## Same Side Interior Angles – Definition, Theorem, Examples

A pair of interior angles on the same side of a transversal are defined as two angles on the interior of (between) two lines that are on the same side of the transversal. The total of the inner angles on the same side equals 180 degrees. Four internal angles are generated when two parallel lines are crossed by a transversal line, as shown in the diagram. The two non-adjacent internal angles that are on the same side of the transversal as the transversal are referred to as supplemental angles.

## What Are Same Side Interior Angles?

When two parallel lines are crossed by a transversal, the result is the formation of eight angles. Identical side interior angles are a pair of non-adjacent interior angles that are located on the same side of the transversal as the transversal itself. As a result, the same side interior angles apply:

- Have no shared vertices or have vertices that are distinct
- Are constructed on the same side as the transversal and are situated between two lines

The “same side interior angles” are sometimes referred to as “co-interior angles” in some circles. There are eight distinct sorts of angles created as a result of this process, which are given below:

- Different Interior Angles, Corresponding Angles, Different Exterior Angles, Same Side Interior Angles or co-interior Angles are all examples of alternate interior angles.

The transversal is represented by the lines AB || CD and l in the illustration. As stated in the “Same Side Interior Angles – Definition,” the following are the pairs of same side interior angles shown in the preceding figure:

## Same Side Interior Angles Theorem

Take, for example, the illustration above. The lines AB and CD in the above diagram are parallel, while the line L is the transversal. We have recently learned that the pairs of similar side interior angles in the preceding image are as follows: The same side interior angle theorem is responsible for determining the relationship between the same side internal angles. If a transversal intersects two parallel lines, the theorem for the “same side interior angle theorem” asserts that each pair of same-side interior angles are supplementary (their total equals 180°).

### Same Side Interior Angles Theorem Proof

Referring to the previous illustration once more: 4 = 8, and 3 = 7. 5 + 8 = 180°, and 6 + 7 = 180°. 5 + 8 = 180°, and 6 + 7 = 180°. According to the previous two equations, 4 + 5 = 180°. In the same way, 3 + 6 Equals 180° As a result, it has been demonstrated that each pair of same-side internal angles is supplementary.

## Converse of Same Side Interior Angles Theorem

A transversal connects two lines in such a way that a pair of same-side interior angles are supplementary, and the two lines are parallel, according to the opposite of the same-side interior angle theorem.

### Converse of Same Side Interior Angles Theorem Proof

If we continue to use the same example from previously, let us assume that 4 + 5 = 180° (1) Since the numbers 5 and 8 constitute a linear pair, 5+8 = 180° (2)From (1) and (2), 4 = 8Thus, a pair of matching angles are equal, which can only occur if the two lines are parallel.5+8 = 180° (2) As a result, the reverse of the same side internal angle theorem is established. Important Points to Keep in Mind Listed below are the most crucial considerations to remember while dealing with the same side interior angles.

- The same-side internal angles are non-adjacent and are generated on the same side of the transversal as the same-side exterior angles are. If and only if the identical side interior angles on both lines are supplementary, two lines are considered parallel.

Articles that are related Check out these interesting articles to learn more about the same side interior angles and the themes that they are associated with.

- Line Segmentation
- Lines and Angles – Fundamental Concepts
- Lines that intersect and do not intersect
- Parallel lines
- Supplementary Angles
- Lines that are intersecting and do not intersect

## Same Side Interior Angles Examples

- Example 1: In the above figure, the side interior angles of 145° and 40° are the same as each other. Check to see if the lines l and m are parallel to each other or not. Solution: In the provided figure, the side interior angles of 145° and 40° are the same as one another. However, the total is not equal to 180° (145° + 40° =185°), as the number indicates. As a result, the angles 145° and 40° are NOT supplementary, and their total does not equal 180°. As a result, according to the “Converse of Same Side Interior Angle Theorem,” the lines in question are NOT parallel to one another. As a result, the lines l and m are not parallel. The side interior angles (4x+4)° and (10x+8)° in the next figure are the same as in the previous image. Example 2: Find out what the value of x is. Solution: Because l || m and t is a transversal, (4x+4)° and (10x+8)° are the identical side interior angles on either side of the equation. The “same side internal angle theorem,” which states that these angles are supplementary, or that their sum is equal to 180°, explains how this may be achieved. As a result, (4x+4) + (10x+8) = 18014x +12 = 18014x = 180 – 1214x = 168x = 12Thus, the value of x equals 12.

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## FAQs on Same Side Interior Angles

Despite having the same side interior angles, they are not congruent. They are meant to be supplemental. When two parallel lines are crossed by a transversal line, the same side interior angles are generated. The same side interior angles can only be congruent if each angle is equal to 90 degrees, since the total of the same side interior angles is equal to 180 degrees when each angle is equal to 90 degrees.

### Are Same Side Interior Angles Adjacent?

Because the angles are created on two distinct lines that are parallel to each other, the same side interior angles are never next to each other on the same side.

### What Is the Sum of the Two Same Side Interior Angles on the Transversal?

When two parallel lines were intersected by a transversal, they generated internal angles on the same side, and the total of these angles equals 180 degrees. Because the total of the internal angles on the same side is 180 degrees, the angles are supplementary in this case.

### What Is the Converse of Same Side Interior Angles?

It is possible to have two lines crossed by a transversal and the angles inside on the same side are additional, or we may say that the total of inside angles on the same side equals 180 degrees, in which case the lines are said to be parallel.

### What Is Another Name of the Same Side Interior Angles?

Interior angles on the same side of the transversal are also known as successive interior angles because the angles are on the same side of the transversal but inside the boundaries of the two parallel lines.

### What Is the Difference Between Same Side Interior Angles and Same Side Exterior Angles?

When two parallel lines are crossed by a transversal line, the result is the formation of eight angles. A parallel line on the same side of the transversal is defined as having an angle inside it, and an angle outside it is defined as having an angle outside it. The same side internal angles are defined as being inside parallel lines on the same side of the transversal.

### What Is the Difference Between the Same Side Interior Angles and Corresponding Angles?

The difference between same side interior angles and corresponding angles is that corresponding angles are congruent, whereas in the case of same side interior angles, the sum of the same side interior angles is equal to 180 degrees only if the transversal line intersects two parallel lines, which is not the case in most cases.

## 3.7: Same Side Interior Angles

angles on the same side of a transversal and inside the lines it intersects are known as transversals. Two angles that are on the same side of the transversal and on the interior of (between) the two lines are known as same side interior angles (or same side interior angles). Drawing (figure) (PageIndex ) If two parallel lines are interrupted by a transversal, then the identical side interior angles of the two parallel lines are additional, as shown in the following formula: Drawing (figure) (PageIndex ) If l is parallel to m, then (mangle 1+mangle 2=180) is the result.

- If If Figure (PageIndex) is more than zero, then (l parallel m) Consider the following scenario: you are given with two angles that are on the same side of a transversal and between the two parallel lines that the transversal crosses.
- As an illustration (PageIndex ) Is (l m parallel to m) true?
- Solution Interior Angles on the same side are what we’re talking about here.
- Due to the fact that (130 + 67 =197) the lines are not parallel, they are not parallel.
- Two of them are (angle 6) and (angle 10), while the other two are (angle 8) and (angle 11).
- Example: (PageIndex )Find the value of the index page (x).
- Due to the fact that the lines are parallel, the angles add up to (180).
- Drawing (figure) (PageIndex ) Solution (y) is an interior angle on the same side as the right angle that has been indicated.
- As an illustration (PageIndex ) If (mangle 3=(3x+12) and (mangle 5=(5x+8)) and (mangle 3=(3x+12)), then find the value of (x).

Keep in mind that the sum of the identical side interior angles equals (180). (((begin(3x+12) +(5x+8)=180+(8x+20)=180+(8x=160)=180+(8x+20)=180+(8x=160)=180+(8x+20)=180+(8x=160)=180+(8x+20)=180+(8x=160)=180+(8x+20)=180+(8x+20)=180+(8x

## Review

To answer questions 1-2, utilize the diagram to determine if each angle pair is congruent, supplementary, or neither of the three. Drawing (figure) (PageIndex )

- (angle 5) and (angle 8)
- (angle 2) and (angle 3)
- (angle 5), and (angle 8)
- Are the lines parallel to one another? Please provide justification for your response. Figure (PageIndex)
- Figure (PageIndex)

the angles (angle 5 and 8); the angles (2 and 3); the angles (2 and 3); and the angles (2 and 3) are all in the same direction. The lines appear to be parallel, correct? Your response must be supported. Table of Contents; Figure (PageIndex);

- The angles (angle AFD) and (angle BDF) are supplementary
- Nonetheless, Both the supplemental angles (angle DIJ) and the supplementary angle (angle FJI) are used.

