Two angles that lie on the opposite sides of the transversal and are placed on two different lines are called alternate exterior angles. These pairs of angles are always equal if the two given lines are parallel. In the given figure below, Line AB || Line CD and they are intersected by the transversal MN.
Full Answer
What are facts about alternate exterior angles?
Alternate Interior Angles Exterior Angles
- Transversal Line. Two separate straight lines, can both be crossed by a third line, called a "Transversal" line. ...
- Equal Alternate Interior Angles. When a Transversal line crosses through 2 parallel straight lines, the interior alternate angles created are equal alternate interior angles.
- Alternate Exterior Angles. ...
- Corresponding Angles. ...
Which angles are pairs of alternate exterior angles?
Properties
- If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
- Consecutive exterior angles are supplementary. ...
- Alternate exterior angles don’t have any specific properties in the case of non – parallel lines.
What are alternate interior and exterior angles?
What are alternate interior and exterior angles? When two lines are crossed by a transversal, the opposite angle pairs on the outside of the lines are alternate exterior angles. One way to identify alternate exterior angles is to see that they are the vertical angles of the alternate interior angles. Alternate exterior angles are equal to one ...
How can you remember alternate interior angles?
To help you remember: the angle pairs are on Alternate sides of the Transversal, and they are on the Interior of the two crossed lines. When the two lines being crossed are Parallel Lines the Alternate Interior Angles are equal. (Click on "Alternate Interior Angles" to have them highlighted for you.)
What are alternate exterior angles always?
Alternate exterior angles always lie outside two lines that are intersected by the transversal and they are located on the opposite sides of the transversal.
How do you know if an angle is alternate exterior?
To find exterior angles, look in the space above and below the crossed lines. To find alternate exterior angles, look at that outside space for each crossed line, on different sides of the transversal. We hope you said ∠1 , ∠2 , ∠7 , and ∠8 are the exterior angles.
Can alternate exterior angles be on the same side?
0:578:37Same Side Exterior Angles - YouTubeYouTubeStart of suggested clipEnd of suggested clipThe same side of the transversal. As one another something to note about same side exterior anglesMoreThe same side of the transversal. As one another something to note about same side exterior angles is that if the lines are parallel lines. Then seam side exterior angles will be supplementary.
What is special about alternate exterior angles?
The term alternate exterior angles is often used when two lines are cut by a third line, a transversal . The Alternate Exterior Angles Theorem states that if k and l are parallel , then the pairs of alternate exterior angles are congruent .
How do you identify an alternate and corresponding angle?
Corresponding angles will be on the same side of the transversal and in the same position relative to one of the parallel lines. Step 2: Identify the alternate interior angles. Alternate interior angles will be on opposite sides of the transversal and in between the lines cut by the transversal.
What is the difference between exterior and alternate interior angles?
Lesson Summary Alternate interior angles are the angles between the two lines being intersected and on opposite sides of the transversal. Alternate exterior angles are alternate angles that are outside the two lines being intersected by the transversal.
Which angle are alternate exterior angles?
Alternate Exterior Angles – Alternate exterior angles are the pair of angles on the outer side of the two parallel lines but on the opposite side of the transversal.
What are the properties of alternate angles?
What are the Properties of Alternate Interior Angles? Alternate Interior angles are congruent. The sum of the angles formed on the same side of the transversal which are inside the two parallel lines is equal to 180°. Alternate interior angles don't have any specific properties, in case of non-parallel lines.
What do alternate exterior angles add up to?
If the transversal cuts across parallel lines (the usual case) then exterior angles are supplementary (add to 180°).
Why are alternate exterior angles congruent?
Alternate exterior angles are congruent if the lines crossed by the transversal are parallel. If alternate exterior angles are congruent, then the lines are parallel. At each intersection, the corresponding angles lie at the same place.
How do you prove that alternate exterior angles are congruent?
Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. The proof of this theorem is very similar to that of the Alternate Interior Angles Theorem.
