Same-Side Interior Angles
In short: Same-side interior angles sit between two lines on one side of a transversal, and when the two lines are parallel their measures add up to 180 degrees.
GeometryGeometryG-CO.C.9
What is Same-Side Interior Angles?
Same-side interior angles appear when one line cuts across two others. The strip between the two lines is the interior, and the pair in question consists of the two corners inside that strip that face the same direction from the cutting line, one corner at each crossing.
For parallel lines the pair is supplementary: its measures total \(180^{\circ}\). Reading the rule backward works too. A total of exactly \(180^{\circ}\) proves the lines parallel, and any other total proves they are not.
How to spot it
- A diagram shows two lines and a cutting line, with two highlighted corners inside the strip, both on the right or both on the left.
- The problem calls the pair consecutive interior angles or co-interior angles instead.
- Two expressions such as \((3x + 20)^{\circ}\) and \((5x – 8)^{\circ}\) are given, with the lines marked parallel.
- You must decide whether two lines are parallel from two measurements taken inside the strip.
The method
- Locate both crossing points and pick the two interior corners on one side of the transversal.
- Confirm the lines are parallel from the figure or the givens.
- Write the sum equation: first measure plus second measure equals \(180\).
- Solve for the variable, then substitute back to get each measure and check that they total \(180\).
Same-Side Interior Angles: solved example
Problem
A diagonal brace crosses two parallel boards in a barn door. The pair of interior corners on one side measure \((3x + 20)^{\circ}\) and \((5x – 8)^{\circ}\). Find \(x\) and both measures.
Solution
Parallel boards make the pair supplementary: \((3x + 20) + (5x – 8) = 180\). Combine terms: \(8x + 12 = 180\), so \(8x = 168\) and \(x = 21\).
The corners are \(3(21) + 20 = 83\) and \(5(21) – 8 = 97\). Check: \(83 + 97 = 180\). The measures are \(83^{\circ}\) and \(97^{\circ}\).
Try it
Question 1
One of the pair measures \(112^{\circ}\) and the lines are parallel. What does the other measure?
Show the answer
\(180 – 112 = 68\), so the other is \(68^{\circ}\).
Question 2
Two interior angles on the same side measure \(104^{\circ}\) and \(78^{\circ}\). Are the two lines parallel?
Show the answer
The sum is \(182^{\circ}\), not \(180^{\circ}\), so the lines are not parallel.
Same-Side Interior Angles vs. Alternate Interior Angles
Alternate interior angles also live inside the strip, but they face opposite directions from the cutting line, and parallel lines make them equal. Same-side interior angles face one direction and total \(180^{\circ}\). Opposite sides give equal; one side gives supplementary.
Keep going
Lessons
Worksheets and quizzes
- Parallel Lines and Transversals – Worksheet
- Parallel Lines and Transversals – Quiz
- Angles in Triangles and Parallel Lines – Worksheet
- Angles in Triangles and Parallel Lines – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry