Polygon Classification
Classifying Quadrilaterals
A quadrilateral is a four-sided polygon. There
are five special quadrilaterals:
A parallelogram is a quadrilateral with two pairs
of parallel sides.
A rectangle is a parallelogram with four right
angles.
A rhombus is a parallelogram with four equallength sides.
A square is a parallelogram with four right
angles and four equal-length sides.
A trapezoid is a quadrilateral with exactly one
pair of parallel sides.
Example:
Classify the following quadrilaterals as
parallelograms, rectangles, rhombi, squares, or
trapezoids.
Quadrilateral 1 is a parallelogram because it has
two pairs of parallel sides. It is also a rectangle
because it has four right angles.
Quadrilateral 2 is a trapezoid because it has
exactly one pair of parallel sides.
Quadrilateral 3 is a parallelogram because it has
two pairs of parallel sides.
Quadrilateral 4 is a parallelogram because it has
two pairs of parallel sides. It is also a rectangle
because it has four right angles. It is also a
rhombus because it has four sides of equal
length. It is also a square.
Quadrilateral 5 is a parallelogram because it has
two pairs of parallel sides. It is also a rhombus
because it has four sides of equal length.
Quadrilateral 6 is a parallelogram because it has
two pairs of parallel sides.
Classifying Polygons
A polygon is a two-dimensional closed figure
that is made by joining three or more line
segments, and where each line segment
intersects exactly two other line segments.
A regular polygon is polygon whose sides all
have the same length and whose interior angles
all have the same measure.
An irregular polygon is a polygon whose sides
do not all have the same length and whose
interior angles do not all have the same measure.
To classify a polygon, count the number of its
sides.
Polygon Number of sides
triangle
3
quadrilateral
4
pentagon
5
hexagon
6
heptagon
7
octagon
8
nonagon
9
decagon
10
Example:
Classify each polygon. Then decide whether
each is regular or irregular.
Polygon 1 is an irregular heptagon because it has
seven sides that do not have the same length.
Polygon 2 is a regular hexagon because it has six
sides that have the same length and six interior
angles that all have the same measure.
Polygon 3 is a regular triangle because it has
three sides that have the same length and three
interior angles that all have the same measure.
Polygon is an irregular quadrilateral because it
has four sides that do not have the same length.
Polygon 5 is an irregular pentagon because it has
5 sides that do not have the same length.
Polygon 6 is a regular quadrilateral because it
has four sides that have the same length and four
interior angles that all have the same measure.
Polygon 7 is an irregular nonagon because it has
nine sides that do not have the same length and
nine interior angles that do not all have the same
measure.
Polygon 8 is an irregular quadrilateral because it
has four sides that do not all have the same
length.
Properties of Quadrilaterals and
Parallelograms
Find a Missing Angle Measure in a
Quadrilateral
A quadrilateral is a four-sided polygon. One of
the properties of a quadrilateral is that the sum
of the measures of the interior angles is 360°.
Example:
In quadrilateral WXYZ, the measure of angle W
is 58°, the measure of angle X is 92°, and the
measure of angle Y is 119°. Find the measure of
angle Z.
The sum of the measures of the interior angles of
a quadrilateral is 360°. You can use this
information to write an equation. Then solve the
equation to find the measure of angle Z.
m∠W + m∠X
= 360°
+ m∠Y + m∠Z
58° + 92° +
= 360°
119° + m∠Z
Substitute given
angle measures.
269° + m∠Z = 360°
Add.
m∠Z =
360° 269°
m∠Z = 91°
Subtract 269°
from both sides.
Simplify.
So, the measure angle Z is 91°.
Find Lengths of Opposite Sides in a
Parallelogram
A parallelogram is a quadrilateral in which the
opposite sides are parallel and congruent, as
shown in the figure below.
Example:
In parallelogram FGHJ, the length of segment
FG is 11.5 inches and the length of segment GH
is 16.8 inches. Find the perimeter of the
parallelogram.
In a parallelogram, the lengths of opposite sides
are congruent. It is given that the length of
segment FG is 11.5 inches, so the side opposite
it, segment JH, is also 11.5 inches. It is given
that the length of segment GH is 16.8 inches, so
the side opposite it, segment FJ, is also 16.8
inches. To find the perimeter P, find the sum of
the lengths of the sides.
FG + GH + JH +
P=
FJ
P=
11.5 + 16.8 + 11.5
+ 16.8
P = 56.6
Substitute the side
lengths.
Add.
So, the perimeter of parallelogram FGHJ is 56.6
inches.
Find Opposite and Consecutive Angle
Measures in a Parallelogram
A property of parallelograms is that the opposite
angles are congruent angles and the consecutive
angles are supplementary angles, as shown in
the figure below.
Example:
In parallelogram UVWX, the measure of angle
U is 117°. Find the measures of angles V, W,
and X.
In a parallelogram, opposite angles are
congruent. It is given that the measure of angle
U is 117°. Angles U and W are opposite angles,
so the measure of angle W is also 117°.
