Classify Polygons, Triangles and Quadrilaterals | Grade 8 Math | LevelKite

In this lesson you sort figures into named families. You start by counting sides, and then you read marks and stated values.

A polygon is a closed figure in a plane. It is made only of three or more straight segments, called its sides. Closed means the chain of sides ends exactly where it began. The sides meet only at their endpoints, exactly two at each corner. Each corner is a vertex (plural vertices). You refer to a polygon by listing its vertices in order around it, such as $ABCDEF$ .

A circle is not a polygon, because its edge is curved. A chain of segments that stops before returning to its start is not a polygon, because it is not closed. To classify a polygon, count its sides and use the matching name in the table.

Polygon names by number of sides

Number of sides	Name
$3$	triangle
$4$	quadrilateral
$5$	pentagon
$6$	hexagon
$7$	heptagon
$8$	octagon
$9$	nonagon
$10$	decagon
$n$ , any whole number of at least $3$	$n$ -gon, such as a $12$ -gon for $12$ sides

Straight sides join the six labelled corners in order,

A \to B \to C \to D \to E \to F

, and the last side returns to

A

.

Worked example

Decide whether figure $ABCDEF$ is a polygon. If it is, classify it by its number of sides.

Check that it is made only of segments. Each side is a straight segment from one labelled corner to the next. Nothing is curved, and the sides meet only at their endpoints.
$\overline{AB},\ \overline{BC},\ \overline{CD},\ \overline{DE},\ \overline{EF},\ \overline{FA}$
Check that it is closed. Go from A through B, C, D, E and F. The last side, FA, ends back at A, so the chain of sides closes up.
$A \to B \to C \to D \to E \to F \to A$
Count the sides. The first step lists six segments. That matches the six vertices, as it should, because each vertex of a polygon starts exactly one side as you go around it.
$6 \text{ sides}$
Classify it. The table gives the name hexagon for 6 sides. The arrow-like shape does not matter, because the side-count name depends only on how many sides a polygon has.

Answer

Yes. Figure $ABCDEF$ is closed and made only of segments, so it is a polygon, and its $6$ sides make it a hexagon.

You classify triangles and quadrilaterals by their given facts. In this lesson's figures, the givens appear as marks:

Sides with the same number of tick marks are given as congruent. Congruent sides have equal lengths, so one side could be moved exactly onto the other by rotations, reflections and translations (turns, flips and slides).
Sides with the same number of arrowheads are given as parallel. They lie on lines in the same plane that never meet.
A right-angle mark is a small square drawn in a corner. It gives that angle as a right angle, which measures exactly $90^\circ$ .

Any other fact, such as a length or an angle measure, appears in the text. The figures are not drawn to scale. So never decide that sides are equal, that sides are parallel or that an angle is right just because it looks that way.

A triangle is a polygon with three sides. You classify it in two separate ways, by its sides and by its angles.

By sides

Scalene: no two sides are congruent.
Isosceles: at least two sides are congruent.
Equilateral: all three sides are congruent.

By angles (an acute angle measures less than $90^\circ$ , and an obtuse angle measures more than $90^\circ$ and less than $180^\circ$ )

Acute triangle: all three angles are acute.
Right triangle: one angle is a right angle.
Obtuse triangle: one angle is obtuse.

At least two includes three, so every equilateral triangle is also isosceles. To call a triangle scalene, you need facts that show no two sides are equal, such as three different stated lengths. A triangle with no tick marks is not scalene for that reason alone.

AB

and

BC

each carry one tick, so

AB = BC

. The right-angle mark at

B

makes angle

ABC

a right angle.

Worked example

Classify triangle $ABC$ by its sides and by its angles. Use only its marks.

Read the side marks. Sides AB and BC each carry one tick, so they are given as congruent.
$AB = BC$
Classify by sides. At least two sides are congruent, so the triangle is isosceles. Side AC has no tick and no stated length. Nothing shows that it matches the other two sides, so you cannot call the triangle equilateral.
Classify by angles. The right-angle mark at B gives angle ABC as a right angle. A triangle with a right angle is a right triangle.
$m\angle ABC = 90^\circ$

Answer

Triangle $ABC$ is an isosceles right triangle.

