Polygons Summary

Geometry · Axiom Academy

SUMMARY Unit 4 Summary: Quadrilaterals & Polygons Everything from parallelograms to regular polygons — the families, their properties, and the angle and area formulas that tie them together. A parallelogram has both pairs of opposite sides parallel and congruent, opposite angles congruent, and diagonals that bisect each other. The rectangle, rhombus, and square are parallelograms with extra structure — and a square is exactly a rectangle and a rhombus at once. A trapezoid has exactly one pair of parallel sides; an isosceles trapezoid adds congruent legs, congruent base angles, and congruent diagonals. The interior angles of any n -gon sum to , while the exterior angles always sum to . A regular polygon has all sides and all angles equal, so each interior angle is and its area is . Core Concept Parallelogram Properties A quadrilateral with both pairs of opposite sides parallel. That single condition forces a whole package: opposite sides congruent , opposite angles congruent , consecutive angles supplementary, and diagonals that bisect each other . When to use: any time you can show one pair of sides is both parallel and congruent. Watch out for: diagonals bisect each other, but they are not congruent in a general parallelogram. Core Concept The Special Parallelograms A rectangle adds four right angles and congruent diagonals. A rhombus adds four congruent sides, with diagonals that are perpendicular and bisect the angles. A square inherits all of those properties at once.

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