What does the value of x have to be in order for the lines to be parallel in numbers 6-8? Drawing (figure) (PageIndex )

- (mangle 3=(3x+25) and (mangle 5=(4x+55))
- (mangle 4=(2x+15) and (mangle 6=(3×5)
- (mangle 3=(x+17) and (mangle 5=(3×5)

Mangle 3=(3x+25) and Mangle 5=(4x+55); Mangle 4=(2x+15) and Mangle 6=(3×5); Mangle 3=(x+17) and Mangle 5=(3×5); Mangle 3=(x+17) and Mangle 5=(3×5)

- (mangle 3=(3x+25)) and (mangle 5=(4x+55))
- (mangle 4=(2x+15)) and (mangle 6=(3×5))
- (mangle 3=(x+17)) and (mangle 5=(3×5))

## Review (Answers)

To read the answers to the Review questions, open this PDF file and go down to section 3.6.

## Vocabulary

Term | Definition |
---|---|

same side interior angles | Same side interior angles are two angles that are on the same side of the transversal and on the interior of the two lines. |

supplementary angles | Two angles that add up to (180^ ). |

transversal | A line that intersects two other lines. |

## Additional Resource

Activities: Same Side Interior Angles Discussion QuestionsVideo: Same Side Interior Angles Principles – FundamentalActivities: Same Side Interior Angles Discussion Questions Angles and Transversals Study Guide is a useful resource for students. Interior angles on the same side should be practiced. Exterior Views from Different Perspectives in the Real World

## Same-Side Interior Angles: Theorem, Proof, and Examples

Basic Principles of the Same Side Interior AnglesVideo: Same Side Interior Angles – BasicActivities: Discussion Questions on the Same Side Interior Angles Angles and Transversals Study Guide is a useful study aid. Same-side interior angles should be practiced. Alternative Exterior Views in the Real World

## The Converse of Same-Side Interior Angles Theorem

For two lines to be parallel, a transversal must cut them both, and if a pair of interior angles on the same side of the transversal is supplementary to each other, the lines are parallel. The converse of the Theorem of Interior Angles on the Same Side Proof Assume that L 1 and L 2 are two lines cut by transversal T in such a way that the numbers 2 and 4 are supplementary, as illustrated in the diagram below. Let us demonstrate that L 1 and L 2 are parallel to one another. Because the numbers 2 and 4 are supplementary, the sum of the numbers 2 and 4 equals 180°.

As a result, 1 + 4 = 180°.

Because of the addition characteristic, 2 = 1.

The converse of the Theorem of Interior Angles on the Same Side John Ray Cuevas is a musician from the Dominican Republic.

## Example 1: Finding the Angle Measures Using Same-Side Interior Angles Theorem

In the following figure, segment AB and segment CD have a difference of 104°, and ray AK bisects ray AB and segment CD. Calculate the size of DAB, DAK, and KAB by multiplying them together. The following is an example of how to use the term “example.” Using the Same-Side Interior Angles Theorem, we can get the angle measures. Solution by John Ray Cuevas Because the sides AB and CD are parallel, the interior angles D and DAB are supplementary to one another.

As a result, DAB = 180° minus 104° = 76°. Additionally, because ray AK bisects ray DAB, ray DAK ray KAB. Answer in its final form As a result, DAK = KAB = (12)(76) = 38.

## Example 2: Determining if Two Lines Cut by Transversal Are Parallel

Determine if lines A and B are parallel based on the internal angles on the same side of the lines, as illustrated in the diagram below. Exemple No. 2: Identifying whether or not two lines cut by a transversal are parallel John Ray Cuevas is a musician from the Dominican Republic. Solution In order to determine if line A is parallel to line B, the Same-Side Interior Angles Theorem must be used. It is said that supplemental same-side interior angles must exist when the lines crossed by a transversal line are parallel, as stated in the theorem above.

127° plus 75° equals 202° Answer in its final form Because the total of the two inside angles is 202°, the lines are not parallel because they are not parallel.

## Example 3: Finding the Value of X of Two Same-Side Interior Angles

Find the value of x that will align L 1 and L 2 such that they are parallel. Example 3: Calculating the value of X for two interior angles that are on the same side John Ray Cuevas is a musician from the Dominican Republic. Solution The internal angles on the same side of the equations are given. Because the lines are assumed to be parallel, the total of the angles must equal 180°. Create an expression that equals the sum of the two equations plus 180°. The product of (3x+45) and (2x+40) is 1805x + 85, which is 180 – 855*95, which equals 19.