In a parallelogram, consecutive angles are
supplementary. Angles U and V are
supplementary, so the sum of the measures of
their angles is 180°. You can use this
information to write an equation. Then solve the
equation to find the measure of angle V.
m∠U +
= 180°
m∠V
117° +
= 180°
m∠V
m∠V =
180° 117°
m∠V = 63°
Substitute given angle
measure.
Subtract 117° from
both sides.
Simplify.
So, the measure of angle V is 63°.
Angles V and X are opposite angles and
opposite angles in a parallelogram are
congruent, so the measure of angle X is 63°.
So, in parallelogram UVWX, the measure of
angle V is 63°, the measure of angle W is 117°,
and the measure of angle X is 63°.
Find Lengths of Diagonals in a Parallelogram
A diagonal of a quadrilateral is a line segment
that connects the opposite angles. In a
parallelogram, the diagonals bisect each other.
This means that the diagonals are divided into
two congruent segments, as shown in the figure
below.
Example:
In parallelogram KLMN, the length of segment
KM is 120 millimeters. The length of segment
LP is 37.5 millimeters. Find the lengths of
segments KP, MP, NP, and LN.
A property of parallelograms is that the
diagonals bisect each other. So, in parallelogram
KLMN, the length of segment KP is equal to the
length of segment MP. It is given that the length
of segment KM is 120 millimeters. The sum of
the lengths of segments KP and MP is equal to
the length of segment KM. Let x represent the
length of segment KP and the length of segment
MP. You can use this information to write an
equation. Then solve the equation for x to find
the lengths of segments KP and MP.
KP + MP = KM
x + x = 120
2x = 120
x = 60
Substitute.
Combine like terms.
Divide both sides by 2.
So, the length of segment KP is 60 millimeters
and the length of segment MP is 60 millimeters.
Because the diagonals of parallelogram KLMN
bisect each other, the length of segment NP is
equal to the length of segment LP. It is given
that the length of segment LP is 37.5
millimeters. So, the length of segment NP is also
37.5 millimeters. The sum of the lengths of
segments NP and LP is equal to the length of
segment NL. So, the length of segment NL is
37.5 + 37.5 = 75 millimeters.
So, in parallelogram KLMN, the length of
segment KP is 60 millimeters, the length of
segment MP is 60 millimeters, the length of
segment NP is 37.5 millimeters, and the length
of segment LN is 75 millimeters.
Find Angle Measures in a Quadrilateral
Using Expressions
Sometimes the exact measure of an angle or side
length of a quadrilateral is unknown. So, an
expression may be used.
Example:
In quadrilateral QRST, the measure of angle Q is
represented by the expression 4x+2, the measure
of angle R is represented by the expression
2x−8, the measure of angle S is represented by
the expression 4x+5, and the measure of angle T
is represented by the expression 6x+9. Find the
measures of all interior angles in the
quadrilateral.
The sum of the measures of the interior angles of
a quadrilateral is 360°. You can use this
information to write an equation. Then solve the
equation for x.
m∠Q + m∠R
= 360
+ m∠S + m∠T
4x+2 + 2x−8 = 360
Substitute the given
+ 4x+5 +
6x+9
16x+8 = 360
expressions.
Combine like terms.
16x = 352
Subtract 8 from both
sides of the equation.
x = 22
Divide both sides of
the equation by 16.
So, the value of x is 22. Substitute this value for
x into the expression for each angle measure.
m∠Q = 4x+2 = 422+2 = 90
m∠R = 2x−8 = 222−8 = 36
m∠S = 4x+5 = 422+5 = 93
m∠T = 6x+9 = 622+9 = 141
So, the measure of angle Q is 90°, the measure
of angle R is 36°, the measure of angle S is 93°,
and the measure of angle T is 141°.
Find Side Lengths in a Parallelogram Using
Expressions
Example:
In parallelogram HJKL, the length of segment
JK is represented by the expression 5x+4 inches,
the length of segment KL is represented by the
expression 4x+9 inches, and the length of
segment LH is represented by the expression
9x−2 inches. Find the lengths of all sides of the
parallelogram.
In a parallelogram, the lengths of opposite sides
are congruent. So, you know that the length of
segment JK is equal to the length of segment
LH. You can use this information to write an
equation. Then solve the equation for x.
JK = LH
5x+4 = 9x−2
6 = 4x
Substitute the given
expressions.
Subtract 5x and add 2 to both
sides.
1.5 = x
Divide both sides by 4.
So, the value of x is 1.5. Substitute this value for
x into the expression for each side length.
length of segment JK = 5x+4 = 51.5+4 = 11.5
length of segment KL = 4x+9 = 41.5+9 = 15
length of segment LH = 9x−2 = 91.5−2 = 11.5
The lengths of opposite sides in a parallelogram
are congruent. So, the length of segment JH
equals the length of segment KL.
So, the length of segment JH is 15 inches, the
length of segment JK is 11.5 inches, the length
of segment KL is 15 inches, and the length of
segment LH is 11.5 inches.