ACT® · Math: Geometry — Transformations, Proof, and Constructions
Drawing Conclusions From a Set of Conditions
When a problem gives you a set of conditions, your job is to find every figure or value that satisfies all of them, not just the first one you picture. The answer is often a whole category, not a single shape.
A list of conditions acts like a filter. Each condition rules out some figures and keeps others. If you pile on enough conditions, eventually only one shape survives — but with just a few, a whole family can pass through. The habit that keeps you honest is this: instead of picturing one figure that fits and stopping, try to sketch a second, different figure that also fits. If you can, the conditions describe a category, not a single shape.
This matters most with quadrilaterals and triangles. For quadrilaterals, each property (equal sides, bisecting diagonals, right angles) locks in a narrower class: parallelograms, rectangles, rhombi, squares. You add properties until only one class is left standing. For triangles, the controlling fact is the triangle inequality: any side must be shorter than the sum of the other two and longer than their difference. That single rule turns 'find the third side' into 'find the range of third sides,' which is really the same skill — describing everything that fits rather than guessing one number.
This topic sits right after you've learned to classify quadrilaterals by their properties and right before formal proofs, where you chain these same conditions into a logical argument for why a figure must belong to a certain class. Getting comfortable naming the whole set of valid figures now makes two-column or paragraph proofs feel like just writing down the filtering steps you already do by eye.
Key terms
- Triangle inequality
- The rule that any side of a triangle must be less than the sum and more than the difference of the other two sides.
- Parallelogram
- A quadrilateral whose diagonals bisect each other, equivalent to having both pairs of opposite sides parallel.
- Rectangle
- A parallelogram whose diagonals are equal in length, giving it four right angles.
- Sufficient condition
- A fact that, once true, guarantees a certain classification, even if other facts about the figure remain unknown.
Watch it explained
- How it worksHow does sketching every figure that fits a set of conditions expose what they all share?
- Worked exampleFind every possible shape of a quadrilateral whose diagonals bisect each other and are equal in length.
- Worked exampleFind all possible lengths of the third side of a triangle whose other two sides measure 7 and 12.
- Common mistakeWhy doesn't a set of conditions that a square satisfies mean the figure has to be a square?
Check yourself
End of topic.