Triangle and Angle Shortcuts

• Apply the triangle angle sum (180°) and the exterior-angle theorem (exterior = sum of two remote interior angles)
• Recall the side ratios for a 30-60-90 triangle (1 : √3 : 2) and a 45-45-90 triangle (1 : 1 : √2)
• Recognize similar triangles by AA, SSS, or SAS similarity, set up proportion equations, and scale areas by the square of the side ratio
• Distinguish congruent triangles (SSS, SAS, ASA, AAS) from similar triangles
• Solve angle-chasing problems using vertical, supplementary, complementary, and parallel-line-with-transversal relationships

Every triangle's interior angles sum to 180°. The exterior angle at any vertex equals the sum of the two non-adjacent (remote) interior angles. This shortcut replaces two steps of angle-chasing with one. Isosceles triangles have two equal sides and two equal base angles (the angles opposite the equal sides), so a 40° vertex angle leaves (180 − 40)/2 = 70° for each base angle; equilateral triangles have three 60° angles. In any triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle, so ordering the angles orders the sides. The two special right triangles are the act's favorite setup. A 45-45-90 triangle has sides in ratio 1 : 1 : √2 (leg : leg : hypotenuse). A 30-60-90 triangle has sides in ratio 1 : √3 : 2 (opposite the 30°, 60°, and 90° angles respectively). If you see a √2 or √3 in the answer choices, you are almost certainly in a special right triangle. Similar triangles have the same shape but different sizes. Corresponding angles are equal and corresponding sides are proportional. The ACT tests this with 'shadow' problems (a 6-ft person casts a 4-ft shadow; how tall is a tree with a 20-ft shadow?) and with triangles inside triangles. Set up a ratio: small side / big side = small side / big side, matching corresponding sides. Areas do not scale like sides: they scale by the square of the side ratio. If two similar right triangles have sides in the ratio 4 : 5, their areas are in the ratio 16 : 25, and a dilation by a scale factor of 1/2 makes every side half as long but leaves only 1/4 of the area. Congruence is different: two triangles are congruent (identical) if SSS, SAS, ASA, or AAS matches. Note that ssa does not guarantee congruence. For angle-chasing: vertical angles are equal, supplementary angles sum to 180°, complementary to 90°. When parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles sum to 180°. Draw every diagram and label every angle you can deduce before reading the question choices.

• If you spot two triangles sharing an angle with parallel sides, they are similar. Set up a ratio immediately.
• When a right triangle has a 45° or 60° angle marked, stop and check if it's a special right triangle before using trig.
• The exterior-angle theorem is the fastest angle shortcut on the ACT. Use it whenever an exterior angle is drawn.
• SSS, SAS, ASA, AAS prove congruence. Ssa and aaa do not. Memorize this so you do not second-guess.
• In shadow/height problems, the triangles are always similar. Write corresponding sides as a fraction and cross-multiply.