## 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)

In 4-5, utilize the information provided to identify whether lines are parallel to one another. If there aren’t any, simply type none. Take each question one at a time and think about it. Drawing (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)

Determine if the statement is true or untrue for questions 9-10.

- Interior angles on the same side of the transversal are on the same side of the transversal. When lines are parallel, the interior angles on the same side are congruent.

## 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

Ray holds a professional engineering license in the Philippines. He enjoys writing about mathematics and civil engineering in particular. A pair of same-side interior angles is a pair of angles that are on the same side of a transversal line and in between two parallel lines that have crossed each other. A transversal line is a straight line that crosses one or more other lines at right angles. The Interior Angles on the Same Side of a Transversal Theorem asserts that if a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are additional to one another.

- The Theorem of Interior Angles on the Same Side ProofAssume that L 1 and L 2 are parallel lines cut by a transversal T in such a way that the interior angles 2 and 3 in the figure below are on the same side of T.
- Because 1 and 2 form a linear pair, they are considered to be supplementary.
- According to the Alternate Interior Angle Theorem, 1 + 3 = 1.
- As a result, numbers 2 and 3 are extra.

## 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°.

Using the transitive property, we get 2 + 4 = 1 + 4 as a result of the addition. Because of the addition characteristic, 2 = 1. As a result, L 1 is parallel to L 2. 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°.

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

Y’s angle measure is the same-side interior angle as 105°, thus we must get the value of y by solving for the angle measure. Finding the value of the variable Y using the Same-Side Interior Angles Theorem is illustrated in Example 5. John Ray Cuevas is a musician from the Dominican Republic. Solution Make certain that y and the obtuse angle 105° are both interior angles on the same side of the equation. For the sake of satisfying the Same-Side Interior Angles Theorem, it simply implies that these two angles must add up to 180°.

y + 105 = 180y = 180 – 105y = 75y + 105 = 180y = 180 – 105y = 75 Answer in its final form When all factors are considered, the ultimate value of x that will fulfill the theorem is 75.

## 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.

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.

## Example 7: Proving Two Lines Are Not Parallel

There is a parallel relationship between the lines L 1 and L 2 in the diagram below. Calculate the angles of m3, m4, and m5 by using the formula below. Calculating the Angle Measure of All Interior Angles on the Same Side (Example 6). Cuevas, John Ray Solution According to the Same-Side Interior Angles Theorem, angles on the same side must be supplementary, hence the lines L 1 and L 2 are parallel. Remember that m5 is an addition to the provided angle measure of 62°, andm5 + 62 = 180m5 + 62m5 = 180-m5 + 62m5 = 118 Because m5 and m3 are supplemental, they are combined.

For the angle measure m4 as well as the supplied angle of 62°, the same notion holds true.

60 + m4 = 180m4 = 180–62m4 = 118m4 = 60 + m4 = 180 – 62m4 = 118m4 It also demonstrates that m5 and m4 are angles with the same measure of the angle axis of rotation.

Final Answer

## 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.

This is because the lines L 1, L 2, and L 3 are parallel and a straight transversal line cuts them. 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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## 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

Using the same side interior angle theorem, we may calculate the relationship between two identical side interior angles. A transversal intersects two parallel lines, and each pair of same-side interior angles are complementary (their total is 180°), according to the theorem for the “same side interior angle theorem”:

## 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.

- Assuming the same diagram as earlier, let us suppose that 4 + 5 = 180° (1) Because the numbers 5 and 8 constitute a linear pair, 5+8 = 180° (2)From (1) and (2), 4 = 8Thus, a pair of matching angles is 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 demonstrated. Consider the following: Important considerations when it comes to the same-side internal angles are listed in the following section:

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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### 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 formed interior angles on the same side, and the sum of these angles equals 180 degrees. Because the sum of the interior 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.

## Same Side Interior and Same Side Exterior Angles – Concept – Geometry Video by Brightstorm

When we apply what we’ve learned about alternate interior and alternate exterior angles to the same side angles, we discover some intriguing facts about the same side angles. Now, what exactly do I mean by “same side”? Interior angles on the same side would be 4 and 5, so take note that we have parallel lines as well as the transversal. The numbers 4 and 5 are on the same side of the transversal as each other. Because the transversal line intersects two parallel lines on the same side, I’ll refer to this as the interior side because it is in between angles that are supplementary.

According to our knowledge of alternate exterior and alternate interior angles, they must be compatible with one another.

This implies that 5+6 must equal 180 degrees, and since 6 and 4 are congruent, we can infer that the same side interior angles are supplementary by using the transitive property, which states that if 5 and 6 are supplementary, then 5 and 4 are supplementary, and thus we can say that the same side interior angles are supplementary.

