Geometric Shapes: Quadrilaterals, Polygons, and Proportions

Quadrilaterals

A quadrilateral is a closed plane figure, a polygon, consisting of four sides and four angles.

A diagonal is a straight line segment joining two non-consecutive vertices.

Types of Quadrilaterals

Square

A square has four equal sides and four 90-degree angles. Its diagonals are equal in length and perpendicular to each other.

Rectangle

A rectangle has opposite sides that are equal and parallel. All its angles are 90 degrees. Its diagonals are equal in length and are oblique (not perpendicular).

Rhombus

A rhombus has four equal sides, with opposite sides parallel. Its diagonals are not equal in length but are perpendicular to each other.

Parallelogram (Rhomboid)

A rhomboid (or parallelogram) has opposite sides that are equal and parallel. Its diagonals are unequal in length and are oblique (not perpendicular).

Trapezoid

A trapezoid has two opposite sides that are parallel, called bases. The other two sides are non-parallel.

Isosceles Trapezoid

An isosceles trapezoid has two parallel sides and two non-parallel sides of equal length. Its base angles are equal in pairs. Its diagonals are equal in length and are oblique.

Right Trapezoid

A right trapezoid has two parallel sides and at least one pair of right angles (90 degrees). Its diagonals are unequal in length and are oblique.

Scalene Trapezoid

A scalene trapezoid has two parallel sides, but all its angles and non-parallel sides are unequal. Its diagonals are unequal and oblique.

Trapezium (Irregular Quadrilateral)

A trapezium (or irregular quadrilateral) has all its sides and angles unequal. Its diagonals are unequal and oblique.

Properties of Quadrilaterals

• The sum of the four interior angles of any quadrilateral is 360 degrees.
• If a quadrilateral is circumscribed about a circle, the sums of its opposite sides are equal.
• If the opposite angles of a quadrilateral are supplementary (add up to 180 degrees), then the quadrilateral can be inscribed in a circle.

Polygons

A polygon is a portion of the plane bounded by straight line segments that intersect two by two at points called vertices. The segment connecting two consecutive vertices is called a side.

Types of Polygons

Inscribed Polygon

An inscribed polygon is one that has all its vertices lying on a circle.

Circumscribed Polygon

A circumscribed polygon is one that has all its sides tangent to a circle.

Equiangular Polygon

An equiangular polygon is one that has all its interior angles equal.

Equilateral Polygon

An equilateral polygon is one that has all its sides equal in length.

Regular Polygon

A regular polygon is one that has all its angles equal and all its sides equal in length.

Key Elements of Regular Polygons

Central Angle

The central angle of a regular polygon is obtained by dividing 360 degrees by the number of sides.

Center

The center of a regular polygon is an interior point that is equidistant from all its vertices.

Diagonal

A diagonal is a segment joining two non-consecutive vertices of a polygon.

Radius

The radius of a regular polygon is a segment joining its center with any of its vertices.

Apothem

The apothem of a regular polygon is a segment connecting its center to the midpoint of one of its sides.

Geometric Ratios and Theorems

Thales' Theorem

Given two concurrent lines that are intersected by a series of parallel lines, the segments intercepted on the concurrent lines are proportional.

Ratio

The ratio of two segments, 'a' and 'b', is the quotient of their magnitudes: a / b.

Proportion

A proportion is an equality between two ratios. For segments a, b, c, and d, it is expressed as a / b = c / d. The elements 'a' and 'd' are called the extremes, and 'b' and 'c' are called the means. The elements 'a' and 'c' are called antecedents, and 'b' and 'd' are called consequents.