## Example 4: Finding the Value of X Given Equations of the Same-Side Interior Angles

Find the value of x given that m4 = (3x + 6)° and m6 = (5x + 12)° are both positive integers. Obtaining the value of X from the Equations of the Same-Side Interior Angles in Example 4. John Ray Cuevas is a musician from the Dominican Republic. Solution The internal angles on the same side of the equations are given. Because the lines are assumed to be parallel, the total of the angles must equal 180°. Formulate an equation that sums the expressions of m4 and m6 to the power of 180 degrees. 180 – 208x is 160x, which equals 20.

Answer in its final form This means that the ultimate value of x will satisfy the equation, which is 20.

## Example 5: Finding the Value of Variable Y Using Same-Side Interior Angles Theorem

To solve for x, take into consideration that m4 = (3x + 6)° and m6 = (5x + 12)° are both positive integers. Obtaining the value of X using the equations of the same-side interior angles in Example 4. Cuevas, John Ray Solution They are the same-side internal angles, as shown by the accompanying formulae. The total of the angles must equal 180° since the lines are considered parallel. Construct an expression that sums the expressions of m4 and m6 to the power of 180. 16x = 180x, 208x = 160x, and so on.

Last and most definitive response The last value of x that will fulfill the equation is the number ten (ten).

## Example 6: Finding the Angle Measure of All Same-Side Interior Angles

The lines L 1 and L 2 in the figure below are parallel to one another. Calculate the angles of m3, m4, and m5 using the formulas above. Example 6: Calculating the angle measure of all interior angles that are on the same side John Ray Cuevas is a musician from the Dominican Republic. Solution The lines L 1 and L 2 are parallel, and according to the Same-Side Interior Angles Theorem, angles on the same side must be supplementary in order for them to be complementary. Make a note of the fact that m5 is additive to the supplied angle measure of 62°, and thatm5 + 62=180m5 = 180– 62m5 = 118.

Make an expression by combining the acquired angle measure of m5 with m3 to equal 180.m5 + m3 = 180118 + m3 = 180m3 = 180 – 118m3 = 62m3 = 180 – 118m3 = 62m3 = 180 – 118m3 = 62 The same notion applies to the angle measure m4 as well as the supplied angle of 62 degrees.

Calculate the total of the two numbers to be 180. 62 + m4 = 180m4 = 180 – 62m4 = 118m4 = 180 – 62m4 = 118 It also demonstrates that m5 and m4 are angles with the same measure of the angle. ConclusionM5 = 118°, 62°, M3, and M4 are the final answers.

## Example 7: Proving Two Lines Are Not Parallel

The lines L 1 and L 2, as seen in the diagram below, are not parallel to one another. Can you explain the angle measure of z? 7th Example: Demonstrating that two lines are not parallel John Ray Cuevas is a musician from the Dominican Republic. Solution Due to the fact that L 1 and L 2 are not parallel, it is not permissible to assume that the angles z and 58° are additional in this situation. Though z cannot be less than or more than 180 degrees and hence equal to 120 degrees, it can be any other measure of greater or lesser magnitude.

Once you’ve established that, it’s simple to make an educated prediction.

## Example 8: Solving for the Angle Measures of Same-Side Interior Angles

The Same-Side Interior Angle Theorem may be used to determine the angles measurements of b, c, f, and g, given that the lines L 1, L 2, and L 3 are parallel to one another. Example number eight: Solving for the Angle Measures of Interior Angles on the Same-Side Surface John Ray Cuevas is a musician from the Dominican Republic. Solution In light of the fact that L 1 and L 2 are parallel, mb and 53° are considered supplementary. Construct an algebraic equation that demonstrates that the sum of mb and 53° equals 180°.

Calculate an algebraic formula that demonstrates that the sum of b and c equals 180°.

mb + mc = 180127 + mc = 180mc = 180 – 127mc = 53mb + mc = 180mb + mc = 180mb + mc = 180mb + mc = 180mb + mc = 180mb + mc = 180 Because the lines L 1, L 2, and L 3 are parallel and a straight transversal line cuts them, all of the same-side interior angles between the lines L 1 and L 2 are the same as the same-side interior angles between the lines L 2 and L 3.

Amounts in milligrams: mf = bmf = 127mg = cmg = 53 mf = bmf = 127 mg = cmg = 53 Answer in its final form mb = 127°, mc = 53°, mf = 127°, mg = 53°, mf = 127°, mg = 53°

## Example 9: Identifying the Same-Side Interior Angles in a Diagram

Give the complicated figure below, and then find three internal angles on the same side. The same-side interior angles of a diagram are identified in the ninth example. Solution by John Ray Cuevas The figure contains a large number of internal angles that are on the same side. It is reasonable to deduce that three out of numerous same-side internal angles are 6 and 10, 7 and 11, and 5 and 9, based on careful observation.

