Quadrilaterals 🔷🟩🟦🟨

quadrilateral

A polygon with four sides

parallelogram

A quadrilateral in which both pairs of opposite sides are parallel

diagonal

Line segment joining two non-adjacent vertices of a polygon

congruent triangles

Triangles that are exactly equal in size and shape

Theorem 8.1

A diagonal of a parallelogram divides it into two congruent triangles

alternate angles

Angles formed when a transversal crosses two parallel lines; they are equal

ASA rule

Angle-Side-Angle congruence rule for triangles

opposite sides

In a parallelogram

Theorem 8.2

In a parallelogram

converse

If the conclusion of a theorem is given as a condition

Theorem 8.3

If each pair of opposite sides of a quadrilateral is equal

opposite angles

Angles that are across from each other in a quadrilateral

Theorem 8.4

In a parallelogram

Theorem 8.5

If in a quadrilateral

diagonals

Line segments joining opposite vertices in a quadrilateral

bisect

To divide into two equal parts

Theorem 8.6

The diagonals of a parallelogram bisect each other

Theorem 8.7

If the diagonals of a quadrilateral bisect each other

rectangle

A parallelogram in which one angle is a right angle

right angle

An angle of 90 degrees

property of rectangle

Each angle of a rectangle is a right angle

rhombus

A parallelogram with all sides equal

property of rhombus

Diagonals of a rhombus are perpendicular to each other

isosceles triangle

A triangle with two equal sides

exterior angle

An angle formed by one side of a polygon and the extension of an adjacent side

alternate angles

Angles on opposite sides of a transversal

property of parallelogram

If both pairs of opposite sides are parallel

property of rectangle

If diagonals of a parallelogram are equal

property of square

Diagonals of a square are equal

property of trapezium

Quadrilateral with one pair of parallel sides

mid-point

The point that divides a line segment into two equal parts

Theorem 8.8

The line segment joining the mid-points of two sides of a triangle is parallel to the third side

Theorem 8.9

The line drawn through the mid-point of one side of a triangle

intercept

The part of a transversal between two parallel lines

congruent triangles in parallelogram

Each diagonal divides a parallelogram into two congruent triangles

pairs of equal sides

AB = DC and AD = BC in parallelogram ABCD

pairs of equal angles

Opposite angles of a parallelogram are equal

bisecting diagonals

Diagonals of a parallelogram cut each other into two equal halves

rectangle diagonals

Diagonals of a rectangle are equal and bisect each other

rhombus diagonals

Diagonals of a rhombus bisect each other at right angles

square diagonals

Diagonals of a square are equal and bisect each other at right angles

mid-point theorem

EF || BC and EF = ½ BC if E and F are mid-points of AB and AC in triangle ABC

converse mid-point theorem

Line through mid-point of one side

parallelogram condition

If both pairs of opposite sides are equal

rectangle condition

If parallelogram has equal diagonals

square condition

If parallelogram has equal diagonals that bisect at right angles

trapezium

Quadrilateral with only one pair of parallel sides

diagonal bisects angle

If diagonal of parallelogram bisects one angle

parallelogram area formula

Area = base × height

rectangle area formula

Area = length × breadth

square area formula

Area = side × side

rhombus area formula

Area = ½ × (product of diagonals)

parallelogram perimeter formula

Perimeter = 2 × (sum of adjacent sides)

rectangle perimeter formula

Perimeter = 2 × (length + breadth)

square perimeter formula

Perimeter = 4 × side

rhombus perimeter formula

Perimeter = 4 × side

properties of parallelogram

Opposite sides equal

properties of rectangle

All angles 90°

properties of rhombus

All sides equal

properties of square

All sides equal

mid-points in quadrilateral

The quadrilateral formed by joining mid-points of sides is a parallelogram

rectangle from bisectors

Bisectors of angles of a parallelogram form a rectangle

rectangle from parallel lines

Quadrilateral formed by bisectors of interior angles of two parallel lines cut by a transversal is a rectangle

congruent triangles in triangle

Joining mid-points of sides of triangle divides it into four congruent triangles

equal intercepts

If three parallel lines cut off equal intercepts on one transversal

angle sum property

Sum of angles of a quadrilateral is 360°

angle sum property of triangle

Sum of angles of a triangle is 180°

linear pair

Two adjacent angles whose non-common sides form a straight line

interior angles on same side

Angles on same side of transversal

alternate interior angles

Angles between two lines on opposite sides of a transversal

transversal

A line that intersects two or more lines at distinct points

Summary

Key properties: parallelogram diagonals bisect