Same-Side Angles

When two lines "r" and "s" are intersected by a transversal "t", two angles are called same-side angles (or consecutive) if they do not share a common vertex, are both either interior or exterior, and are located on the same side of the transversal.

Same-side angles can be classified as:

• Same-side interior angles (or consecutive interior angles), if they are between the two lines.
• Same-side exterior angles (or consecutive exterior angles), if they are outside the two lines.

Here’s a practical example:

Consider two lines "r" and "s" intersected by a transversal line "t".

The intersection of these lines creates eight angles.

The pairs of angles (β, α') and (γ, δ') are known as same-side interior angles.

They are called "interior" because they lie between the two lines "r" and "s".

For instance, the red angles (β, α') do not share a common vertex, are both interior, and are on the same side of the transversal. The same holds true for the blue angles (γ, δ').

The pairs of angles (α, β') and (δ, γ') are known as same-side exterior angles.

They are called "exterior" because they lie outside the two lines "r" and "s".

For example, the red exterior angles (α, β') do not share a common vertex, are both exterior, and are on the same side of the transversal. The same is true for the blue angles (δ, γ').

The Case of Parallel Lines

When lines "r" and "s" are parallel, same-side angles are also supplementary angles because their sum is equal to a straight angle (180°).

For instance, the same-side interior angles (β, α') are also supplementary angles.

This is because angle α' is congruent with angle α, meaning they have the same measure, and angles α and β form a straight angle (180°).

The same reasoning applies to the other pairs of same-side angles (α, β'), (δ, γ'), and (γ, δ').