Essential Geometry Theorems and Properties

Posted on Mar 15, 2026 in Mathematics

Quadrilaterals

  • The sum of the interior angles of a quadrilateral is 360°.
  • Properties of a parallelogram:
    • Opposite sides are equal.
    • Opposite angles are equal.
    • Consecutive angles are supplementary (sum = 180°).
    • Diagonals bisect each other.
    • A diagonal divides it into two congruent triangles.
  • Conditions for a parallelogram:
    • Both pairs of opposite sides are equal.
    • Both pairs of opposite angles are equal.
    • Diagonals bisect each other.
    • One pair of opposite sides is equal and parallel.
  • Mid-Point Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it.

Circles

  • Equal chords of a circle subtend equal angles at the centre.
  • If two chords of a circle subtend equal angles at the centre, then the chords are equal.
  • The perpendicular drawn from the centre of a circle to a chord bisects the chord.
  • The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
  • There is one and only one circle passing through three non-collinear points.
  • Equal chords of a circle are equidistant from the centre.
  • Chords equidistant from the centre of a circle are equal.
  • The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
  • Angles in the same segment of a circle are equal.
  • The angle in a semicircle is a right angle.
  • If a line segment joining two points subtends equal angles at two other points on the same side of the line, then the four points lie on a circle.
  • The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Triangles

  • In a triangle, the angle opposite to the longer side is greater.
  • In a triangle, the side opposite to the greater angle is longer.
  • The sum of the angles of a triangle is 180°.
  • An exterior angle of a triangle is equal to the sum of its two interior opposite angles.
  • Triangle Inequality Property: The sum of any two sides of a triangle is greater than the third side.
  • Isosceles Triangle Theorem: If two sides of a triangle are equal, then the angles opposite to them are equal.
  • If two angles of a triangle are equal, then the sides opposite to them are equal.
  • Pythagoras Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
  • The angles opposite to equal sides of a triangle are equal.
  • The sides opposite to equal angles of a triangle are equal.

Lines and Angles

  • Linear Pair: If a ray stands on a line, then the sum of the two adjacent angles so formed is 180°.
  • When two lines intersect each other, vertically opposite angles are equal.
  • Parallel lines cut by a transversal:
    • Corresponding angles are equal.
    • Alternate interior angles are equal.
    • Interior angles on the same side of the transversal are supplementary (sum = 180°).
  • Conditions for parallel lines: If a transversal cuts two lines such that corresponding angles are equal, alternate interior angles are equal, or interior angles on the same side are supplementary, then the two lines are parallel.