What is an example of an alternate angle?
The term alternate interior angles is often used when two lines are cut by a third line, a transversal . The Alternate Interior Angles Theorem states that if k and l are parallel , then the pairs of alternate interior angles are congruent . That is, ∠2≅∠8 and ∠3≅∠5 .
What are Alternate Exterior Angles?
Alternate exterior angles are created when two or more lines are intersected by a transversal. These angles are formed on the outer side of the transversal on the different sides.
Why is an exterior angle called an exterior angle?
It is an exterior angle because it is located at the exterior position of the triangle. When two lines are intersected by a transversal, the angles on the outer side of these two lines are called exterior angles.
What are the two exterior angles that lie on two different lines cut by a transversal called?
According to the angles theory, the two exterior angles that lie on two different lines cut by a transversal and are placed on the opposite sides of the transversal are called alternate exterior angles.
What is the converse of alternate exterior angle theorem?
The converse of alternate exterior angle theorem states that, if the alternate exterior angles formed by two lines, which are cut by a transversal, are congruent, then the lines are parallel.#N#Hence , in the above figure, if it is given that ∠1 = ∠7 then line AB || line CD.
What is a pair of exterior angles that satisfy these conditions called?
Now, a pair of angles that satisfy these conditions is called a pair of alternate exterior angles . Two exterior angles that lie on two different lines cut by a transversal and are placed on the opposite sides of the transversal are called alternate exterior angles.
What are the two opposite angles of a transversal?
Two angles that lie on opposite sides of the transversal and are placed on two different lines, both either inside the two lines or outside, are called alternate angles. In the given figure below, the alternate exterior angle pairs are 1 and 7; 2 and 8. Angle 2 is on the right side of the transversal M and 8 is on the left side;
What are the pairs of angles that are always equal?
These pairs of angles are always equal if the two given lines are parallel. In the given figure below, Line AB || Line CD and they are intersected by the transversal MN. Here, the alternate exterior angle pairs are ∠1 and ∠7; ∠2 and ∠8. This means ∠1 = ∠7; ∠2 = ∠8
What is an alternate exterior angle?
Alternate Exterior Angles are a pair of angles on the outer side of each of those two lines but on opposite sides of the transversal. To help you remember: the angle pairs are on Alternate sides of the Transversal, and they are on the Exterior of the two crossed lines.
Where are angle pairs on a transversal?
To help you remember: the angle pairs are on Alternate sides of the Transversal, and they are on the Exterior of the two crossed lines.
What are Parallel Lines?
Before we understand what we mean by alternate interior angles, we must first recall the basic concepts and terms that are integral to the understanding of alternate interior angles.
What is a Transversal?
A transversal is defined as a line intersecting two or more given lines in a plane at different points. For example, in the figure below, the lines l and m are parallel while the line p is intersecting both the line l and the line m. Hence, the line p is a transversal to the lines l and m.
Angles Made by a Transversal with Two Lines
Now, we shall learn about the angles made by a transversal with two given lines. Some of these angles can be paired together by virtue of the positions they occupy.
Definition
A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms are directed in opposite directions and do not include segment PQ is called alternate exterior angles in a transversal.
Alternate Exterior Angles Theorem
The alternate exterior angle theorem states that if two lines are parallel and are intersected by a transversal, then the alternate exterior angles are considered as congruent angles or angles of equal measure. What does this mean?
Degrees
The angles formed by the alternate exterior angles lie on different sides of a transversal and are termed to be congruent if the lines are parallel. These angles can range from being more than 0 o to being less than 180 o .
Real Life Applications of Alternate Exterior Angles
Alternate exterior angles are not just used to geometrical calculations in mathematics. They are used extensively in real life situations as well. For instance, alternate exterior angles are used to design the fittings of the sofas and chair in your room.
What are Alternate Exterior Angles?
Alternate exterior angles are the pair of angles that lie on the outer side of the two parallel lines but on either side of the transversal line.
