutive interior angles, are two angles on the same side of the transversal. The internal angles are co-inside angles, which add up to 180 degrees. It signifies that the total of two internal angles on the same transversal side is additional. Co-interior angles resemble a “C” form, yet neither angle is equal to the other. The co-interior angle is also known as successive interior angles or interior angles on the same side.

Proof

When the transversal crosses the two parallel lines, the total of the co-interior angles equals 180 degrees (supplementary angles).

Solution:

Take a look at the image above:

Angles 3 and 5 in the diagram are co-interior angles, whereas angles 4 and 6 are co-interior angles.

To demonstrate, angle 3 and angle 5 are supplementary, while angle 4 and angle 6 are supplementary.

As a result, a and b are parallel, and t is the transversal.

According to the definition of a linear pair,

The linear pair is formed by angle 1 and angle 3.

Similarly, the linear pair is formed by angle 2 and angle 4.

Making use of the supplement postulate, angle 1 and angle 3 are extraneous.

(For example, angle 1 + angle 3 Equals 180)

Similarly,

angle 2 and angle 4 are extraneous. (For example, angle 2 + angle 4 Equals 180)

We may formulate this using the related angles theorem.

angle 1 approximately equal to angle 5 and angle 2 approximately equal to angle 6.

As a result of the substitution feature, we may state, Angle 3 and angle 5 are supplemental, while angle 4 and angle 6 are also supplementary.

As a result, the co-interior angle theorem (consecutive interior angle) is established.

This theorem’s converse is “if a transversal intersects two lines such that the pair of co-interior angles are supplementary, then the two lines are parallel.”

Alternate Interior Angles property

An internal angle is an angle that exists within the form. The triangle has three inner angles, as seen in the diagram above. x, y, and z are all internal angles in this triangle. The fact that the total of the internal angles is always 180° means that x + y + z = 180°. Let us examine the evidence for this claim.

Equation:

The internal angle total is always 180°.

Consider the example of an ABC, as indicated in the image above. To demonstrate the aforementioned property of triangles, draw a line PQ parallel to the given triangle’s side BC. Because PQ is a straight line, we may assume that: angle PAB + angle BAC + angle QAC = 180…… (1)

Because PQ || BC, AB, and AC are transverse, angle QAC = angle ACB (a pair of alternate angles)

Furthermore, angle PAB = angle CBA (a pair of alternate angles)

Now, in equation (1), replace the values of angle QAC and angle PAB by angle ACB + angle BAC + angle CBA = 180°.

As a result, the total of the internal angles is always 180°.

Alternate Interior Angles are equal

When two parallel lines are crossed by a transversal, the angles created inside them are equal to their alternative pairings. These are known as alternate interior angles.

As seen in the diagram above, two parallel lines are crossed by a transversal. As a result, the alternative angles inside the parallel lines are equal.

i.e. angle A = angle D and angle B = angle C

These angles are parallel. The total of the angles created on the same side of the transversal that are within the two parallel lines is always 180°. Alternate interior angles have no special features in the case of non-parallel lines.

Alternate Interior Angles have one common

When a transversal travels between two lines, alternate interior angles are generated. Alternate interior angles are generated on opposite sides of the transversal and inside the two lines. When the lines are parallel, the alternate interior angles are equal, according to the theorem. Alternative inner angles are congruent, which means they have the same measure. When two parallel lines are crossed by a transversal, and my parallel lines are identified by using the same number of arrows, two special angles, alternate interior angles, are congruent.

Alternate Interior Angles are supplementary

Yes, alternate interior angles are supplementary. When the transversal intersects two parallel lines, matching angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, and successive interior angles become supplementary, meaning they have a total of 180 degrees.

Alternate Interior Angles in triangles

The alternate interior angles are the angles created within two parallel lines when they are crossed by a transversal and are equal to their alternate pairings. The alternate interior angles of two parallel lines are equal, according to the alternate interior angles theorem. This knowledge is used to calculate alternate interior angles. Let me illustrate this with an example.

Read Also:Linear Interpolation Formula:

For example, a road called Sixth Avenue that runs perpendicular to the adjacent streets First and Second. Another route, Maple Avenue, forms a 40° angle with 2nd Street. Can you calculate the angle x?

Using the alternate interior angles property, get angle x.

Solution:

If the two streets are parallel and Maple Avenue is regarded as the transversal, then x and 40° are the alternate interior angles, according to the alternate interior angles theorem. As a result, both angles are equal. As a result, x = 40°.

  • First, each pair of alternate interior angles has the same value.
  • Then, each co-interior angle pair is extra.
  • Then, each corresponding angle pair is equal.
  • Atlast, each alternating pair of external angles is equal.

Alternate Interior Angles are equal or not

The alternate interior angles are the angles created within two parallel lines when they are crossed by a transversal and are equal to their alternate pairings.

Example 1

Determine the value of x given that the interior angles (3x + 20) ° and 2x° are consecutive.

Solution

As a result, consecutive interior angles are supplementary;

(3 x + 20) ° + 2 x ° = 180 °

Then, 3 x 20 Plus 2 x = 180

Then, 5 x 20 = 180

Take 20 off both sides.

5x = 160

Take each side and multiply it by 8.

