Quadrilaterals and Polygons: Chapter Review

Vocabulary and notation

Polygon and boundary order

A polygon is a simple closed chain of at least three straight sides. Sides meet only at consecutive endpoints, and consecutive sides are not collinear. Letters follow the boundary; the final vertex is adjacent to the first.

Diagonal

A segment joining two nonadjacent vertices. A side joins adjacent vertices; adding a point on a straight side creates no new polygon vertex.

Convex and concave

In a convex polygon, every segment between points of its region stays inside or on it; every interior angle is strictly less than a straight angle. A simple concave polygon has a reflex interior angle. A crossed chain is not a polygon here.

Equilateral, equiangular, regular

Equilateral means all sides congruent; equiangular means all interior angles congruent. Regular requires both. Regular polygons considered here are convex.

Parallelogram

A quadrilateral with both opposite-side pairs parallel. A rectangle is a parallelogram with four right angles; a rhombus has four congruent sides; a square has both features.

Trapezoid, bases, legs

A trapezoid has at least one opposite parallel pair. Choose such a pair as bases; the remaining two sides are legs. Parallelograms remain included.

Isosceles trapezoid

Relative to chosen parallel bases, the two angles at one base are congruent. Rectangles qualify; equal legs alone are not the definition.

Kite

A convex quadrilateral with two disjoint adjacent congruent-side pairs, each side used once. The pair lengths may agree, so rhombi and squares qualify.

Bisect and midsegment

To bisect a segment is to divide it into equal halves. A trapezoid midsegment joins its leg midpoints, rather than its base midpoints.

Necessary and sufficient

A necessary condition must hold for every member of a family. A sufficient condition guarantees membership. Reversing a proved implication needs its own proof.

Tessellation

A covering of the plane with no gaps or overlapping interiors. A regular tessellation uses congruent copies of one convex regular polygon, meeting whole side to whole side.

Symmetry counts

A symmetry matches the whole figure. Count reflection axes separately from rotations; count identity once, with rotation angles from zero degrees up to but excluding a full turn.

Formulas and theorems

• Interior angle sum

  $$S=(n-2)180^\circ$$

  Text description

  Convex hexagon ABCDEF is named around its boundary. Diagonals from A reach C, D and E. They remain inside and meet only at A, forming four triangles without a new interior vertex. This is the six-sided instance of the general two-fewer-triangles count.

  Use when A convex polygon has an integer side count $n\ge3$. A fan from one vertex has $n-2$ triangles and $n-3$ diagonals; regularity is unnecessary.

  e.g. A convex 17-gon is given; find its interior sum: $15(180^\circ)=2700^\circ$.

• One exterior turn at every vertex

  $$\begin{gathered} i+e=180^\circ\\ E=360^\circ \end{gathered}$$

  Use when Traverse a convex polygon in one direction, selecting exactly one exterior turn per vertex. Here $i,e$ are neighboring interior and exterior measures, and $E$ is the total exterior turn.

• Regular polygon angles and side count

  $$\begin{gathered} e=\frac{360^\circ}{n}\\ i=180^\circ-e\\ n=\frac{360^\circ}{e} \end{gathered}$$

  Use when The polygon is convex and regular. The computed $n$ must be an integer at least three. In particular, $0<e\le120^\circ$; do not round a noninteger side count.

  e.g. A convex regular 36-gon is given; find one exterior turn: $e=10^\circ$.

• Regular polygon symmetries

  $$\begin{gathered} \text{reflection axes}=n\\ \text{rotations, with identity}=n\\ \text{least positive turn}=\frac{360^\circ}{n} \end{gathered}$$

  Use when The unmarked polygon is convex and regular, $n\ge3$. Matching turns are multiples of the least step. Odd $n$: each axis joins a vertex to an opposite side midpoint. Even $n$: half the axes join opposite vertices, half join opposite side midpoints; a half-turn matches exactly when $n$ is even.

