AA Similarity (Angle-Angle Similarity, AA~)

AA Similarity (Angle-Angle Similarity, AA~)

How to solve AA similarity problems: postulate, example, and its solution.

Definition of Similarity

Similar triangles have the same shape, but have different size. This means their interior angles are congruent. And their sides are proportional. '~' means 'is congruent to'.

Similar triangles have the same shape,
but have different size.

This means their interior angles are congruent.
But their sides are proportional.
(= The ratios of their corresponding sides are equal.)

'~' means 'is similar to'.

Postulate

If two angles of a triangle are congruent to two angles of another triangle, then those two triangles are similar.

If two angles of a triangle are congruent to
two angles of another triangle,
then those two triangles are similar.

Example

Given: line segment AB is parallel to line segment CD. Prove: triangle PAB is similar to triangle PDC.

Start from the given statement.

Two column proof

Blue angles are alternate interior angles in parallel lines.

So the blue angles are congruent.

By the same way,
Red angles are alternate interior angles in parallel lines.

So the red angles are congruent.

By the AA similarity postulate,
PAB is similar to △PDC.

Example: Another Solution

Given: line segment AB is parallel to line segment CD. Prove: triangle PAB is similar to triangle PDC.

There's another way to solve this example.

Start from the given statement.

Two column proof

Blue angles are alternate interior angles in parallel lines.

So the blue angles are congruent.

Red angles are vertical angles.

So the red angles are congruent.

By the AA similarity postulate,
PAB is similar to △PDC.