Then, 32 = x

As a result, the value of x is 32 degrees.

As a result, the successive internal angles are 60° and 120°.

Example 2

Given two angles, (4x – 19) degree and (3x + 16) degree are alternate interior angles that are equivalent. Determine the value of x as well as the value of the other pair of alternate interior angles.

Solution

3x + 16 = 4x – 19

4 x – 3 x = 19 + 16

35 = x

As a result, x = 350.

(4x – 19)0 = 1210 4(35) – 19

Because extra angles are produced on the same side of the transversal. The value of the other pair of alternate interior angles is then;

1800 – 1210 equals 590.

Alternate Interior Angles equations

Example 1

To find out if the lines sliced by the transversal are parallel, use the Alternate Interior Angles Theorem.

Because angle A and 56 degrees make a straight line, we know that A + 56 = 180. As a result, angle A has a measurement of 124 degrees. Similarly, Angle B and 124 degrees make a straight angle, therefore we know that B + 124 = 180, and hence Angle B is 56 degrees in length. Angle A and the initial 124 degree angle are equal and alternating inner angles. Angle B and the initial 56° angle are both equal alternate interior angles. As a result, the lines sliced by the transversal must be parallel according to the Alternate Interior Angles Theorem.

Example 2

The transversal line cuts parallel lines. Determine the dimensions of angles A, B, and C.

Angle A and 156 degrees are alternative internal angles, and since the lines cut by the transversal are parallel, the measure of angle A must be 156 degrees as well. A + B Equals 180 because angle A and angle B create a straight angle. Because A = 156, we must have B = 24, and so the angle B is 24 degrees. Then, because angle B and angle C are opposite interior angles, we must have angle C’s measure to be 24 degrees as well.

Example 3

Is it possible for the transversal to intersect parallel lines? Explain.

The alternate interior angles must be equal if the lines are parallel. This would result in 4x + 2 = 3x – 2. When we solve for x, we get x = -4. Substituting this number back into the angle measurements yields 4(-4) + 2 = -14 degrees for one angle and 3(-4) – 2 = -14 degrees for the other. As a result, the angles are only equal if they add up to a negative number of degrees, which is not possible. As a result, the lines cannot be parallel.

Alternate Interior Angles are congruent

Alternative inner angles are congruent, which means they have the same measure. When two parallel lines are crossed by a transversal, and my parallel lines are identified by using the same number of arrows, two special angles, alternate interior angles, are congruent.

Alternate Interior Angles are on the dash side of the transversal

Consider that m and n are straight lines, and that transversal l overlaps both m and n.

However, as a result, the angles a and b are alternate angles because of their position, that is, on the opposite side of the transversal, and because they both lie on the inside of an area spanned by lines m and n.

Then, as a result, they are opposite internal angles.

As a result, alternate interior angles are on different sides of the transversal.

Some frequently asked questions

Do alternate interior angles make 180?

They will not add up to 180° unless the alternate internal vertical angles are 90°. If the alternate interior angles are obtuse, combining them together will provide a value greater than 180°. As a consequence, if the alternate interior angles are sharp, adding them together will provide a value less than 180°.

What are alternate interior and exterior angles?

Angles in the space between the parallel lines, such as angles 2 and 8, are termed internal angles, but angles on the outside of the two parallel lines, such as 1 and 6, are called exterior angles. Angles on opposing sides of the transversal are known as alternative angles, for example, 1 + 8.

What is the rule for alternate angles?

When two parallel lines intersect, the alternative angles (as indicated above) are always equal. Because the angles form a Z shape, this rule is commonly referred to as “Z angles.”

What are the 3 interior angles?

Shape Sides Sum of Interior Angles
Triangle 3 180°
Quadrilateral 4 360°
Pentagon 5 540°
Hexagon 6 720°

Why do all triangles add up to 180?

The angles of a triangle add up to 180 degrees because one outer angle equals the total of the triangle’s other two angles. In other words, the triangle’s other two angles (the ones that add up to generate the outer angle) must join with the third angle to form a 180-degree angle.

What Are Alternate Interior Angles in Geometry?

To determine alternate interior angles in geometry, we must consider the following factors:

  • The alternate interior angles are all the same size.
  • They are located on opposite sides of the transversal.
  • They are located between the two lines’ interiors.
  • In other words, the angles on the inner side of the parallel lines but on opposing sides of the transversal.

Is Alternate interior supplementary or complementary?

When the transversal intersects two parallel lines, matching angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, and successive interior angles become supplementary, meaning they have a total of 180 degrees.

What are alternate corresponding and interior angles?

 Corresponding angles are two angles that are on the same side of the transversal, one inner and one exterior. The alternative angles are two angles on opposing sides of the transversal.

What is the difference between alternate interior angles and alternate exterior angles, same side interior angles and same side exterior angles?

If two lines are parallel, their corresponding angles are congruent, as are alternate interior angles and alternate exterior angles. When lines are parallel, same side interior angles and same side exterior angles are supplementary.

How many interior angles are there?

Polygon Name

 

Number of Interior Angles Sum of Interior Angles = (n-2) x 180°
Triangle 3 180°
Quadrilateral 4 360°
Pentagon 5 540°
Hexagon 6 720°