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📖 Mathematics — Part 1

Std 8
Books Book Guide
1 2 3 4 5 6 7
Chapters
1 A Square and a Cube 2 Power Play (Exponents) 3 A Story of Numbers 4 Quadrilaterals 5 Number Play 6 We Distribute Yet Things Multiply 7 Proportional Reasoning - I
4
Chapter 4
Skill: 30%

Quadrilaterals

Quadrilaterals

Chapter at a Glance

This chapter details the properties and definitions of special four-sided polygons, known as quadrilaterals. By utilizing geometric deduction—specifically triangle congruence conditions—and hands-on constructions, it explores rectangles, squares, parallelograms, rhombuses, kites, and trapeziums. The chapter outlines the specific behavior of diagonals in each shape, explains the Angle Sum Property of quadrilaterals, establishes their classification using Venn diagrams, and details practical applications in construction and carpentry.

Key Definitions & Terminology

  • Quadrilateral: A closed planar polygon with four straight sides and four interior angles.
  • Rectangle: A quadrilateral in which all four interior angles are right angles ($90^\circ$).
  • Square: A special rectangle in which all four sides are of equal length and all interior angles are right angles ($90^\circ$).
  • Parallelogram: A quadrilateral with both pairs of opposite sides parallel to each other.
  • Rhombus: A quadrilateral in which all four sides have equal length.
  • Kite: A quadrilateral with two distinct pairs of equal adjacent sides.
  • Trapezium: A quadrilateral with at least one pair of parallel opposite sides.
  • Isosceles Trapezium: A trapezium in which the non-parallel opposite sides are equal in length.
  • Bisection: Dividing a geometric segment or angle into two equal parts.
  • Conjecture: A mathematical statement believed to be true based on observation or measurement, but not yet formally proven.
  • Deduction: The process of using logical reasoning and established theorems (like triangle congruence) to prove geometric properties.

Formulas, Rules & Properties

  • Angle Sum Property:
  • The sum of all interior angles of any quadrilateral is always $360^\circ$:
    $$\angle A + \angle B + \angle C + \angle D = 360^\circ$$
  • Transversal Angle Rule:
  • If two lines are parallel and cut by a transversal line, the sum of the interior angles on the same side of the transversal is $180^\circ$ (supplementary).
  • Triangle Congruence Conditions:
  • SAS (Side-Angle-Side), AAS (Angle-Angle-Side), ASA (Angle-Side-Angle), and SSS (Side-Side-Side) are used to deduce diagonal and side properties.
  • Hierarchical Classification (Venn Diagram Relations):
  • Every square is a rectangle, and every rectangle is a parallelogram.
  • Every square is a rhombus, and every rhombus is a parallelogram.
  • Every rhombus is a kite.
  • Every parallelogram is a trapezium.
  • Diagonal Properties:
  • Diagonals Bisect Each Other: True for all parallelograms (including rectangles, rhombuses, and squares).
  • Diagonals are Equal: True for rectangles, squares, and isosceles trapeziums.
  • Diagonals are Perpendicular: True for squares, rhombuses, and kites.
  • Diagonals Bisect Interior Angles: True for squares and rhombuses.

Core Concepts & Topics

  • Diagonal Properties Summary Chart:
    | Quadrilateral | Equal Length? | Bisect Each Other? | Perpendicular? | Bisect Angles? |
    | :--- | :---: | :---: | :---: | :---: |
    | Parallelogram | No | Yes | No | No |
    | Rectangle | Yes | Yes | No | No |
    | Rhombus | No | Yes | Yes | Yes |
    | Square | Yes | Yes | Yes | Yes |
    | Kite | No | One bisected | Yes | One diagonal |
    | Trapezium | No | No | No | No |
    | Isosceles Trapezium | Yes | No | No | No |
  • The Carpenter's Problem:
  • To construct a rectangular frame, a carpenter joins two strips of equal length ($AC = BD$) at their midpoints ($O$). Joining them at their midpoints ensures they bisect each other, guaranteeing that the resulting shape is a rectangle, regardless of the angle of intersection. If joined at exactly $90^\circ$, it forms a square.
  • Properties of Isosceles Trapeziums:
  • The base angles adjacent to the parallel sides and opposite the equal non-parallel sides are equal ($\angle U = \angle V$ and $\angle X = \angle W$).

Worked Examples

  • Calculating Angles inside a Rectangle (Figure it Out Q1, Page 119):
  • Problem (i): In a rectangle $ABCD$, diagonal $AC$ makes $\angle BAC = 30^\circ$. Find all other angles.
    • Solution:
    • Since the corner angle is $90^\circ$, $\angle DAC = 90^\circ - 30^\circ = 60^\circ$.
    • Opposite sides are parallel ($AD \parallel BC$). Alternate interior angles are equal: $\angle ACB = \angle DAC = 60^\circ$, and $\angle ACD = \angle BAC = 30^\circ$.
    • Diagonals of a rectangle are equal and bisect each other ($OA = OB$). Thus, $\triangle AOB$ is isosceles: $\angle ABD = \angle BAC = 30^\circ$.
    • $\angle ADB = 90^\circ - \angle ABD = 60^\circ$. Alternate angle: $\angle BDC = 30^\circ$.
  • Problem (ii): In rectangle $PQRS$, diagonals intersect at $O$ with $\angle POS = 110^\circ$. Find all angles.
    • Solution:
    • $\angle POS = 110^\circ \rightarrow \angle QOR = 110^\circ$ (vertically opposite).
    • $\angle POQ = 180^\circ - 110^\circ = 70^\circ$ (linear pair), and $\angle ROS = 70^\circ$.
    • Since $OP = OQ$, $\triangle POQ$ is isosceles: $\angle OPQ = \angle OQP = \frac{180^\circ - 70^\circ}{2} = 55^\circ$.
    • Since $OP = OS$, $\triangle POS$ is isosceles: $\angle OPS = \angle OSP = \frac{180^\circ - 110^\circ}{2} = 35^\circ$.
    • Alternate angles yield: $\angle OQR = \angle ORQ = 35^\circ$ and $\angle OSR = \angle ORS = 55^\circ$.
  • Sum of Angles in a Concave Quadrilateral (Figure it Out Q10):
  • Problem: Will the sum of angles in a concave quadrilateral $ADCB$ (dart shape) also be $360^\circ$?
  • Solution: Yes. Draw the diagonal $BD$, which splits the figure into two triangles: $\triangle ADB$ and $\triangle BDC$.
    • Sum of angles of $\triangle ADB = 180^\circ$.
    • Sum of angles of $\triangle BDC = 180^\circ$.
    • Combining these, the sum of all angles in the quadrilateral is $180^\circ + 180^\circ = 360^\circ$.
  • Angles in an Isosceles Trapezium (Page 132 Q3(ii)):
  • Problem: In a trapezium $ABCD$, $\angle D = 100^\circ$ and the non-parallel sides are equal ($AD = BC$). Find the remaining angles.
    • Solution:
    • Because $AD = BC$, it is an isosceles trapezium, meaning the base angles are equal: $\angle C = \angle D = 100^\circ$.
    • Adjacent angles along the transversal of parallel lines are supplementary: $\angle A = 180^\circ - \angle D = 180^\circ - 100^\circ = 80^\circ$.
    • Since the base angles are equal, $\angle B = \angle A = 80^\circ$.
  • Square within a Square Proof (Figure it Out Q7):
  • Problem: In square $CASE$, points $U, V, W,$ and $X$ are the midpoints of the sides. Prove that $UVWX$ is a square.
    • Solution:
    • Let the outer square side be $x$. Midpoints split the sides into segments of length $\frac{x}{2}$.
    • The outer corners form four congruent right-angled triangles (e.g., $\triangle CUV$).
    • By Pythagoras theorem, the hypotenuse is $UV = \sqrt{(\frac{x}{2})^2 + (\frac{x}{2})^2} = \frac{x}{\sqrt{2}}$.
    • Since all four triangles are congruent, $UV = VW = WX = XU = \frac{x}{\sqrt{2}}$, so all four sides of $UVWX$ are equal.
    • In $\triangle CUV$, since $CU = CV$, the base angles are $\angle 1 = \angle 2 = 45^\circ$.
    • In $\triangle E UX$, since $EU = EX$, the base angles are $\angle 3 = \angle 4 = 45^\circ$.
    • Along the straight line $CUE$: $\angle 2 + \angle 5 + \angle 4 = 180^\circ \rightarrow 45^\circ + \angle 5 + 45^\circ = 180^\circ \rightarrow \angle 5 = 90^\circ$.
    • Since all sides are equal and all angles are $90^\circ$, $UVWX$ is a square.
  • Classification True or False (Figure it Out Q11):
  • (i) A quadrilateral whose diagonals are equal and bisect each other must be a square $\rightarrow$ False (It can be a rectangle with unequal adjacent sides).
  • (ii) A quadrilateral having three right angles must be a rectangle $\rightarrow$ True (The 4th angle is $360^\circ - 3 \times 90^\circ = 90^\circ$, making all angles $90^\circ$).
  • (iii) A quadrilateral whose diagonals bisect each other must be a parallelogram $\rightarrow$ True (Bisection of diagonals is a necessary and sufficient condition for a parallelogram).
  • (iv) A quadrilateral whose diagonals are perpendicular must be a rhombus $\rightarrow$ False (It could be a kite with unequal side pairs).
  • (v) Isosceles trapeziums are parallelograms $\rightarrow$ False (Only one pair of opposite sides is parallel, whereas a parallelogram requires both pairs to be parallel).

Practical Activities & Experiments

  • Carpentry Rectangle Assembly: Connect two equal sticks ($8\text{ cm}$) at their midpoints using a pin, allowing them to rotate. Tie a thread connecting the four endpoints. Measure the side lengths and corner angles at various rotations to verify that the shape remains a rectangle, regardless of the angle of intersection.
  • Geoboard Diagonal Configurations: Use a geoboard (or dot grid paper) to build quadrilaterals. Fix rubber bands to represent diagonals and study the resulting boundary shapes when the diagonals are: (1) equal and perpendicular, (2) unequal and perpendicular, and (3) bisecting but unequal.
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