• Parallelogram sides and angles

  $$\begin{gathered} AB=CD\\ BC=AD\\ \angle A\cong\angle C\\ \angle B\cong\angle D\\ m\angle A+m\angle B=180^\circ \end{gathered}$$

  Use when $ABCD$ is a parallelogram in boundary order. Every consecutive angle pair is supplementary; the displayed pair is one instance.

• Parallelogram diagonals and half-turn

  $$\begin{gathered} AO=OC\\ BO=OD\\ A\leftrightarrow C\\ B\leftrightarrow D \end{gathered}$$

  Text description

  The diagonals of parallelogram ABCD meet at O. Single ticks match AO with OC, while double ticks match BO with OD. Each equality concerns the parts of one diagonal. The different patterns do not assert that either diagonal, or its halves, equals the other diagonal. No right angle is marked.

  Use when $O$ is the diagonal intersection of parallelogram $ABCD$. Each diagonal bisects the other, and the vertex exchanges describe the half-turn about $O$. Equal or perpendicular whole diagonals are not guaranteed.

• Four complete parallelogram tests

  Any one complete record suffices:

  • Both opposite-side pairs are congruent.
  • Both opposite-angle pairs are congruent.
  • The diagonals bisect each other.
  • One opposite-side pair is both parallel and congruent.

  Use when The quadrilateral is convex and named in boundary order. The last test puts both facts on the same side pair. One side pair, one angle pair, or one diagonal midpoint alone is insufficient.

• Rectangle properties and tests

  A rectangle has congruent diagonals. Within parallelograms, congruent diagonals or one right angle each suffice to prove rectangle.

  Use when Establish the parallelogram before using either test. Its diagonal-bisection property remains available; four equal half-diagonals do not alone establish a square.

• Rhombus properties and tests

  A rhombus has perpendicular diagonals, and each diagonal bisects its two endpoint vertex angles. Within parallelograms, perpendicular diagonals, a diagonal bisecting an endpoint angle, or two congruent adjacent sides each suffice to prove rhombus.

  Use when The converse tests require a known parallelogram. For side calculations, the perpendicular half-diagonals are legs of a right triangle; the boundary side is its hypotenuse.

  e.g. A rhombus has half-diagonals $11$ cm and $60$ cm; find a side: $\sqrt{11^2+60^2}=61$ cm.

• Square combines both families

  A square is both a rectangle and a rhombus. Within parallelograms, congruent and perpendicular diagonals together characterize squares.

  Use when For a general convex quadrilateral, first establish a parallelogram, for example by mutual diagonal bisection. In a square, each diagonal splits its endpoint right angles into $45^\circ$ parts.

• Isosceles trapezoid relationships

  $$\begin{gathered} \angle A\cong\angle B\\ \angle D\cong\angle C\\ AD=BC\\ AC=BD \end{gathered}$$

  Use when $ABCD$ is a convex isosceles trapezoid with chosen bases $AB\parallel CD$. Angles along each leg are supplementary. The common perpendicular bisector of the bases is a reflection axis; rectangles remain included.

• Restricted equal-legs test

  Congruent legs imply congruent base angles when exactly one opposite-side pair is parallel.

  Use when The trapezoid is convex. Without the exactly-one hypothesis, an oblique parallelogram has congruent legs but unequal angles at a chosen base.

• Trapezoid midsegment

  $$\begin{gathered} MN\parallel AB\parallel CD\\ MN=\frac{a+b}{2} \end{gathered}$$

  Text description

  Convex trapezoid ABCD has horizontal parallel bases AB of length a and CD of length b. M lies halfway along leg AD, shown by matching single ticks. N lies halfway along leg BC, shown by matching double ticks. The segment MN, labeled m, joins those leg midpoints and is parallel to both bases. The base midpoints are not its endpoints.

  Use when $ABCD$ is convex, with chosen parallel bases $AB=a$ and $CD=b$. Points $M,N$ are the midpoints of legs $AD,BC$. Positive lengths use common units. Parallelograms are included.

  e.g. A trapezoid has bases $19$ cm and $43$ cm; find its midsegment: $(19+43)/2=31$ cm.