In terms of the same side external, if I delete these marks, exterior implies outside of the parallel lines, which is what I’m referring to here.

As a result, angle 2 and angle 7 are supplementary in the same way that angle 1 and angle 8 are. These two are on the same side of the field and will serve as a backup pair.

## 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 |

## Angles and Parallel Lines – MathBitsNotebook(Geo

When atransversalintersects two or more lines in the same plane, a series of angles are formed. Certain pairs of angles are given specific “names” based upon their locations in relation to the lines. These specific names may be used whether the lines involved are parallel or not parallel. |

alternate interior angles | |

alternate exterior angles | |

corresponding angles | |

interior angles on the same side of the transversal |

Look at these angles in respect to parallel lines and see what you come up with.

Alternate Interior Angles: The word “alternate” means “alternating sides” of the transversal.This name clearly describes the “location” of these angles.When the lines are parallel, the measures areequal. |

∠1and∠2are alternate interior angles∠3and∠4are alternate interior angles | Alternate interior angles are ” interior ” (between the parallel lines), and they ” alternate ” sides of the transversal. Notice that they are not adjacent angles (next to one another sharing a vertex).When the lines are parallel, the alternate interior anglesare equal in measure.m∠ 1 =m∠ 2andm∠ 3 =m∠ 4 |

If two parallel lines are cut by a transversal, the alternate interior angles are congruent. |

Converse | If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. |

Alternate Exterior Angles: The word “alternate” means “alternating sides” of the transversal.The name clearly describes the “location” of these angles.When the lines are parallel, the measures areequal. |

∠1and∠2are alternate exterior angles∠3and∠4are alternate exterior angles | Alternate exterior angles are ” exterior ” (outside the parallel lines), and they ” alternate ” sides of the transversal. Notice that, like the alternate interior angles, these angles are not adjacent.When the lines are parallel, the alternate exterior anglesare equal in measure. m∠ 1 =m∠ 2andm∠ 3 =m∠ 4 |

If two parallel lines are cut by a transversal, the alternate exterior angles are congruent. |

Converse | If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel. |

Corresponding Angles:The name does not clearly describe the “location” of these angles. The angles are on the SAME SIDE of the transversal, one INTERIOR and one EXTERIOR, but not adjacent.The angles lie on the same side of the transversal in “corresponding” positions. When the lines are parallel, the measures areequal. |

∠1and∠2are corresponding angles∠3and∠4are corresponding angles∠5and∠6are corresponding angles∠7and∠8are corresponding angles | If you copy one of the corresponding angles and you translate it along the transversal, it will coincide with the other corresponding angle. For example, slide ∠ 1 down the transversal and it will coincide with ∠2.When the lines are parallel, the corresponding anglesare equal in measure. m∠ 1 =m∠ 2andm∠ 3 =m∠ 4 m∠ 5 =m∠ 6 andm∠ 7 =m∠ 8 |

If two parallel lines are cut by a transversal, the corresponding angles are congruent. |

Converse | If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel. |

InteriorAngles on the Same Side of the Transversal: The name is adescription of the “location” of the these angles.When the lines are parallel, the measures aresupplementary. |

∠1and∠2are interior angles on the same side of transversal∠3and∠4are interior angles on the same side of transversal | These angles are located exactly as their name describes. They are” interior ” (between the parallel lines), and they are on thesame side of the transversal.When the lines are parallel, the interior angles on the same side of the transversal are supplementary. m∠ 1 +m∠ 2 = 180m∠ 3 +m∠ 4 = 180 |

If two parallel lines are cut by a transversal, theinterior angles on the same side of the transversal are supplementary. |

Converse | If two lines are cut by a transversal and theinterior angles on the same side of the transversal are supplementary, the lines are parallel. |

In addition to the four pairs of named angles that are used when working with parallel lines (as described above), there are a few pairs of “old buddies” that are also utilized when working with parallel lines (as listed below).

Vertical Angles: When straight lines intersect, vertical angles appear.Vertical angles areALWAYS equalin measure,whether the lines are parallel or not. |

There are 4 sets of vertical angles in this diagram!∠1and∠2 ∠3and∠4 ∠5and∠6 ∠7and∠8 Remember:the lines need not be parallel to have vertical angles of equal measure. |

Linear Pair Angles: A linear pair are two adjacent angles forming a straight line.Angles forming a linear pair areALWAYS supplementary. |

Since a straight angle contains 180º, the two angles forming a linear pair also contain 180º when their measures are added (making them supplementary).m ∠1 +m ∠4 = 180m ∠1 +m ∠3 = 180m ∠2 +m ∠4 = 180m ∠2 +m ∠3 = 180m ∠5 +m ∠8 = 180m ∠5 +m ∠7 = 180m ∠6 +m ∠8 = 180m ∠6 +m ∠7 = 180 |

If two angles form a linear pair, they are supplementary. |

## Alternate Interior & Exterior Angles (video lessons, examples, step-by-step solutions)

Pages that are related Corresponding AnglesCo-Interior AnglesCorresponding Angles Angles on the inside of a triangle Help with Geometry Pairs of angles can be related to one another in a variety of ways in geometrical terms. We’ll have a look at it here.

- The alternate interior angle theorem is proven, as is the converse interior angle theorem
- Alternate exterior angles are proven, as is the converse of the alternate exterior angle theorem
- Alternate interior angles are proven, as is the converse of the alternate interior angle theorem

Alternate Interior Angles and Alternate Exterior Angles are two different types of angles. The alternate interior and alternate exterior angle pairs shown in the accompanying figure are some instances of alternate interior and alternate exterior angle pairs. If you require further examples and explanations of alternate interior angles and alternate external angles, please continue reading this page.

### Alternate Interior Angles

The formation of alternating internal angles on opposing sides of a transversal occurs when a line (referred to as a transversal) contacts a pair of lines (referred to as an intersection). Whenever a pair of lines is parallel to one another, the alternate interior angles are equal to one another. TheAlternate Interior Angles Theorem asserts that when two parallel lines are intersected by a transversal, the alternative interior angles formed are congruent with one another. What are Alternate Interior Angles and How Do I Identify Them?

- Dandeare alternates between internal angles in the figures above.
- In a Z-shaped figure, the angles are found.
- The following is an example of how to use the term “example.” Determining the angles b, c, d, e, f, g, and h from the figure below is a difficult task because the angles are all different.
- As a result, b + 60° = 180° b = 180° b = 180° b = 180° b = 120° Step 2: The angles b and c are vertical in nature.
- Vertical angles are represented by the numbers 3:d and 60°.
- Step 4:d and e are alternate internal angles that are used in this step.
- Step 5: The additional angles f and e are defined.

As a result, g = f = 120° is obtained.

As a result, h = e = 60° is obtained.

You may have seen from the above example that an angle can be either 60° or 120° in degree.

In general, the diagram will look somewhat like the one illustrated below.

You would be able to calculate out the values of all the other angles if you were given only one angle.

Any line that crosses two other lines is referred to as a transversal.

If the two lines are parallel, then the alternative interior angles of the two lines are congruent with one another.

If two lines are parallel, the same side interior angles are supplementary, and the same side exterior angles are supplementary if the two lines are parallel. In this example, we will use Alternate Interior Angles to solve for x.

Using various internal angles to get the measurements of angles, what is the procedure? The Alternate Interior Angle Theorem is shown. Using the Corresponding Angle Postulate, this video will demonstrate that Alternate Interior Angles are congruent when compared to one another. How do you establish the Alternate Interior Angle Theorem, which states that when lines are parallel, alternate interior angles are congruent? What are the steps involved? What is the inverse of the Alternate Interior Angles Theorem and why is it important?

TheConverse of the Alternate Interior Angle Theorem states that if two lines are cut by a transversal and their alternate interior angles are congruent, then the lines are said to be parallel.

This video demonstrates an example of the alternative interior angle converse in action.

### Alternate Exterior Angles

Whenever a transversal connects two parallel lines, the opposing angle pairs on either side of the lines are referred to as alternative external angles. If the two lines are parallel, then the alternate exterior angles are congruent if the two lines are not parallel. To discover alternate outside angles, look for the vertical angles of the alternate interior angles. This is one method of identifying alternate exterior angles. Exterior angles that are alternated are equal to one another. The Alternate Exterior Viewing Positions Theorem asserts that when two parallel lines are intersected by a transversal, the alternate exterior angles formed are congruent with one another.

They are identical to one another and alternate external angles in the same manner as bandgare Definition and characteristics of Alternate Exterior Angles This video demonstrates how to recognize different outside views as well as their characteristics.

If the two lines are parallel, then the alternate exterior angles are congruent if the two lines are not parallel.

Is it possible to obtain an angle using the other exterior angle method?

When a transversal is cut through a pair of lines and their alternative angles are congruent, the lines are said to be parallel. This video demonstrates an example of the alternative exterior angle converse in action.

You can practice many arithmetic concepts by using the freeMathway calculator and problem solution provided below. Make use of the examples provided, or put in your own issue and cross-reference your answer with the step-by-step instructions. If you have any suggestions, comments, or concerns regarding this site or page, please let us know. Thank you for taking the time to provide comments or inquiries on ourFeedbackpage.

