To get the best deal on Tutoring, call 1-855-666-7440 (Toll Free)

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.

Related Calculators
Bayes Theorem Calculator Binomial Theorem Calculator
Calculate Pythagorean Theorem De Moivre's Theorem Calculator

Alternate Exterior Angles Theorem Proof

Back to Top

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


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

alternate exterior angles theorem


1p | | q with the transversal l
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

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


S.No. Statement Reasons
Step 1: $\angle$2 is congruent to $\angle$8
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

Back to Top
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
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$^o$

Related Topics
Math Help Online Online Math Tutor
*AP and SAT are registered trademarks of the College Board.