Are Alternate Exterior Angles Supplementary
Alternate Exterior Angles
Alternate exterior angles are formed when a transversal intersects two or more than parallel lines at distinct points. The term exterior means something which is situated externally. Alternate exterior angles ever lie outside two lines that are intersected by the transversal and they are located on the contrary sides of the transversal. Thus, the two exterior angles which form at the alternating ends of the transversals in the outside office are considered as the pair of alternate outside angles and they are always equal. We become two such pairs of alternating exterior angles when a transversal cuts two parallel lines. Let usa larn more almost them.
| ane. | What are Alternate Exterior Angles? |
| 2. | Alternate Exterior Angles Theorem |
| 3. | Converse of Alternate Exterior Angles Theorem |
| four. | FAQs on Alternate Exterior Angles |
What are Alternate Exterior Angles?
Alternating outside angles are created when two or more lines are intersected past a transversal. These angles are formed on the outer side of the transversal on the different sides.
When whatever two parallel lines are intersected past a transversal, they create some pairs of angles with the transversal. Interior angles are created in the space inside the parallel lines whereas alternate exterior angles are created in the space outside the parallel lines.
Alternate Exterior Angles Definition
Two angles that lie on the opposite sides of the transversal and are placed on two different lines are called alternating exterior angles. These pairs of angles are ever equal if the two given lines are parallel. In the given figure below, Line AB || Line CD and they are intersected past the transversal MN. Here, the alternating exterior bending pairs are ∠one and ∠seven; ∠2 and ∠viii. This means ∠ane = ∠7; ∠2 = ∠8
We can run into that ∠two is on the right side of the transversal M and eight is on the left side; ∠2 is above the line AB whereas ∠8 is below the line CD. The aforementioned rule applies for the other pair of angles (∠1 and ∠7). Therefore, the pair of angles that satisfy these conditions are called alternating exterior angles.
According to the figure, we tin define alternate exterior angles equally:
Two exterior angles that prevarication on two different lines cut by a transversal and are placed on the opposite sides of the transversal are chosen alternate exterior angles.
In the figure, we can observe that ∠1 and ∠seven are alternate exterior angles and ∠2 and ∠8 are alternate outside angles considering:
- Both lie on the exterior side of the lines; and
- They are placed on the opposite sides of the transversal.
Alternate Exterior Angles Theorem
The alternating exterior angle theorem states that if 2 lines are parallel and are intersected by a transversal, and so the alternate outside angles are considered as coinciding angles or angles of equal mensurate.
Following the same figure given above, nosotros tin can observe that ∠1 and ∠7; ∠2 and ∠8 are pairs of alternating exterior angles. Let the states see how they are congruent (equal).
Given: Lines AB and CD are parallel lines on a transversal M.
[Hint: To prove the above theorem, nosotros volition be using the following axioms:
Corresponding Angle Axiom: When two lines are parallel the corresponding angles are congruent angles.
The converse of Corresponding Bending Axiom: When the corresponding angles made by ii lines are congruent, then those two lines are parallel.]
To Prove: ∠one = ∠7
Proof:
To evidence this result, we will consider the vertically reverse angle of ∠ane. Here, the vertically opposite angle of ∠1 is ∠3.
So, ∠one = ∠3 (vertically opposite angles)
Here, line AB || CD,
∠3 = ∠7 (corresponding angles axiom)
∴∠i = ∠7 (transitivity) Hence, proved.
Converse of Alternating Exterior Angles Theorem
The converse of alternate exterior bending theorem states that, if the alternate outside angles formed past two lines, which are cut by a transversal, are congruent, and then the lines are parallel.
Hence, in the above effigy, if it is given that ∠i = ∠7 so line AB || line CD.
Note: Apart from alternate outside angles, there is one more blazon of outside angles pair. This is called the pair of consecutive exterior angles. In that location are two pairs of consecutive outside angles in the higher up effigy.
∠2 and ∠vii
∠ane and ∠8
Consecutive Outside Angles are supplementary, i.eastward., ∠2 + ∠7 = 180° and ∠1 + ∠eight = 180°
Topics Related to Alternate Outside Angles
- Angle between a Line and a Plane
- Multiple Angle Formulas
- Sum of Exterior Angles Formula
- Outside Angle Theorem
- Exterior Angle Formula
Alternating Exterior Angles Examples
-
Example i: Using the alternate outside angle theorem solve the given problem: Given: Line RS || Line PQ.
a) Find 10.
b) Also check if line XY || line RS.
Solution:
i) Given: Line RS || Line PQ; ∠XBA = 30° and ∠QCD = 30°
Here, ∠QCD = ∠EDS (corresponding angles)
∴ 10 = 30°⋯(1)
2) At present let u.s.a. check if lines XY and RS are parallel or not.
It is given that Line RS || Line PQ
AE equally a transversal bisecting three lines XY, PQ, and RS
The pair of alternate outside angles have the aforementioned measure i.due east., ∠XBA = ∠EDS = 30°
∠XBA = ∠EDS are alternate exterior angles.
If ∠XBA and ∠EDS are alternate exterior angles so, by the converse of alternate exterior bending theorem, we become to know that line XY is parallel to line RS
∴ a) ten = xxx° , b) line XY || line RS
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Practice Questions on Alternate Outside Angles
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FAQs on Alternate Exterior Angles
What are Alternate Exterior Angles in Math?
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. In other words, the two exterior angles that prevarication on two dissimilar lines cut by a transversal and are placed on the contrary sides of the transversal are chosen alternate exterior angles.
What is the Alternate Exterior Angles Theorem?
The alternate outside angle theorem states that when two parallel lines are intersected by a transversal, then the outside angles formed on either side of the transversal are equal.
Do Alternate Exterior Angles Prove that the Lines are Parallel?
Alternate exterior angles prove that the lines are parallel only if the alternate exterior angles are congruent. This issue is known equally the converse of the alternate exterior angle theorem.
How are Alternate Exterior Angles Related?
Co-ordinate to the alternate exterior angle theorem, when two lines are parallel and are intersected by a transversal, then the alternate exterior angles are considered as coinciding angles. In other words, alternate exterior angles that are formed by two parallel lines and a transversal are ever equal in mensurate.
Practise Alternating Outside Angles Add upwardly to 180?
No, alternate exterior angles do non add upwards to 180°. In fact, they are coinciding to each other. But in the case when they are 90° each, and so they will sum upwardly to ninety° + 90° = 180°.
Are Alternating Exterior Angles Congruent?
Alternating exterior angles are congruent when the lines are parallel. If the lines are non parallel, the alternate outside angles are not congruent.
What are Alternating Interior and Alternate Exterior Angles?
When any ii parallel lines are intersected past a transversal, they create some pairs of angles with the transversal. Alternating Interior angles are ever equal and are formed on the inner side of the parallel lines, simply are located on the reverse sides of the transversal. Alternating exterior angles are those angles that take different vertices, they lie on the alternate sides of the transversal and are exterior to the lines.
When are Alternate Exterior Angles Coinciding?
Alternate Outside angles are congruent only when the two lines that are intersected by a transversal are parallel.
Are Alternate Exterior Angles Supplementary,
Source: https://www.cuemath.com/geometry/alternate-exterior-angles/
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