How to Solve Lines, Angles and Triangles on the SAT

Welcome to SAT Climb. Lines, angles, and triangles feel like a hundred separate rules. They're really just a handful, and the digital SAT reuses the same few over and over. Learn the family, and these become some of the fastest points on the test.

Here's your whole toolkit. A straight line is one hundred eighty degrees. Vertical angles are equal. Parallel lines cut by a transversal make two families of angles, some equal, some supplementary. And every triangle sums to one hundred eighty. That's it. Let's use it. Two parallel lines are cut by a transversal. One angle is fifty degrees. Find x.

The whole move is deciding which family x belongs to. The fifty and the x are both interior angles, and both on the same side of the transversal. Same-side interior angles are supplementary. They add to one hundred eighty. So x plus fifty equals one hundred eighty, and x is one hundred thirty. Here's the trap. It is so tempting to say x equals fifty because the angles look like they match. But matching only works for the equal family, corresponding and alternate. Same side means supplementary. One hundred thirty.

Now triangles. The three angles always add to one hundred eighty. If two of them are fifty and sixty, the third is seventy. But here's the shortcut the SAT loves. The exterior angle at a vertex equals the two remote interior angles added together. So the exterior angle here is fifty plus sixty, one hundred ten, and you never even had to find the seventy. Angle sum, plus the exterior-angle shortcut, solves most triangle questions on sight.

Your turn, and this one is on your reference sheet. A thirty, sixty, ninety triangle has a hypotenuse of ten. The ratio is one, root three, two. The hypotenuse is the two, so the scale is five. The short leg, across from thirty, is five. The long leg, across from sixty, is five root three.

Three traps. One: treating same-side interior angles as equal, when they're supplementary. Two: losing track of the one hundred eighty sum, or calling an exterior angle by its interior. Three: swapping the special-right legs. The root three always rides the longer leg, the one across from sixty. Sort the angle into its family, keep the sum, and place the root three correctly. That's the whole topic.

Lines, angles, and triangles: solved. Sort each angle into its family, keep every triangle at one hundred eighty, and place your special-right sides correctly. Start practicing free at S.A.T. climb dot com. Your SAT is closer than you think.