Where do the pairs of alternating exterior angles lie?
Notice how the pairs of alternating exterior angles lie on opposite sides of the transversal but outside the two parallel lines.
What is an alternate angle?
Alternate angles are non-adjacent and pair angles that lie on the opposite sides of the transversal. In this article, we are going to discuss alternate exterior angles and their theorem.
What is the purpose of alternate exterior angles in trigonometry?
In trigonometry, alternate exterior angles can be used to calculate the height of tall structures such as buildings.
What are the two angles that are congruent?
Angle (2x + 26) ° and (3x – 33) ° are alternate interior angles. Since L1 and L2 are parallel, the two angles are therefore congruent. So, we have;
Where do the corresponding angles lie at each intersection?
At each intersection, the corresponding angles lie at the same place.
Do we make different angles in different postures while doing yoga and exercises?
We even make different angles in different postures while doing yoga and exercises.
Where are alternate exterior angles located?
Just like the alternate interior angles, then alternate exterior angles are also located on the opposite sides of the transversal line but the only difference being that they are located on the exterior of the transverse. A classic example of the alternate exterior angle is shown in the below-mentioned figure where angles 1 and 8 along with angles 2 and 7 are known to be the alternate exterior angle.
What are alternate interior angles?
The alternate interior angles are generally on the opposite sides but in the interior of the transversal lines. Angles 3 and 6, as well as angles 5 and 4 in the below-given figure, are classic examples of alternate interior angles.
What happens when a transversal line cuts a parallel line?
When a transversal line cuts the running parallel line, in this case, the measure of both the exterior angles will have identical angle values. However, if in case, the lines fail to run parallelly, the measure of the angle changes to being different. If parallel lines are cut by a transversal then the alternate exterior angles will have the same measures. If the lines are not parallel the alternate exterior angles will not have the same measure. If the alternate exterior angles which are formed when the transversal line cuts two parallel lines are congruent, it simply indicates that the lines are parallel.
What are the parallel lines PQ and RS?
These parallel lines PQ and RS are intersected by another transversal line LM. This line of intersection has further formed a few angles which are W, X, Y, Z.
What is the case of angles on the opposite side of the transverse?
In the case of angles on the opposite side of the transverse, the angles are known to be alternate.
What is alternate angle?
In mathematical geometry, alternate angles are one of the most intriguing topics for students. Typically, the alternate angles can be considered as pairs of non-adjacent angles on either side of the transversal line. In an effort to understand alternate angles, here is all the information you need to know thoroughly.
When two parallel lines are cut by a transversal, the resulting alternate interior angles or alternate exterior angles are?
Alternate angle theorem states that when two parallel lines are cut by a transversal, then the resulting alternate interior angles or alternate exterior angles are congruent.
What is an alternate exterior angle?
Alternate exterior angles are similar to vertex angles, in that they are opposite angles (on either side of the transversal). Alternate exterior angles are on opposite sides of the transversal (that's the alternate part) and outside the parallel lines (that's the exterior part). Can you find the two pairs of alternate exterior angles in our drawing?
What is an exterior angle?
An exterior angle among line constructions (not polygons) is one that lies outside the parallel lines. You can see two types of exterior angle relationships:
What happens when two parallel lines intersect?
When two parallel lines are intersected by a transversal, complex angle relationships form, such as alternating interior angles, corresponding angles, and so on. Being able to spot angle relationships, and confidently find congruent angles when lines intersect, will make you a better, geometry student. You will solve complex problems faster ...
What angles can you learn from geometry?
Your geometry studies have shown you acute, right and obtuse angles . You may even have learned about straight and reflex angles, but if you are angling to learn even more, you can investigate many other kinds of angles like exterior and interior angles. You can learn about congruent, adjacent, vertical, corresponding, and alternating angles, too.
What are congruent angles?
