ACT® · Math: Geometry — Geometric Measurement and Modeling
Vertical Angles and Linear Pairs
When two lines cross, they make two pairs of equal "vertical" angles, and any two angles next to each other form a "linear pair" that adds to 180 degrees. Those two facts together let you find every angle at the intersection from just one.
Two straight lines crossing at a point create four angles. Angles directly across from each other are vertical angles, and they are always equal, no matter how sharp or wide the crossing is. Angles sitting next to each other, sharing a ray, form a linear pair, and a linear pair always adds to 180 degrees because together the two angles sweep out a straight line.
The equality of vertical angles isn't a separate rule to memorize, it falls right out of the linear pair rule. Say one angle is A. Its neighbor on one side, call it B, must satisfy A + B = 180. The neighbor on the other side, call it C, also forms a straight line with A, so A + C = 180 too. Since B and C both equal 180 - A, B and C are equal. And B and C are the two angles adjacent to A, which makes the angle vertical to A equal to A itself, by the same chain of logic run from the other side.
This pattern shows up constantly once lines start crossing inside triangles, parallel-line setups, and polygon diagrams. Spot one angle, and the other three are already determined: the vertical angle matches it exactly, and the two adjacent angles each fill in the rest of 180 degrees. That's often the fastest way into a problem that looks like it needs a diagram full of unknowns but actually only has one.
Key terms
- Vertical angles
- The two angles directly across from each other at a crossing of two lines, always equal in measure.
- Linear pair
- Two adjacent angles that together form a straight line, so their measures always add to 180 degrees.
- Adjacent angles
- Two angles that share a vertex and one common ray, sitting right next to each other.
- Intersection
- The single point where two lines cross, creating four angles around it.
Watch it explained
- How it worksHow does a linear pair summing to 180 force the two angles across the intersection to match?
- Worked exampleFind all four angles at an intersection where one of them measures 47 degrees.
- Worked exampleFind x when two vertical angles are labeled 3x + 10 and 5x - 24.
- Common mistakeWhy aren't two angles that merely look equal in a rough sketch actually vertical angles?
Check yourself
End of topic.