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Classifying Shapes
by Properties

Learn to identify and classify geometric shapes based on their sides, angles, and special properties to solve quantitative reasoning problems.

ISEE Middle Level

What Is a Shape Property?

A property is a specific characteristic that is always true about a shape. Properties include the number of sides, angle measures, whether sides are parallel or equal in length, and the types of symmetry a shape has.

The Four Key Properties

Sides
How many sides? Are any sides equal in length?
Angles
Are angles right (\(90^\circ\)), acute (\(<90^\circ\)), or obtuse (\(>90^\circ\))?
Parallel Sides
Do any pairs of sides run in the same direction and never meet?
Symmetry
Can you fold or rotate the shape so it looks the same?

Why Does Classification Matter?

On the ISEE, knowing a shape's classification tells you which formulas and rules apply. For example, if you recognize a shape is a parallelogram, you immediately know its opposite sides are equal and its opposite angles are equal: powerful shortcuts for solving problems.

Triangles by Sides

\(5\)
\(5\)
\(5\)
 Equilateral
\(6\)
\(6\)
\(7\)
 Isosceles
\(5\)
\(8\)
\(6\)
 Scalene
Equilateral
All 3 sides equal, all angles \(= 60^\circ\)
Isosceles
At least 2 sides equal, 2 base angles equal
Scalene
No sides equal, no angles equal

Triangles by Angles

\(70^\circ\)
 Acute Right
\(120^\circ\)
 Obtuse

The Quadrilateral Family

Quadrilateral Parallelogram Trapezoid Rectangle Rhombus Square Kite

Rectangle vs. Rhombus vs. Square

Property Rectangle Rhombus Square
All sides equal ✕ ✓ ✓
All angles \(90^\circ\) ✓ ✕ ✓
Opposite sides parallel ✓ ✓ ✓
Diagonals equal ✓ ✕ ✓
Diagonals perpendicular ✕ ✓ ✓

Parallelogram Properties

A B C D
1
Opposite sides are parallel and equal in length: \(AB = CD\) and \(BC = AD\).
2
Opposite angles are equal: \(\angle A = \angle C\) and \(\angle B = \angle D\).
3
Consecutive angles are supplementary: they add up to \(180^\circ\).

How to Classify Any Shape

1
Count the sides. This tells you the polygon family: \(3\) = triangle, \(4\) = quadrilateral, \(5\) = pentagon, and so on.
2
Check for equal sides. Mark sides that are equal in length. Use tick marks or given measurements.
3
Check for parallel sides. Look for arrows or matching slope. Parallel sides define parallelograms and trapezoids.
4
Check the angles. Are they right angles (\(90^\circ\))? Are they all equal? Use given values or the angle sum rule.
5
Name the most specific shape. Use the family tree: a square is more specific than a rectangle.

Example 1: Identify the Quadrilateral

\(12\) cm
\(12\) cm
\(7\) cm
\(7\) cm
 P Q R S

Example 2: Triangle Classification

\(5\) cm
\(5\) cm
\(8\) cm
\(106^\circ\)
\(37^\circ\)
\(37^\circ\)

Example 3: Find the Missing Angle

\(65^\circ\)
?
 A B C D

Regular Polygons

A regular polygon has all sides equal in length and all interior angles equal. The interior angle of a regular \(n\)-sided polygon is:

Equilateral Triangle
\(n=3\): each angle \(= 60^\circ\)
Square
\(n=4\): each angle \(= 90^\circ\)
Regular Hexagon
\(n=6\): each angle \(= 120^\circ\)

Common Mistakes to Avoid

✕ Judging by Appearance
A shape that "looks like" a square may actually be a rhombus if its angles are not \(90^\circ\). Always check the given properties.
✕ Forgetting the Hierarchy
Every square is a rectangle, but not every rectangle is a square. The ISEE may test whether you know these relationships.
✕ Confusing Regular and Irregular
"Regular" means all sides and all angles are equal. A rectangle is not regular unless it is also a square.

Quick Reference Table

Shape Sides Angles Parallel Sides
Equilateral Triangle 3 equal All \(60^\circ\) \(0\)
Trapezoid 4 Varies Exactly \(1\) pair
Parallelogram 4, opp. equal Opp. equal \(2\) pairs
Rectangle 4, opp. equal All \(90^\circ\) \(2\) pairs
Rhombus 4, all equal Opp. equal \(2\) pairs
Square 4, all equal All \(90^\circ\) \(2\) pairs

Question 1 of 5

A quadrilateral has all four sides equal in length but its angles are not \(90^\circ\). What is this shape?

Question 2 of 5

Which of the following is always true about a parallelogram?

Question 3 of 5

A triangle has sides of length 6 cm, 6 cm, and 6 cm. It is classified as:

Question 4 of 5

In parallelogram WXYZ, \(\angle W = 72^\circ\). What is the measure of \(\angle X\)?

Question 5 of 5

Which statement is false?