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Alternate Exterior Angles

When two lines segments are crossed by another line segments (which is called the Transversal), the pairs of angles on opposite sides of the transversal which are outside the two lines are called Alternate Exterior Angles.

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.

alternate exterior angles

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.

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Alternate Exterior Angles Definition

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The couple of angles which are in the opposite sides of the transversal and created outside the two parallel lines is determined as alternate exterior angles.

Alternate Exterior Angles Diagram

  • $\angle$1 and $\angle$2 are alternate interior angles
  • $\angle$a and $\angle$b are alternate interior angles

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 and cut 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.

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Alternate Exterior Angles Examples

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

Solved Example

Question: Find the values of the angles b, c, d, e, f, g and h in the figure given below.
alternate exterior angles example


Step 1: b is a supplement of 45$^o$.

Therefore, b + 45$^o$ =180$^o$ => b = 180$^o$ - 45$^o$ = 135$^o$

Step 2: b and c are vertical angles.

Therefore, c = b = 135$^o$

Step 3: d and 45$^o$ are vertical angles.

Therefore, d = 45$^o$

Step 4: d and e are alternate interior angles.

Therefore, e = d = 45$^o$

Step 5: f and e are supplementary angles.

Therefore, f + 45$^o$ = 180$^o$ => f = 180$^o$ - 45$^o$ = 135$^o$

Step 6: g and f are vertical angles.

Therefore, g = f = 135$^o$

Step 7: h and e are vertical angles.

Therefore, h = e = 45$^o$

More topics in Alternate Exterior Angles
Alternate Exterior Angles Theorem
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