Classifying Polygons

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Learning goals

Identify simple, convex, and concave polygons and name their sides, vertices, and diagonals
Distinguish equilateral, equiangular, and regular polygons using definitions and counterexamples
Classify quadrilaterals from stated evidence within an inclusive hierarchy
Justify reflection and rotation counts for convex regular polygons

Follow the boundary

Trace five straight segments, returning to the starting point without crossing an earlier segment. You have made a pentagon. The five meeting points are its vertices, and the five segments are its sides.

A polygon here is a simple closed chain of at least three straight sides in a plane. Simple means that sides meet only at consecutive endpoints, with no crossing or repeated vertex except the closing return. Consecutive sides are not collinear. A redundant point along a straight side is not counted as a new vertex. An open chain or a boundary with a curved piece is not a polygon under this definition.

Write vertices in boundary order, the order you meet them on one trip around the boundary. In polygon $ABCDE$ , the sides are $AB$ , $BC$ , $CD$ , $DE$ , and $EA$ . Adjacent sides share a vertex. A diagonal joins two nonadjacent vertices, such as $AC$ .

Consecutive letters name sides; the dashed segment joins nonadjacent vertices.

Number of sides	Name
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
9	Nonagon
10	Decagon
Any whole number $n$ at least 3	$n$ -gon

A convex polygon has every segment joining two points of the polygonal region inside or on that region. Every interior angle is strictly less than $180^\circ$ . A simple polygon is concave when at least one interior angle exceeds $180^\circ$ . This is an inward corner, and some joining segments leave the region. A diagonal need not lie inside a concave polygon.

The dashed diagonal leaves the region at the inward corner.

A crossed closed chain fails the simple-boundary condition.

Worked example 1 Use the named order

Problem. A simple hexagon has vertices $P,Q,R,S,T,U$ in boundary order. It has one interior angle greater than $180^\circ$ . Name the sides adjacent to $P$ , list every diagonal from $P$ , and classify the polygon as convex or concave.

Solution. The sides that meet at $P$ are $PQ$ and $PU$ . The last listed vertex is adjacent to the first because the chain closes.

The other three vertices are nonadjacent to $P$ , giving diagonals $PR$ , $PS$ , and $PT$ . Segment $PQ$ is a side, not a diagonal.

The reflex interior angle makes the polygon concave. It is still simple because an inward corner does not require a crossing.

The solution distinguishes sides at P from the three diagonals from P.

Check your understanding

A simple pentagon is named ABCDE in boundary order. Which segment is a diagonal?

AB

AE

CD

BD

Equal sides and equal angles are different conditions

Four equal rods can make a slanted parallelogram. Their lengths stay equal while the corners change. Equal sides alone do not make a polygon regular.

Matching ticks give equal side lengths, not equal angles.

An equilateral polygon has all sides congruent. An equiangular polygon has all interior angles congruent. A regular polygon has both properties. We study convex regular polygons here, with at least three sides.

A nonsquare rectangle gives the reverse warning: all its angles are right angles, but its side lengths are not all equal. It is equiangular without being equilateral. A square satisfies both conditions, so it is regular.

Worked example 2 Check both conditions separately

Problem. Classify each convex polygon as equilateral, equiangular, both, or neither. Decide whether it is regular.

Case A. An oblique parallelogram, meaning one with no right angle, has four congruent sides and two different interior angle measures.

Case B. A rectangle has adjacent side lengths 3 cm and 8 cm.

Solution. Case A is equilateral because all four sides are congruent. Its unequal angles make it nonequiangular and therefore nonregular.

Case B is equiangular because each corner is a right angle. Its adjacent sides have different lengths, so it is not equilateral and is not regular.

The four square marks supply equal angles; the lengths differ.

Check your understanding

A convex polygon has all sides congruent. What further evidence makes it regular?

A It is drawn upright B All interior angles are congruent C It has at least four vertices D One diagonal lies inside it

Quadrilateral names form a hierarchy

We use names that include their special cases. A square keeps the name rectangle because it still meets the rectangle definition.

The special quadrilaterals in the table are convex. Parallel sides below always mean opposite sides.