Any two angles, no matter their orientation, that have equal measures (in radians or degrees) are congruent. They show the same "openness" between the two rays, line segments or lines that form them. So these two 35° 35 ° angles are congruent, even if they are not identically presented, and are formed with different constructions:
Why do we talk about angle relationships?
We talk of angle relationships because we are comparing position, measurement, and congruence between two or more angles. For example, when two lines or line segments intersect, they form two pairs of vertical angles. When two parallel lines are intersected by a transversal, complex angle relationships form, such as alternating interior angles, ...
When a transversal crosses two parallel lines, we get corresponding angles?
Anytime a transversal crosses two other lines, we get corresponding angles. The more restrictive our intersecting lines get, the more restrictive are their angle relationships. When a line crosses two parallel lines (a transversal), a whole new level of angle relationships opens up:
What are exterior angles of a triangle?
The exterior angles of a triangle are the angles that form an adjacent pair with the interior angles by extending the sides of the triangle.
What is the measure of exterior angle adjacent to angle A?
Measure of exterior angle adjacent to angle A = m∠B + m∠C = 45° + 75° = 120°.
What is the angle 4?
Angle 4 is the exterior angle adjacent to the angle 3.
Which angle is always greater than the interior angle?
Exterior angle is always greater than the either of the two remote interior angles.
Is the exterior angle equal to the interior angle?
So, the exterior angle is equal to the sum of the two non-adjacent interior angles.
What Are Parallel lines?
What Is A transversal?
Angles Made by A Transversal with Two Lines
- Now, we shall learn about the angles made by a transversal with two given lines. Some of these angles can be paired together by virtue of the positions they occupy. Let l and m be two lines and let n be the transversal intersecting them at P and Q respectively as shown below –
Definition
- A pair of angles in which one arm of each of the angles is on opposite sides of the transversal and whose other arms are directed in opposite directions and do not include segment PQ is called alternate exterior angles in a transversal. In the above figure, ∠ 2 and ∠ 8 form a pair of alternate exterior angles. Another pair of alternate exterior angles in this figure is ∠ 1 and ∠ 7. Now, we sh…
Alternate Exterior Angles Theorem
- The alternate exterior angle theorem states that if two lines are parallel and are intersected by a transversal, then the alternate exterior angles are considered as congruent angles or angles of equal measure. What does this mean? Let us consider the following figure – We can clearly see that the line l is parallel to line m. The two lines have be...
Degrees
- The angles formed by the alternate exterior angles lie on different sides of a transversal and are termed to be congruent if the lines are parallel. These angles can range from being more than 0o to being less than 180o.
Properties
- The following are the properties of alternate exterior angles – 1. If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 2. Consecutive exterior angles are supplementary. Here, it is important to note that consecutive exterior angles are exterior angles that are on the same side of the transversal line. 3. Alternate exterior angles don’…
Real Life Applications of Alternate Exterior Angles
- Alternate exterior angles are not just used to geometrical calculations in mathematics. They are used extensively in real life situations as well. For instance, alternate exterior angles are used to design the fittings of the sofas and chair in your room. Another area where Alternate exterior angles are extensively used is for the designing of buildings, roads and bridges. Different polygo…
Using External Alternate Angle Theorem to Find Missing angles?
- We have seen that eight angles are formed when two lines are intersected by a transversal. So, how can use alternate exterior angles to find the remaining angles? We shall understand this by an example. Suppose we have the figure below where ∠ 2 = 60o. We shall use this value to find the remaining seven angles formed by the intersection of the two lines by a transversal. Let us s…
Examples
- Example 1 Find the value of x in the given figure, where the line L1 and L2are parallel. Solution We have been given that the lines L1 and L2 are parallel. Therefore, by the alternate external angle theorem, we can say that the angles ( 3x – 33 ) o and ( 2x + 26 ) oare equal. Hence, we have, 3x – 33 = 2x + 26 ⇒ 3x – 2x = 26 + 33 ⇒ x = 59 Hence, in the above figure, x = 59o Can we verify our a…