Comprehensive theory, key formulas, diagrams, and memory aids for Quadrilaterals.
If you look around, many objects you see are shaped like quadrilaterals—books, doors, computer screens, and windows. A quadrilateral is a closed two-dimensional polygon bounded by four line segments. In this chapter, we will study the various types of quadrilaterals, with a special focus on the properties of parallelograms and the vital Mid-point Theorem.
A figure formed by joining four points in an order is called a quadrilateral. A quadrilateral has four sides, four angles, and four vertices.
Let the vertices be $A, B, C,$ and $D$. * Sides: $AB, BC, CD,$ and $DA$. * Angles: $\angle A, \angle B, \angle C,$ and $\angle D$. * Diagonals: The line segments joining the opposite vertices, $AC$ and $BD$, are called diagonals.
Theorem: The sum of the angles of a quadrilateral is $360^\circ$. Proof: Let $ABCD$ be a quadrilateral and $AC$ be a diagonal. The diagonal $AC$ divides the quadrilateral into two triangles, $\triangle ABC$ and $\triangle ADC$. We know the sum of angles in a triangle is $180^\circ$. In $\triangle ABC$, $\angle BAC + \angle ABC + \angle BCA = 180^\circ$ In $\triangle ADC$, $\angle DAC + \angle ADC + \angle DCA = 180^\circ$ Adding both equations, the sum of all angles in the two triangles equals $360^\circ$. Since $\angle BAC + \angle DAC = \angle A$ and $\angle BCA + \angle DCA = \angle C$, the sum of angles of the quadrilateral $ABCD$ is $360^\circ$.
Quadrilaterals are classified based on the nature of their sides and angles.
Note: A square, rectangle, and rhombus are all special types of parallelograms. A square is a rectangle and also a rhombus. A parallelogram is a trapezium. A kite is not a parallelogram.
Since the parallelogram is the most fundamental of the special quadrilaterals, its properties are extensively studied.
Theorem 1: A diagonal of a parallelogram divides it into two congruent triangles. Proof Concept: Let $ABCD$ be a parallelogram and $AC$ be a diagonal. Because $AB \parallel DC$ and $AD \parallel BC$, the alternate interior angles formed by the transversal $AC$ are equal. Using the ASA congruence rule (Angle-Side-Angle) with the common side $AC$, we can prove $\triangle ABC \cong \triangle CDA$.
Because of this congruence, several other properties naturally follow (via CPCTC): Theorem 2: In a parallelogram, opposite sides are equal. Theorem 3: In a parallelogram, opposite angles are equal.
The converses of these theorems are also true and are used to prove that a given quadrilateral is a parallelogram: * If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. * If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
Theorem 4: The diagonals of a parallelogram bisect each other. Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Theorem: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel. If $AB \parallel DC$ and $AB = DC$, then $ABCD$ is a parallelogram.
The Mid-point theorem establishes a beautiful relationship between a triangle and a line segment joining the mid-points of its two sides.
Theorem (Mid-point Theorem): The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it. Proof Concept: Let $\triangle ABC$ have mid-points $E$ and $F$ on sides $AB$ and $AC$ respectively. Extend $EF$ to $D$ such that $EF = FD$. By joining $C$ to $D$, we can prove $\triangle AEF \cong \triangle CDF$. This implies $CD$ is equal and parallel to $AE$ (and thus $EB$). Since one pair of opposite sides ($EB$ and $CD$) is equal and parallel, $EBCD$ is a parallelogram. Therefore, $ED \parallel BC$ and $ED = BC$. Since $EF$ is half of $ED$, it is also half of $BC$.
Converse of Mid-point Theorem: The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.
The mid-point theorem is extremely powerful. A classic application is proving that if you join the mid-points of the adjacent sides of any quadrilateral, the resulting figure is a parallelogram.
By drawing a diagonal in the quadrilateral, it is split into two triangles. Applying the mid-point theorem to both triangles shows that the line segments formed by the mid-points are both parallel to the diagonal and half its length. Because they are parallel to the same diagonal and equal in length to each other, they form a parallelogram.
Quadrilaterals surround us in our daily lives. Understanding their properties, especially the hierarchy from trapeziums to parallelograms, and down to squares, is vital. The Mid-point Theorem serves as a crucial link showing how triangle properties extend to quadrilaterals, laying the groundwork for complex geometric proofs.