Proven: The Congruence Of Alternate Exterior Angles Theorem.

alternate exterior angles are congruent

if two parallel lines are cut by a transversal, then alternate interior angles are congruent

Alternate exterior angles are defined as a pair of angles that are on opposite sides of a transversal and outside the two parallel lines. Congruent means that two things are exactly the same in shape and size.

In the context of geometry, alternate exterior angles are always congruent when the two parallel lines are intersected by a transversal. This is a theorem known as the Alternate Exterior Angles Theorem.

To prove this theorem, we can draw two parallel lines and a transversal that intersects them. Label the angles formed by the transversal and parallel lines as shown in the diagram below:

1 2
________
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________
3 4

From this diagram, we can see that angles 1 and 3 are on opposite sides of the transversal and outside the parallel lines. Similarly, angles 2 and 4 are also on opposite sides of the transversal and outside the parallel lines.

Now, since the two parallel lines are cut by a transversal, we know that:

∠1 + ∠2 = 180° (Interior Angle Sum Theorem)
∠3 + ∠4 = 180° (Interior Angle Sum Theorem)

Rearranging these equations, we get:

∠1 = 180° – ∠2
∠3 = 180° – ∠4

Substituting these expressions in the equation ∠1 = ∠3, we get:

180° – ∠2 = 180° – ∠4

Simplifying this equation, we get:

∠2 = ∠4

Therefore, we have proved that alternate exterior angles are congruent.

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