Same Side Interior Angles: Theorem & Examples

Same Side Interior Angles: Theorem & Examples

What Are Same Side Interior Angles?

Same side interior angles are a pair of angles that satisfy two conditions at once: they lie in the interior region (between the two lines that a transversal crosses), and they sit on the same side of the transversal. A transversal is simply a line that cuts across two or more other lines.

When a transversal crosses two lines, it creates eight angles, four at each crossing. The four "interior" ones are the angles trapped between the two lines. Of those four, the two that share a side of the transversal, one from the upper crossing and one from the lower, are the same side interior angles. The naming is literal: same side of the transversal, in the interior region.

Same side interior angles are also known as co-interior angles and as consecutive interior angles. They appear in NCERT Class 7, Chapter 5 (Lines and Angles) and across CCSS-M 8.G.A.5, where parallel-line angle relationships are first formalized.

The Same Side Interior Angles Theorem

The relationship that makes these angles useful only holds when the two cut lines are parallel. The same side interior angles theorem states it precisely:

If a transversal intersects two parallel lines, then each pair of same side interior angles is supplementary, meaning their measures add to 180°.

Are same side interior angles supplementary? Only when the two lines are parallel. If the lines are not parallel, the two angles still sit inside, on the same side, but their sum can be anything, less than or greater than 180°. The 180° guarantee is what parallelism buys you.

Why they are supplementary, a one-line reason

The supplementary result is not an extra rule to memorize; it falls straight out of two facts:

Thus, ∠3 + ∠5 = 180°.

The Converse: Using Them to Prove Lines Parallel

If a transversal intersects two lines such that a pair of same side interior angles is supplementary, then the two lines are parallel.

So when you measure or compute two same side interior angles and they add to exactly 180°, you have proved the lines are parallel without checking anything else. If they add to something other than 180°, the lines are not parallel.

Examples of Same Side Interior Angles

Example 1 - Two parallel lines are cut by a transversal. One same side interior angle is 115°. Find the other

By the theorem, same side interior angles on parallel lines are supplementary:

180°−115°=65°. The other angle is 65°.

Example 2 - A transversal cuts two parallel lines. Two same side interior angles measure 70° and 70°. Is the reasoning correct?

Same side interior angles are supplementary when the lines are parallel, not equal. Two equal 70° angles sum to 140°, which is not 180°.

Example 3 - Lines m and n are parallel. A same side interior angle measures (3x)° and its partner measures (x + 40)°. Find x.

The pair is supplementary: 3x + (x + 40) = 180; 4x + 40 = 180; 4x = 140; x = 35.

Example 4 - A transversal cut forms same side interior angles of 105° and 80°. Are the lines parallel?

105° + 80° = 185° ≠ 180°. The sum is not 180°, so the lines are not parallel.

Example 5 - Two parallel lines have same side interior angles of (2y + 10)° and (4y - 30)°. Find both angles.

(2y + 10) + (4y - 30) = 180; 6y - 20 = 180; 6y = 200; y = 100/3.

Example 6 - Lines p and q are parallel, and one same side interior angle is twice the other. Find both angles.

Let the smaller angle be a and the larger be 2a. They are supplementary:

a + 2a = 180°; 3a = 180°; a = 60°.

The angles are 60° and 120°.

Key Takeaways