Name	Defining conditions
Trapezoid	A quadrilateral with at least one pair of opposite parallel sides
Parallelogram	A quadrilateral with both pairs of opposite sides parallel
Rectangle	A parallelogram with four right angles
Rhombus	A parallelogram with four congruent sides
Square	A parallelogram with four right angles and four congruent sides
Kite	A convex quadrilateral with two disjoint pairs of adjacent congruent sides; each side is used once

The familiar definition of a rectangle as a quadrilateral with four right angles agrees with this table: opposite sides are perpendicular to the same side, so they are parallel by the earlier perpendicular-line result. No diagonal property is part of any definition here.

For a kite $ABCD$ , one allowed pairing is $AB$ with $AD$ , and $BC$ with $CD$ . Another is $AB$ with $BC$ , and $CD$ with $DA$ . The pairs must use all four sides. All four sides may have the same length, so rhombi and squares are kites.

Follow arrows upward. A family can have more than one parent.

An arrow means every member of the lower family belongs to the upper family. A square is both a rectangle and a rhombus. Every parallelogram is a trapezoid because two parallel pairs include at least one. Rhombi belong to both the kite and parallelogram branches. The kite definition itself does not require parallel sides.

For an isosceles trapezoid, choose a pair of parallel sides as bases and require the two angles at one base to be congruent. This definition includes rectangles. Congruent legs alone are not the definition under our inclusive convention; their use as a test needs an extra hypothesis, proved in the later trapezoid lesson.

Worked example 3 Classify the turned drawing

Problem. Convex quadrilateral $ABCD$ is drawn below in boundary order. Both pairs of opposite sides are parallel, all four sides are congruent, and all four angles are right angles. List every applicable name from the table, and decide whether it is a regular polygon.

The written givens and marks supply the evidence, regardless of orientation.

Solution. Both parallel pairs give a parallelogram, which is also a trapezoid. The right angles give a rectangle, and the congruent sides give a rhombus. Having both conditions gives a square.

It is also a kite: pair $AB$ with $AD$ , and $BC$ with $CD$ . These pairs are adjacent and disjoint. It is regular because all sides and all interior angles are congruent. Turning the page changes none of these conditions.

Check your understanding

Under these definitions, which statement holds for every square?

A It is a rectangle, rhombus, trapezoid, and kite B It is a rectangle but cannot be a rhombus C It is a kite only if two side lengths differ D It is not a trapezoid because it has two parallel pairs

Classify the evidence, not the sketch

A statement or conventional mark supplies evidence. A side that looks parallel or an angle that looks right does not. In Definitions, Conjectures, and Counterexamples we distinguished a condition that is given from a condition that merely seems true. A counterexample disproves a universal claim by meeting its hypothesis and failing its conclusion.

Review definitions and counterexamples.

Worked example 4 Decide what the givens establish

Problem. A convex quadrilateral $KLMN$ is known to have both pairs of opposite sides parallel. No lengths or angle measures are supplied. Which table names are guaranteed? Give an example showing why a rectangle is not guaranteed.

Solution. The parallel pairs directly establish a parallelogram and therefore a trapezoid. It is also a quadrilateral. The data do not establish a rectangle, rhombus, square, or kite.

An oblique parallelogram with adjacent side lengths 4 cm and 7 cm and one angle $60^\circ$ meets the hypothesis but is not a rectangle. This is a counterexample to the proposed guarantee, even if a particular sketch happens to look rectangular.

The counterexample meets the parallel givens and fails the right-angle condition.

Check your understanding

A convex quadrilateral has AB congruent to AD and BC congruent to CD. No other evidence is supplied. Which definition can be verified directly?

A Square B Rectangle C Kite D Parallelogram

Why regular polygons have predictable symmetries

A symmetry must match the whole boundary, not just one corner. Identity is counted once among rotational symmetries, using angles from $0^\circ$ up to but excluding $360^\circ$ . A half-turn is a $180^\circ$ rotation. These are the conventions from Symmetry.

Start with a regular pentagon. Reflect across the line bisecting one vertex angle. The two adjacent sides exchange because their lengths and angles match. Moving outward along the boundary pairs the next equal sides and equal corners too, until the whole pentagon matches. The axis passes through that vertex and the midpoint of the opposite side.