## Example 10: Determining Which Lines Are Parallel Given a Condition

Determine which lines in the figure are parallel based on the fact that AFD and BDF are supplemental. Example 10: Identifying whether lines are parallel in the presence of a certain condition John Ray Cuevas is a musician from the Dominican Republic.

Solution By careful examination, it can be determined that the parallel lines are line AFJM and line BDI, under the condition that AFD and BDF are supplementary.

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## Interior Angles of a transversal definition

When atransversal crosses two (typically parallel) lines, a transversal is formed. Each pair of inner angles is contained inside the parallel lines and is on the same side of the transversal as the previous pair. Consider the following: A or B can be selected by dragging an orange dot. It is important to note that the two interior angles given are supplementary (they add up to 180°) if the lines PQ and RS are parallel to one another. If we look at the diagram above, the transversalAB crosses the two lines PQ and RS, resulting in the creation of junctions at E and F, respectively.

As a result, there are two pairs of these angles.

Keep in mind that interior refers to the area between the parallel lines.

## The parallel case

If the transversal runs over parallel lines (which is the most common scenario), then the interior angles are additional (they add up to 180°) to the transversal. If a result, as you move points A or B in the diagram above, the two internal angles depicted always sum up to 180°.Try it and you’ll see what I mean.Click on ‘Other angle pair’ to see both pairs of inner angles in succession.

## The non-parallel case

Whenever a transversal runs over parallel lines (as is typically the case), the interior angles are supplementary (that is, they are added on top of 180°). So as you change points A or B in the diagram above, the two inside angles depicted always sum up to 180°. Try it and you’ll see what I mean.Click on “Other angle pair” to go through both pairs of inner angles in turn.

## Other parallel topics

- Equivalent interior and exterior angles
- Corresponding interior and exterior triangles
- Transversal triangles and transversal triangles
- Interior and exterior triangles and transversal triangles and transversal triangles.

2011 Copyright Math Open Reference. All rights reserved. (C) 2011 Copyright Math Open Reference.

## Proof of Same Side Interior Angles

Universityof Georgia students Kristina Dunbar and Michelle Corey and Russell Kennedy and Floyd Rinehart demonstrated that the same side interior angles are the same on both sides. Interior angles on the same side are the same as on the outside. Assume that the lines L, M, and T are different from one another. Then L and M are parallel if and only if the identical side interior angles of the intersection of L and T and the intersection of M and T are supplementary, otherwise they are not parallel.

- Assume L||Mand the angle assignments listed above.
- This is because A and C are two points on the parallel line L on opposite sides of the transversal T, and B is the intersection of the parallel lines L and T.
- As a result, we know that + 180 equals 180, and we can swap for to get + 180 equals 180.
- In order to achieve += 180, substitute for in the equation.
- =Assuming that the identical side interior angles are supplementary, demonstrate that L and M are parallel.
- L and M do not cross since their interior angles on each side of the transversal equal180, which is, of course, more than 180 according to the parallel axiom.
- In addition, if we know that + =+= 180, then we know that there can only be two possibilities: either the lines do not intersect at all (and so are parallel), or the lines overlap on both sides (and hence are parallel).
- This would be impossible since a line is determined by two points.

Therefore, L||M.|| Parallels Main Page|| Kristina Dunbar’s Main Page|| Dr. McCrory’s GeometryPage|| L||M.|| Parallels Main Page|| Kristina Dunbar’s Main Page|| Dr. McCrory’s GeometryPage|| L||M.|| Parallels Main Page|| L||M.|| Parallels Main Page|| L||M.|| Parallels

## On the same side interior angles?

Same side interior angles are two angles that are on the interior of (between) the two lines and are especially on the same side of a transversal. Asked by: Nickolas Herman Score: 4.3/5(74 votes) The internal angles on the same side add up to a total of 180 degrees. Four internal angles are generated when two parallel lines are crossed by a transversal line, as shown in the diagram.

## What are interior angles on the same side called?

The term “Consecutive Interior Angles” refers to a pair of angles that are on one side of the transversal but are contained inside the two lines.

## Are same side interior angles supplementary?

The term “Consecutive Interior Angles” refers to a pair of angles that are on one side of the transversal yet within the two lines.

