HK Math · Geometry

Quadrilaterals

Learn the definitions, properties, and proof conditions for special quadrilaterals, then connect them to the midpoint theorem and the intercept theorem.

Jump ToParallelogramsJump ToRectanglesJump ToRhombusesJump ToSquares

Flow Of Ideas

Start with parallelograms. Rectangles and rhombuses are special parallelograms. A square is both a rectangle and a rhombus.

  • Parallelogram: opposite sides are parallel.
  • Rectangle: a parallelogram with four right angles.
  • Rhombus: a parallelogram with four equal sides.
  • Square: a quadrilateral with four equal sides and four right angles.

Parallelograms

A parallelogram is a quadrilateral with two pairs of parallel opposite sides.

Properties

  • Opposite sides are equal: AB = DC and AD = BC.
  • Opposite angles are equal: angle A = angle C and angle B = angle D.
  • Diagonals bisect each other: the intersection point divides each diagonal into two equal line segments.
Source diagram listing the three properties of parallelograms: opposite sides equal, opposite angles equal, and diagonals bisect each other.

Rectangles

A rectangle is a quadrilateral with four right angles.

Properties

  • Has all the properties of parallelograms.
  • Diagonals are equal: AC = BD.
  • The diagonals bisect each other into four equal line segments: AE = BE = CE = DE.
Source diagram listing the properties of rectangles with equal diagonals AC and BD and equal segments AE, BE, CE, and DE.

Rhombuses

A rhombus is a quadrilateral with four equal sides.

Properties

  • Has all the properties of parallelograms.
  • Diagonals bisect the opposite angles, for example angle BAC = angle DAC.
  • Diagonals are perpendicular to each other: AC is perpendicular to BD.
Source diagram listing the properties of rhombuses: diagonals bisect opposite angles, diagonals are perpendicular, and rhombuses have all parallelogram properties.

Squares

A square is a quadrilateral with four equal sides and four right angles.

Properties

  • Has all the properties of rectangles.
  • Has all the properties of rhombuses.
  • The angle between each diagonal and each side is 45 degrees.
Source diagram listing the properties of squares, including rectangle properties, rhombus properties, and 45 degree angles between diagonals and sides.

Conditions To Prove A Parallelogram

  • Both pairs of opposite sides are equal.
  • Both pairs of opposite angles are equal.
  • The diagonals bisect each other.
  • One pair of opposite sides is both equal and parallel.
Source diagram listing four conditions for proving that a quadrilateral is a parallelogram.

Midpoint Theorem

In triangle ABC, if M is the midpoint of AB and N is the midpoint of AC, then MN is parallel to BC and MN is half of BC.

  • Given: AM = MB and AN = NC.
  • Conclusion: MN is parallel to BC.
  • Conclusion: MN = 1/2 BC.
  • Use it to find unknown lengths or to prove two lines are parallel.
Source diagram for the midpoint theorem showing M and N as midpoints of AB and AC, MN parallel BC, and MN equal to one half of BC.

Intercept Theorem

When parallel lines cut two transversals, equal intercepts on one transversal give equal corresponding intercepts on the other transversal.

  • In the source figure, AB, CD, and EF are parallel.
  • ACE and BDF are straight lines.
  • If AC = CE, then BD = DF.
  • Use it when several parallel lines cross two slanted lines.
Source diagram for the intercept theorem showing parallel lines AB, CD, and EF cut by straight lines ACE and BDF.

Practice Checks

Check: Which Conditions Prove A Parallelogram?
  • Both pairs of opposite sides are equal.
  • Both pairs of opposite angles are equal.
  • The diagonals bisect each other.
  • One pair of opposite sides is both equal and parallel.
Check: How Are Rectangle, Rhombus, And Square Related?
  • A rectangle is a parallelogram with four right angles.
  • A rhombus is a parallelogram with four equal sides.
  • A square has the properties of both rectangles and rhombuses.