Triangles Summary

Geometry · Axiom Academy

SUMMARY Unit 3 Summary: Triangles Everything that makes a triangle tick — how to name them, the angle and side rules they always obey, and when two of them are truly the same. Every triangle has a name by sides (scalene, isosceles, equilateral) and a name by angles (acute, right, obtuse) — the two classifications are independent. The three interior angles always sum to . This single fact unlocks most "find the missing angle" problems. The Triangle Inequality decides whether three lengths can even form a triangle: the two shorter sides must out-reach the longest. In a right triangle, the Pythagorean Theorem a^2+b^2=c^2 ties the legs to the hypotenuse — and the and triangles give exact side ratios with no calculator. Congruence (SSS, SAS, ASA, AAS, HL) proves two triangles are identical ; similarity (AA, SSS~, SAS~) proves they are the same shape, scaled . Core Concept Classifying Triangles Name a triangle two ways at once . By sides: scalene (no equal sides), isosceles (two equal), equilateral (all three). By angles: acute (all ), right (one ), obtuse (one ). Key idea: equilateral equiangular (all angles ). Watch out for: isosceles has a "two equal sides" rule, so equilateral counts as isosceles too. Core Concept Triangle Angle Sum The interior angles of every triangle add to a straight angle. Given any two angles, the third is forced — subtract from . When to use: any "find the missing angle" problem. Watch out for: it's , not — that's quadrilaterals.

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