Key takeaway: Same-side interior angles are two angles that lie between two lines on the same side of a transversal, and when the two lines are parallel they are supplementary.
Grade 10 – GeometryG-CO.C.9Geometry and Measurement
Definition of Same-Side Interior Angles
Same-side interior angles appear when a transversal cuts two lines and makes eight angles. These two sit in the strip between the lines and stay on one side of the cutting line. Some books call them consecutive interior angles.
With parallel lines, their measures total \(180^{\circ}\): one is acute and one obtuse, or both are right angles. The reverse works as a test, too. A pair totaling \(180^{\circ}\) proves the lines are parallel.
Formula and notation
For lines \(a \parallel b\) with numbered angles \(\angle 3\) and \(\angle 5\) in such a pair:
\[ m\angle 3 + m\angle 5 = 180^{\circ} \]
The pair is supplementary, not congruent, unless both angles measure \(90^{\circ}\).
Key facts
- There are two such pairs in a figure with one transversal and two lines.
- Both angles must be inside the strip and on the same side of the transversal.
- A sum of 180 degrees proves the two lines are parallel.
- Alternate interior angles are equal, but this pair adds up instead.
Same-Side Interior Angles example
Two parallel rails \(a\) and \(b\) are crossed by a fence board. The angles between the rails on the right of the board measure \((4x + 17)^{\circ}\) and \((5x + 10)^{\circ}\). Find \(x\) and both sizes.
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Because the lines are parallel, the pair is supplementary: \((4x + 17) + (5x + 10) = 180\).
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Combine: \(9x + 27 = 180\), so \(9x = 153\) and \(x = 17\).
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Angles: \(4(17) + 17 = 85\) and \(5(17) + 10 = 95\).
\(x = 17\), and the angles measure \(85^{\circ}\) and \(95^{\circ}\), which add to \(180^{\circ}\).
Common mistake
Watch out
Isla sets the two expressions equal, \(4x + 17 = 5x + 10\), and gets \(x = 7\), giving \(45^{\circ}\) each.
Why it is wrong
Equal angles belong to alternate interior or corresponding pairs. For same-side interior angles the total, not each value, is fixed at \(180^{\circ}\). Her two \(45^{\circ}\) angles total only \(90^{\circ}\), so the setup is wrong.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement