Jacobs Geometry
Contents 7 sections

Lesson 2 — Polygons and Congruence

Big idea: A polygon is a closed flat figure made of straight sides. Congruent figures have the same size and shape, and the order of letters tells which parts match.

Words to know

Word Meaning
Polygon A connected set of at least three line segments in the same plane; each segment meets exactly two others, one at each endpoint.
Sides The line segments that form a polygon.
Vertices The endpoints where sides meet (singular: vertex).
n-gon A polygon with n sides.
Congruent figures Figures with the same size and shape; a tracing fits exactly.
Congruence correspondence The matching of vertices from one figure to another.
Congruent triangles Triangles with a vertex correspondence such that all corresponding sides and angles are equal.

Concepts to know

  • A polygon has the same number of sides as vertices.
  • A polygon cannot have a gap, a loose end, or sides that cross.
  • Names by sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), octagon (8).
  • Congruent figures can be turned or flipped; they still match exactly.
  • In a congruence statement, the order of the letters is the correspondence.
  • Corollary: If two triangles are each congruent to a third triangle, they are congruent to each other.

The order rule

ΔABC ≅ ΔEFD

means:

ABC EFD
A E
B F
C D

So:

∠A = ∠E      AB = EF
∠B = ∠F      BC = FD
∠C = ∠D      AC = ED

Homework helper

If the homework asks… Do this…
Is this figure a polygon? Check: closed? straight sides? each side meets exactly two others, one at each endpoint?
Name a polygon by its sides Count the sides: 3 triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 8 octagon.
Write a congruence statement Match corresponding vertices first, then write the letters in the same order.
Name equal parts from a congruence Use the letter order to pair angles and sides.
Decide whether two figures are congruent Check size and shape, not just area.

Worked example

Given ΔABC ≅ ΔEFD, name all six pairs of equal parts.

  • Angles: ∠A = ∠E, ∠B = ∠F, ∠C = ∠D.
  • Sides: AB = EF, BC = FD, AC = ED.

Watch out

  • Order matters. ΔABC ≅ ΔDEF means A↔︎D, B↔︎E, C↔︎F — a different correspondence.
  • Equal area does not mean congruent. A long thin rectangle and a square can have equal areas but different shapes.
  • A figure with a loose end is not a polygon.
  • Match sides with sides and angles with angles.

Before you turn it in