• Kite diagonal and angle relationships

  $$\begin{gathered} AC\perp BD\\ BO=OD\\ \angle BAC\cong\angle DAC\\ \angle BCA\cong\angle DCA\\ \angle ABC\cong\angle ADC \end{gathered}$$

  Text description

  Convex kite ABCD has one matching tick on AB and AD and two matching ticks on BC and CD. The equal-side pairs meet at A and C, making AC the distinguished diagonal. AC meets BD at O at a marked right angle. Matching triple ticks show BO equals DO. The unequal lengths AO and OC have no matching ticks, illustrating that the reverse bisection is not guaranteed.

  Use when Convex kite $ABCD$ has congruent sides $AB$ and $AD$, congruent sides $BC$ and $CD$, and diagonal intersection $O$. The equal pairs meet at $A,C$, so $AC$ is the reflection axis and perpendicular bisector of $BD$. Reverse bisection is extra: $AO=OC$ makes the kite a rhombus.

• Quadrilateral symmetry cases

  • Non-parallelogram isosceles trapezoid: 1 axis, 1 rotation.
  • Unequal-legged trapezoid with exactly one parallel pair: 0 axes, 1 rotation.
  • Oblique parallelogram with unequal adjacent sides: 0 axes, 2 rotations.
  • Nonsquare rectangle or rhombus: 2 axes, 2 rotations.
  • Square: 4 axes, 4 rotations.
  • Non-rhombus kite: 1 axis, 1 rotation.

  Use when All counts include identity once among rotations. Rectangle axes join opposite side midpoints; rhombus axes are its diagonals. Square axes include both types. A non-parallelogram isosceles trapezoid uses the common base bisector; a non-rhombus kite uses its distinguished diagonal.

Problem types, step by step

Exam traps

• Trap Treating an inward corner as a crossing

  Fix A concave polygon is still simple. A crossed boundary fails the polygon definition; a straight-side extra point is not another vertex.

• Trap Equating equal sides with regularity

  Fix Check equal interior angles as well. A nonsquare rhombus or nonsquare rectangle satisfies only one of the two conditions.

• Trap Excluding special cases from a family

  Fix At least one parallel pair includes two. Squares remain rectangles, rhombi, trapezoids, and kites; rhombi remain kites.

• Trap Sharing an angle total without equality evidence

  Fix An interior sum does not give equal angles in a general convex polygon. Regularity or another stated equality is needed.

• Trap Counting both exterior choices or rounding a side count

  Fix Select one boundary turn per vertex. An exact fractional regular side count means impossible, not approximately possible.

• Trap Using a local corner fit as a whole tiling proof

  Fix A full-turn fit is necessary. Matching edges and a continuing placement must establish plane coverage too.

• Trap Reversing a property or using the desired conclusion

  Fix A direct theorem assumes a known family. Establish a proved converse or the definition before using that family’s properties.

• Trap Mixing bisection, congruence, and perpendicularity

  Fix Bisection compares halves of the same diagonal; congruence compares whole lengths; perpendicularity concerns the crossing angle. None substitutes for the others.

• Trap Combining parallelism and congruence from different pairs

  Fix The one-pair parallelogram test attaches both facts to the same opposite-side pair. Both-pair tests need both pairs.

• Trap Dropping hypotheses from a special-shape test

  Fix Equal or perpendicular diagonals alone do not force rectangle or rhombus. Equal legs require exactly one parallel pair for the restricted trapezoid converse.

• Trap Bisecting the wrong kite diagonal or angle

  Fix Name the equal-pair meeting vertices first. The joining diagonal bisects the other diagonal and its own endpoint angles; a second bisection forces the rhombus case.

• Trap Assigning one symmetry count to an inclusive family

  Fix State the special case. A half-turn does not automatically give a reflection; a nonsquare rectangle and a non-parallelogram isosceles trapezoid have different counts. Count identity once.