This figure has no tick marks. The stated values are

DE = 3

cm,

EF = 5

cm and

DF = 7

cm, and angle

DEF

measures

120^\circ

.

Worked example

In triangle $DEF$ , you are given $DE = 3$ cm, $EF = 5$ cm and $DF = 7$ cm. Angle $DEF$ , the angle at $E$ , measures $120^\circ$ . Classify the triangle by its sides and by its angles.

Compare the stated lengths. No two of them are equal, so no two sides are congruent. The triangle is scalene.
$3 \ne 5,\quad 5 \ne 7,\quad 3 \ne 7$
Compare the stated angle with 90° and 180°. Angle DEF lies between them, so it is obtuse. A triangle with an obtuse angle is an obtuse triangle.
$90^\circ < 120^\circ < 180^\circ$

Answer

Triangle $DEF$ is a scalene obtuse triangle.

A quadrilateral is a polygon with four sides. Two of its sides that share no vertex are opposite sides. Each special quadrilateral has a list of required properties:

Trapezoid: a quadrilateral with at least one pair of parallel sides.
Parallelogram: a quadrilateral in which both pairs of opposite sides are 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.

A parallelogram has two pairs of parallel sides, which counts as at least one pair. So every parallelogram is also a trapezoid, and that includes every rectangle, rhombus and square.

Key concept

Classify by given facts, and keep every name that fits

A figure earns a name when its given facts meet every requirement in that name's definition. The given facts are its number of sides, its marks and its stated values. How its sides and angles look never counts. A figure keeps every name it earns, so one figure can have several names. An equilateral triangle is also isosceles. A square is also a rectangle, a rhombus, a parallelogram and a trapezoid.

Single arrowheads give

AB \parallel DC

, and double arrowheads give

AD \parallel BC

. One tick on every side makes all four sides congruent.

Worked example

Use only its marks to decide which of these names quadrilateral $ABCD$ earns: trapezoid, parallelogram, rectangle, rhombus, square.

Read the arrows. AB and DC each carry one arrowhead, and AD and BC each carry two. So both pairs of opposite sides are given as parallel.
$AB \parallel DC,\quad AD \parallel BC$
Apply the definitions that use parallel sides. Both pairs of opposite sides are parallel, so ABCD is a parallelogram. That gives it at least one pair of parallel sides, so ABCD is also a trapezoid.
Read the ticks. All four sides carry one tick, so all four sides are congruent. A parallelogram with four congruent sides is a rhombus.
$AB = BC = DC = AD$
Check the names that need right angles. A rectangle and a square each require four right angles. The figure has no right-angle marks and no stated angle measures, so the facts do not support either name, however the figure looks.

Answer

Quadrilateral $ABCD$ is a trapezoid, a parallelogram and a rhombus. Its given facts do not show that it is a rectangle or a square.

Only

EF

and

HG

carry arrows, so they are the only pair of sides given as parallel.

In quadrilateral $EFGH$ , the only marks are the arrows on $EF$ and $HG$ . So the only parallel fact you have is $EF \parallel HG$ . A trapezoid needs only one pair of parallel sides, so $EFGH$ is a trapezoid. Nothing states that $EH$ and $FG$ are parallel, so you cannot call $EFGH$ a parallelogram. It also cannot earn the names rectangle, rhombus or square, because each of those must first be a parallelogram. Its parallel sides run up and down the page instead of across it, and that changes nothing. The marks decide the name, not the way the figure is turned.

Takeaway: Classify a figure only from its facts. To classify a polygon, check that it is closed and made only of segments, and then count its sides. To classify a triangle or a quadrilateral, read its tick marks, arrows, right-angle marks and stated values. Then give it every name whose full list of requirements those facts meet. Next, you will find missing angles in quadrilaterals and interior angles of polygons.