For odd side counts, each mirror joins a vertex to an opposite side midpoint.

For a regular hexagon, a vertex fold reaches the opposite vertex. A second kind of fold exchanges the endpoints of one side and passes through that side’s midpoint and the opposite side’s midpoint.

For even side counts, the two kinds of mirror occur equally often.

Counting the symmetries of a convex regular polygon#

For the pentagon, the first side and next corner determine the following side: its length is fixed, and convexity chooses the direction of its turn. Repeating that fact determines every vertex. Thus the vertex and side folds described above match the entire boundary. The five vertex-to-opposite-midpoint lines give five mirrors. Compose the fold exchanging side endpoints $A,B$ with the vertex fold at $B$ . It sends $A$ to $B$ and each following vertex one place forward. As the general argument below explains, this is a rotation. Its five powers give five matching positions, including identity, and its least positive step is $72^\circ$ .

Nothing in this boundary-matching argument depended on choosing five sides. For any convex regular $n$ -gon, reflection in a vertex-angle bisector exchanges its two neighboring vertices, then the next pair, and continues around. Reflection in the perpendicular bisector of a side exchanges its endpoints, then continues around using the equal angles and side lengths. Convexity makes the continuation stay on the chosen boundary side at every step.

If $n$ is odd, each mirror crosses one vertex and the midpoint of the opposite side. There are $n$ choices of vertex and hence $n$ distinct mirrors. If $n$ is even, opposite vertices share a mirror, giving $n/2$ mirrors of that kind. Opposite side midpoints give another $n/2$ . These are all the mirrors: any mirror of a convex polygon crosses its boundary at two places, and at each crossing it must either fix a vertex and bisect its angle or exchange a side’s endpoints and bisect that side perpendicularly.

For rotations, compose the reflection that exchanges the endpoints $A,B$ of a side with the vertex reflection at $B$ . The composition sends $A$ to $B$ and each vertex to the next vertex in boundary order. Two distinct mirrors either intersect and compose to a rotation, or are parallel and compose to a nonzero translation, as proved in Compositions and Congruence. A nonzero translation cannot preserve a finite polygon: repeated translations would move a vertex arbitrarily far from the original bounded figure. So this composition is a rotation.

Its first $n$ powers send $A$ to the $n$ different vertices and give $n$ different matching rotations, including identity. There cannot be more: a matching rotation preserves boundary order, and the image of one directed side determines the whole rigid motion. There are just $n$ possible image sides. These rotations are powers of the same turn, so they share its center. The finite-order result from Symmetry then gives a least positive angle of $360^\circ/n$ .

The proof uses equal sides, equal angles, and rigid motions; it does not assume a circle or a center theorem. Its counts concern the unmarked polygon. Names and dashed guides in our diagrams are not decorations on the tested shape.

Worked example 5 Count matching motions

Problem. An unmarked convex regular decagon has 10 sides. Find its reflection-axis count, rotational-symmetry count including identity, and least positive matching angle. Decide whether a half-turn matches.

Solution. Its 10 reflection axes split into five through opposite vertices and five through opposite side midpoints. It has 10 rotational symmetries, including identity once.

The least positive matching angle is

360^\circ\div10=36^\circ.

A half-turn is five such steps:

5\times36^\circ=180^\circ.

Thus a half-turn matches. More generally, a convex regular polygon has a half-turn exactly when its number of sides is even: then half the number of equal rotation steps completes half a full turn.

Check your understanding

An unmarked convex regular nonagon has 9 sides. Which symmetry description is correct?

A 9 mirrors, 8 rotations including identity, least angle 40 degrees B 9 mirrors, 9 rotations including identity, least angle 40 degrees C 18 mirrors, 9 rotations including identity, least angle 20 degrees D 9 mirrors, 9 rotations including identity, including a half-turn

Classifying Polygons

Learning goals

Follow the boundary

Equal sides and equal angles are different conditions

Quadrilateral names form a hierarchy

Classify the evidence, not the sketch

Why regular polygons have predictable symmetries

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Core practice

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