Lesson Alternate interior angles and Alternate exterior angles

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Alternate interior angles

Alternate interior angles are two congruent (equal in measurement) interior angles
that lie on different parallel lines and on opposite sides of a transversal.

In the figure shown,
PQ is the transversal that cut the parallel lines AB and CD. Angles w and x
and angles y and z are alternate interior angles.

Angle

proof
for the proof we will modify the figure as follows

In the proof we will use the property of the corresponding angles that they
are congruent.

When two lines are crossed by another line (Transversal), the angles in matching
corners are called corresponding angles. in the figure, corresponding angles
are shown by same name i.e w, x, y and z. Hence by using this property we will
proof that alternate interior angles are equal.

At the intersection point of straight lines AB and PQ
Angle

..................(1) (AB is a straight line)

and

Angle

..................(2) (PQ is a straight line)

from equation (1) and (2)

Angle

Again applying the same concept at the intersection point of straight lines AB and PQ

Angle

..................(1) (AB is a straight line)

and

Angle

..................(2) (PQ is a straight line)

from equation (1) and (2)

Angle

Hence Proved that alternate interior angles are congruent.

Alternate exterior angles

Alternate exterior angles are two congruent exterior angles
that lie on different parallel lines and on opposite sides of a transversal.

In the figure shown,
PQ is the transversal that cut the parallel lines AB and CD. Angles w and x
and angles y and z are alternate exterior angles.

Angle

Proof is same as "Alternate interior angles"
For more information refer to
Interior Angles.

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