Triangles — Properties, Area, Special Triangles

GRE Quantitative · Axiom Academy

LESSON Triangles — Properties, Area, Special Triangles Angle sum, the triangle inequality, area, the Pythagorean theorem, and the two special right-triangle ratios every GRE test-taker should know cold. 1. Every Triangle's Angles Sum to 180° Triangles are classified by their sides (equilateral, isosceles, scalene) or their angles (acute, right, obtuse) — but no matter the shape, one fact never changes: the three interior angles always add up to exactly 180° . 2. Area and the Triangle Inequality The area of any triangle uses the base and the perpendicular height to that base: where b is any side (the base) and h is the perpendicular distance from that side to the opposite vertex. A triangle has base 10 and height 6. Not every three lengths can form a triangle. The triangle inequality requires the sum of any two sides to exceed the third: , which is not greater than 8 — these lengths cannot form a triangle. 5,\ 3+5>4,\ 4+5>3"> — all three checks pass, so this is a valid triangle. In a right triangle, the square of the hypotenuse always equals the sum of the squares of the other two sides: Pythagorean triples are whole-number side sets that satisfy this relationship exactly — recognizing them instantly saves calculation time on the GRE: Two right triangles show up so often that their side ratios are worth memorizing outright — no Pythagorean theorem required once you recognize them. Legs equal, length x . Hypotenuse = . Ratio: .

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