## Alternate Interior Angles – Explanation & Examples

The purpose of this article is to introduce you to another sort of angle that may be generated when two parallel or non-parallel lines are crossed by a transverse line. As you are probably aware, parallel lines are two or more lines that never meet, but a transversal line is a straight line that intersects two or more parallel lines and is therefore called a transversal line. You can go to the previous articles if you want to learn about additional angles and their definitions as well as the numerous sorts of angles.

## 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.

- Alternate interior angles are consistent with one another. Interior angles that follow one another are supplementary. Contingent interior angles are those that are on the same side of the transversal line as the preceding interior angle. The features of alternate interior angles in the case of non-parallel lines are not as distinct as they are in the case of parallel lines.

## Alternate Interior Angles Theorem

The alternative interior angles theorem asserts that the alternate interior angles are congruent when a transversal meets two parallel lines, which is true in many situations. The alternate internal angles theorem is demonstrated. Given: PQ/RS is a line of code. To demonstrate that a = d and b = c Knowing that corresponding angles and vertical angles are identical to each other when a transversal crosses any two parallel lines, we may use this to our advantage. As a result, g = c. (i)g = b.

- (ii) We may deduce the following from equations I and (ii): b = c In a similar vein, a = d As a result, it has been established.
- Exemple No.
- Find the value of x, as well as the value of the other pair of alternate interior angles, and then multiply the two values together.
- Four times nineteen equals three times sixteen.
- The value of the other pair of alternative interior angles is thus 180 0–121 0 = 59 0 (180 0–121 0 = 59 0).
- 2 The inner angles of (2x + 10) ° and (x + 5) ° are two successive interior angles.
- Solution Interior angles that follow one another are supplementary.

((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((( Divide both sides by 3.x = 55 to get the answer.

3 Find the measurement of the two interior angles if (2x + 26) ° and (3x – 33) ° are alternate inner angles that are congruent with each other.

Therefore, 2x + 26 = 3x-33 = 59, which is equal to The angles are measured at a distance of 144°.

4 Suppose that (3x + 20)° and 2x° are successive interior angles.

Solution Consecutive interior angles are supplementary, as follows: (3x + 20)° + 2x° = 180°3x + 20 + 2x = 180°3x + 20 + 2x = 180°3x + 20 + 2x = 180°5x + 20 = 180 Subtract 20 from both sides of the equation 5x = 160 Divide each side by 8.x = 32 to get the total.

As a result, the value of x is equal to 32 degrees. As a result, the internal angles that follow each other are 60° and 120°.

### 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.

## 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.

Parallels Main Page||

Dr.

L||M.||

Kristina Dunbar’s Main Page||

McCrory’s GeometryPage||

Parallels Main Page||

Parallels Main Page||

Parallels

## Section 3-3

Transversals and parallel lines are both types of transversals. In yesterday’s session, we discussed the pairs of angles that are generated when two lines are sliced by a transversal line. What happens to the unusual pairings of angles when two parallel lines are cut by a transversal will be the topic of today’s discussion. If a transversal is used to intersect two parallel lines, each pair of angles generated will be either supplementary or congruent in nature. The congruence and supplementary nature of vertical angles is obvious, as is the case with linear pairings.

Four will be sharp, and four will be obtuse.

The four obtuseangles that are generated will be of equal size.

Supplementary angles are simple to memorize since they are supplementary to the primary angle.

When two parallel lines are split by a transversal, same-side interior and same-side exterior angles are supplementary.

Angles that are congruent- the angles between the remaining pairs of angles must be congruent.

Look for the letter “Z.” The angles formed by the parallel lines that create the “Z” must be consistent with one another.

3.If two parallel lines are intersected by a transversal, then the inner angles of the two parallel lines are equivalent.

5.If two parallel lines are intersected by a transversal, the corresponding angles are incongruent with one another.

Determine the values of the variables x, y, z, u, and p.

Solution: 1.x = 130 is a mathematical expression (AlternateInterior Angles) the square root of 2x is 5y (Vertical Angles) The number 130 equals 5yy = 263.x + z = 180.

Because the lines are parallel, the alternateinterior angles are congruent because the lines are parallel.

Because the lines are parallel, the angles formed by the lines are congruent.

Because the lines are parallel, interior angles on the same side are additional. Therefore: (3 times – 10) o+ (5x + 30) o+ (5x + 30) o+ (5x + 30) o+ (5x + 30) o= 180 o3x – 10 + 5x + 30 = 1808x + 20 = 1808x = 160x = 20 o= 180 o3x – 10 + 5x + 30