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Alternate Exterior Angles Theorem

Alternate Exterior Angles Theorem states that "Alternate Exterior Angles formed by two parallel lines and a transversal are always congruent".

alternate exterior angles theorem

In the figure given above, angles 2 and 8 are alternate exterior angles. Angles 1 and 7 are also alternate exterior angles. Therefore, $\angle$2 = $\angle$8 and $\angle$1 = $\angle$7.

One way to easily find the alternate exterior angles is that they are the vertical angles of the alternate interior angles. Alternate exterior angles are equal to one another.

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Alternate Exterior Angles Theorem Proof

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If two parallel line segments or rays are cut by a transversal, the alternate exterior angles are congruent.

Given:

Line p is parallel to line q with the transversal l, as shown in the figure given below.

alternate exterior angles theorem

Proof:

S.No.StatementReasons
1p | | q with the transversal l
Given
2$\angle$2 is congruent to $\angle$6
Parallel Lines Postulate
3$\angle$6 is congruent to $\angle$8Vertical Angle Theorem
4$\angle$2 is congruent to $\angle$8
Using Transitive Property

Therefore, the alternate exterior angles are congruent.

Hence Proved.

Converse of the Alternate Exterior Angles Theorem

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If any two lines are cut by a transversal and alternate exterior angles are congruent, then both the lines are parallel.
alternate exterior angle

Proof:

S.No. Statement Reasons
Step 1: $\angle$2 is congruent to $\angle$8
Given
Step 2: $\angle$6 is congruent to $\angle$8
Vertical Angle Theorem
Step 3: $\angle$2 is congruent to $\angle$6 Corresponding Angles
Step 4:
p | | q with the transversal l
From Step 2 and Step 3
Parallel Line Postulate

Therefore, the lines are parallel.

Hence Proved.

Alternate Exterior Angles Theorem Examples

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Given below are some of the examples on alternate exterior angles theorem.

Solved Example

Question: Find the values of an angles b, c, d, e, f, g and h in the figure mentioned below.

alternate exterior angles theorem example
Solution:
Step 1: b is a supplement of 65$^o$.

Therefore, b + 65$^o$ = 180$^o$ => b = 180$^o$ - 65$^o$ = 115$^o$

Step 2: b and c are vertical angles.

Therefore, c = b = 115$^o$

Step 3: d and 65$^o$ are vertical angles.

Therefore, d = 65$^o$

Step 4: d and e are alternate interior angles.

Therefore, e = d = 65$^o$

Step 5: f and e are supplementary angles.

Therefore, f + 65$^o$ = 180$^o$ => f = 180$^o$ - 65$^o$ = 115$^o$

Step 6: g and f are vertical angles.

Therefore, g = f = 115$^o$

Step 7: h and e are vertical angles.

Therefore, h = e = 65$^o$


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