Top

# Alternate Exterior Angles Theorem

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

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

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.

Proof:

 S.No. Statement Reasons 1 p | | q with the transversal l Given 2 $\angle$2 is congruent to $\angle$6 Parallel Lines Postulate 3 $\angle$6 is congruent to $\angle$8 Vertical 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

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

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 3Parallel Line Postulate

Therefore, the lines are parallel.

Hence Proved.

## Alternate Exterior Angles Theorem Examples

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.

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$^o$

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