ACT® · Math: Geometry — Geometric Measurement and Modeling
Finding an Angle Using Several Properties at Once
Some angle problems don't give you a direct path to the angle you want. You solve them by naming every angle you can reach, one property at a time, until the chain lands on the target.
These items hand you a figure with one or two labeled angles and ask for some angle x that has no direct rule connecting it to what's given. That's the point. The test is checking whether you can chain several facts together, not whether you know one formula. The labeled angle is rarely next to x. It's three or four relationships away, and the only way through is to work forward.
Run an angle chase: start at the angle you know, and name every angle you can find from it using one property at a time. Vertical angles are equal. Angles on a straight line sum to 180. A triangle's angles sum to 180. An exterior angle equals the sum of the two remote interior angles. Parallel lines cut by a transversal give equal corresponding or alternate angles. After each step you have a new known angle. Keep going outward until x shows up in your list. Don't hunt backward from x trying to guess which rule applies — you won't see the chain until you've built it.
This sits right after single-property angle problems, where one rule gets you from given to target in one step. Here the setup is the same vocabulary, just linked: the exterior-angle rule and the straight-line rule might both fire in the same figure, or a triangle's angles might feed into a second triangle sharing a side. The same chase technique carries forward into perimeter and area problems where you first need an angle to find a missing side length before you can measure anything.
Key terms
- Angle chase
- Working forward from a known angle, naming every angle reachable by a rule, until the target angle appears.
- Exterior angle
- The angle formed outside a triangle between one side and the extension of an adjacent side.
- Remote interior angles
- The two triangle angles not adjacent to a given exterior angle, whose sum equals that exterior angle.
- Vertical angles
- The pair of equal angles formed opposite each other when two lines cross.
- Transversal
- A line that crosses two other lines, creating pairs of equal angles when those lines are parallel.
Watch it explained
- How it worksHow do you run an angle chase forward, naming every angle you can before hunting the target?
- Worked exampleFind angle x when a triangle's exterior angle is 125 degrees and one remote interior angle is 48 degrees.
- Worked exampleFind the angle between two lines when one rises at 35 degrees and the other at 100 degrees from horizontal.
- Common mistakeWhy isn't the angle you want always reachable from whichever angle happens to be labeled nearest it?
Check yourself
End of topic.