## Do same side interior angles have the same measure?

Interior angles on the same side are not necessarily congruent. They are only congruent (i.e., have the same measure) when a transversal cuts across them at an angle of 90 degrees to the parallel lines; otherwise, they are not congruent.

## What do same side interior angles look like?

Interior angles on the same side are not necessarily consistent with one another. As a matter of fact, the only time they are congruent (meaning they have the same measure) is when the transversal cutting across the parallel lines is perpendicular to the parallel lines themselves.

### Why are same side interior angles not always supplementary?

The same-side interior angles theorem claims that when the lines crossed by the transversal line are parallel, the same-side internal angles are supplementary, and this is supported by the data. Secondly, because the lines A and B are parallel, the theorem of same-side interior angles indicates that same-side interior angles will be supplementary in this situation.

### What relationship exists between interior angles on the same side of a transversal line?

A transversal separates two parallel lines, and the interior angles on the same side of the transversal are additional angles, as shown in Theorem 10.4. According to Theorem 10.5: If two parallel lines are intersected by a transversal, then the exterior angles on either side of the transversal are considered to be additional angles.

### Do alternate interior angles add up to 180?

A transversal cuts two parallel lines, and the interior angles on the same side of each parallel line are supplementary angles, according to Theorem 10.4. According to Theorem 10.5: If two parallel lines are intersected by a transversal, then the exterior angles on either side of the transversal are considered additional angles.

### Do consecutive interior angles equal each other?

A transversal separates two parallel lines, and the interior angles on the same side of the transversal are additional angles, according to Theorem 10.4. As a result of a transversal cutting two parallel lines, the external angles on each side of the transverse are additional angles.

### What is the difference between alternate interior angles and consecutive interior angles?

Interior angles that are different from one another are congruent. Interior angles that are consecutive are supplementary. Contiguous interior angles are interior angles that are on the same side of the transversal line as the previous interior angle. In the case of non-parallel lines, alternate interior angles don’t have any special qualities that distinguish them from one another.

### Are corresponding interior angles congruent?

Those angles that are congruent with one another are those that are either exterior or interior angles, opposite angles, or comparable angles.

The image above depicts two parallel lines with a transversal in between them.

### What pair of nonadjacent interior angles are on the opposite side of a transversal?

An alternate angle is any pair of angles that is not contiguous to the transversal and that is located on either side of the transversal

### What is the meaning of interior angles on the same side of the transversal?

In the transversal plane, angles that are on the same side of the line are angles that are in one of the half-planes generated by the transversal. Interior and exterior angles on the same side of the transversal are often defined as “interior angles on the same side of the transversal” and “exterior angles on the same side of the transversal,” respectively.

### Which pair of angles are alternate interior angles?

What are Alternate Interior Angles, and how do they work? In the case of two parallel lines being spanned by a transversal, the pair of angles generated on the inner side of the parallel lines, but on opposing sides of the transversal, is referred to as alternative interior angles. In the case of two parallel lines being crossed by a transversal, These angles are always in the same proportion.

### What are the 5 special angle relationship?

When it comes to geometry, there are five essential angle pair relationships to be aware of: Angles that are complementary to one another. Angles that aren’t as important as the primary ones. Angles that are next to each other.

### What is the relationship between co interior angles?

Co-interior angles are those that are formed by two lines intersecting on the same side of a transversal. The two angles that are depicted in each figure are referred to as co-interior angles. If the two lines are parallel, then the co-interior angles sum up to 180 degrees and are hence supplementary to each other.

### What are the examples of alternate angles?

When a transversal cuts across two straight lines, the angles created on the opposite side of the transversal with respect to both lines are referred to as alternative angles. The following are the pairs of alternative angles shown in the above figure:

### What do alternate interior angles mean?

Whenever a transversal is used to join two straight lines together, the angles created on either side of the transversal with respect to both lines are referred to as alternative angles. Angles 1 and 2 in the above illustration are alternative angles.

### Which of the following pairs of angles are consecutive interior angles?

An angle pair on one side of a transversal that is contained inside two lines is referred to as a pair of successive internal angles when two lines are cut by a transversal. When two lines are severed by a transversal, a transversal is referred to as a transversal. The inner angles 3 and 5 are shown in the illustration as successive interior angles.

### How many interior angles does a Heptagon have?

A heptagon is a polygon having seven inner angles, and the name heptagon is used to describe it. Every heptagon contains seven angles, seven sides, and seven vertices, and they are all the same size.

## Angles and parallel lines (Pre-Algebra, Introducing geometry) – Mathplanet

When two lines cross, they generate two sets of angles that are diametrically opposed to one another, A + C and B + D. Vertical angles are another term for the angles that are opposed to each other. Whenever two vertical angles are identical in size, this is known as being congruent. Adjacent angles are angles that originate from the same vertex as their parent angle. Adjacent angles are connected by a single ray and do not overlap. xzy is the sum of the angles A and B in the figure above, and its size is represented by xzy.

- Whenever the total of the two angles equals 180°, two angles are said to be supplementary.
- The crossing line is referred to as a transversal line.
- The eight angles will come together to produce four pairs of comparable angles when they are added together.
- Angles that correspond to one other called congruent.
- Angles that are within the region between the parallel lines, such as angles 2 and 8 in the example above, are referred to as internal angles, whereas angles that are outside of the two parallel lines, such as angles 1 and 6, are referred to as exterior angles.
- For example, 1 + 8 is an alternate angle.
- Example The image above depicts two parallel lines with a transversal in between them.
- Is there another angle that measures 65 degrees as well?

As a result, 6 and 2 have equivalent angles and are congruent, which indicates that angle 2 is 65° in both directions. Angles 6 and 4 are alternative exterior angles and are thus congruent, which indicates that angle 4 is 65 degrees.

## Video lesson

Calculate the sum of all the angles in the illustration.

## Consecutive Interior Angles Theorem

An angle pair on one side of a transversal that is contained inside two lines is referred to as a pair of successive internal angles when two lines are cut by a transversal. When two lines are severed by a transversal, a transversal is referred to as a transversal. The angles 3 and 5 in the illustration are successive internal angles. Angles 4 and 6 are also successive internal angles, as are angles 5 and 6.

### Consecutive Interior Angles Theorem

When a transversal is used to intersect two parallel lines, the pairs of successive interior angles that are generated are known as additional internal angles. Proof: Given that kl is a transversal verb, Clearly demonstrate that numbers 3 and 5 are supplementary and that numbers 4 and 6 are supplementary.

Statement | Reason | |

1 | k∥l,tis a traversal. | Given |

2 | ∠1and∠3form a linear pair and∠2and∠4form a linear pair. | Definition oflinear pair |

3 | ∠1and∠3are supplementarym∠1+m∠3=180°∠2and∠4aresupplementarym∠2+m∠4=180° | Supplement Postulate |

4 | ∠1≅∠5and∠2≅∠6 | Corresponding Angles Theorem |

5 | ∠3and∠5are supplementary∠4and∠6are supplementary. | Substitution Property |

## What Are Alternate Interior Angles Definition & Examples

When you are learning calculus in class or studying it from a textbook, it may be a very intimidating subject. Some concepts can be difficult to grasp because of the way they are taught in class or written down. After hours of investigation, you may discover that your knowledge has not increased in any manner from where you started. Math might be challenging, but don’t be discouraged! For those of you who are having difficulty comprehending various interior angles, this post is for you. Following are some examples and commonly asked questions to help you understand what these angles are and how they may be used effectively.

## What are Alternate Interior Angles?

Simply said, alternate interior angles are generated when two lines are crossed by a third line, resulting in a right angle. The transversal line is the third line that connects the first and second lines. The formation of alternative internal angles is always guaranteed when two parallel or non-parallel lines are crossed by a transversal line. All of the angles are on the transversal line on opposite sides of it, and they are on the inside of the parallel and non-parallel lines respectively.

It is important to note that on the transversal line, you will discover two sets of alternative interior angles.

If the lines crossed by the transversal are parallel, the alternate interior angles are the same as the first interior angle.

## Properties of Alternate Interior Angles

Consider the following characteristics in order to have a better understanding of what alternate interior angles actually are:

- In contrast to each other, alternate interior angles with the same measure are congruent (have the same length). Consecutive interior angles are alternate interior angles that are on the same side of the transversal line as the transversal line. Alternative interior angles have no special qualities and are not congruent when there are non-parallel sides
- Nevertheless, when there are parallel sides, alternate interior angles have unique properties and are congruent.

## Alternate Interior Angles Theorem

A theorem is a hypothesis that can be proven correct through the use of mathematical proof.

A proof is a technique for establishing the correctness of a theorem or statement. According to the Alternate Interior Angles Theorem, if two parallel lines are cut by a transversal, the alternate interior angles formed are congruent with each other on both sides.

## Examples of Alternate Interior Angles

We have two parallel lines, and our objective at this point is to demonstrate that Y= 122° by applying the theorem previously discussed.

- If we consider the equivalent angles of the two sides of a straight line, the angle X is the same as the angle K. This indicates that 122° plus Angle X equals 180°. This is true for angles K and Y as well. Due to the fact that angle X is the same as angle K, subtracting angle K (58°) from 180° will result in the measurement of angle Y. This is illustrated in the following way:

Angle X (58°) is equal to 180° – 122°. And Angle X is the same as Angle K180° – 58° (Angle K) = Angle Y (122°)4 (180° – 58° (Angle K) = Angle Y (122°)4. Consequently, both angles are 122°, which means that the alternate internal angles are congruent with one another.

## Frequently Asked Questions

Despite the fact that it is quite rare. When two parallel lines are linked by a line that is perpendicular to them, the resulting alternate interior angles are 90 degrees on either side of the intersection. Please keep in mind, however, that this is extremely unlikely to be noticed in class or on an examination.

### Do alternate interior angles add up to 180°?

Angles that sum up to 180 degrees are referred to as supplementary angles. We all know that neighboring angles on a straight line always add up to 180°, but did you realize that interior angles on a straight line always add up to 180° as well? What do you think about different interior angles? Unless the alternative interior vertical angles are 90 degrees, the sum of the alternate interior vertical angles will not equal 180 degrees. Adding the alternate interior angles together will result in a value greater than 180° if the alternate interior angles are obtuse.

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## Are same side interior angles supplementary or congruent?

The same – side interior angletheorem asserts that when two parallel lines are crossed by a transversal line, the same – side interior angles that are created are supplementary, meaning that they add up to a total of 180 degrees. The same is true for side interior angles: they are not necessarily congruent. In reality, the only time they are congruent (meaning they have the samemeasure) is when the alternate interior angles are congruent. Secondly, are alternate interior angles supplementary or congruent?

- When the transversal intersects two parallel lines, corresponding angles and alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary.
- Angles on the same side of the transversal and on the interiorof (between) two lines are referred to as same side interiorangles or same side interior angles.
- Two lines are parallel if they are intersected by a transversal and the same side interior angles are supplementary on both sides of the line.
- Exterior angles on the same side of the transversal line that are on the same side of the parallel lines are referred to as same – side exterior angles.

The theory asserts that same – side exterior angles are supplementary, which means that they have a sum of 180 degrees when they are added together.

## Alternate Interior Angles – Explanation & Examples

Specifically, according to the same – side interior angletheorem, when two lines that are parallel are crossed by a transversal line, the same – side interior angles that are created are supplementary, meaning they add up to 180 degrees. Side interior angles are not necessarily consistent on the same side. The only time they are congruent (meaning they have the samemeasure) is when they are alternate interior angles. Secondly, are alternate interior angles congruent or supplementary when they are alternate interior angles.

- When the two intersecting lines are parallel, corresponding angles are congruent, alternate interior angles are congruent, alternate exteriorangles are congruent, and consecutiveinterior angles become supplementary.
- Those who are on the same side of the room as you are will appreciate this.
- Is it possible to have supplemental external angles on the same side as the interior angles?
- The theory indicates that the same – side exterior angles are supplementary, which means that they have a sum of 180 degrees when they are added together.

## What are the Alternate Interior Angles?

Internal angles generated when two parallel or non-parallel lines are crossed by a transversal are known as alternate interior angles. The angles are located in the inner corners of the intersections and are on opposite sides of the transversal plane of the transversal plane. It is possible to have alternate interior angles that are equal if the lines crossed by the transversal are parallel to one another. There is no geometrical relationship between the alternate interior angles created when a transversal crosses two non-parallel lines.

Alternate interior views are illustrated as follows: Take a look at the illustration above.

As a result, the interior angle pairs that alternate are as follows: As a result, a=d and b=c are equivalent.

- So the equations a=d and b=c are equivalent. In terms of alternate interior angles, we can derive the following conclusions:

## Alternate Interior Angles Theorem

As a result, a=dand b=c are equivalent. The following are some observations we may make concerning alternate interior angles:

### Applications of Alternate Interior Angles

- Alternate interior angles are most famously used to show the Earth’s roundness by a great Greek scientific writer, Eratosthenes, who used alternate interior angles to argue that the Earth is round. The inner angles of the windows, which have panes split by mun-tins, alternate on each side
- The horizontal lines at the top and bottom of the letter Z are parallel, while the diagonal line is transversal in the letter Z. There are two different internal angles in a letter Z, as a result of this.