4. Quadrilaterals

Part I, printed Chapter 410 topics2 h

Chapter 4 of NCERT Mathematics for Class 8 (Part I, printed Chapter 4): Quadrilaterals. Four sections — Rectangles and squares, Angles, and sides that stay parallel, Sides that stay equal and Building and naming quadrilaterals. Ten videos, 2 h in all.

Worked answers to the 19 exercise questions in this chapter →

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Part 1

Rectangles and squares

9 minThe carpenter's problem: how to be sure a frame really is rectangularA carpenter making a rectangular frame never has to measure a corner. That is not a saving of effort — it is the only way to be sure.9 min · 12 partsहिंदी10 minHow a property gets deduced, and why you still check it against a real shapeDraw a rectangle, draw both diagonals, measure where they cross. Do it a thousand times and you still have not proved anything.10 min · 12 partsहिंदी10 minProperties of a rectangle, and why the square is the special caseA rectangle is a four-sided shape whose corners are all square. That is the whole definition — everything else has to be earned.10 min · 12 partsहिंदी

Part 2

Angles, and sides that stay parallel

10 minWhy the angles of any quadrilateral add to 360°Can a four-sided figure have three square corners and a fourth that is not? Almost everybody says yes, then finds the last side will not close.10 min · 12 partsहिंदी10 minThe parallelogram: everything that follows from "opposite sides parallel"Push a rectangle over and most of its habits survive. Exactly one does not, and that one is the whole difference.10 min · 12 partsहिंदी

Part 3

Sides that stay equal

10 minThe rhombus, and what its diagonals doFour equal sides says nothing about direction — you could check it with a piece of string. And yet it forces the diagonals to cross square.10 min · 12 partsहिंदी

Part 4

Building and naming quadrilaterals

9 minWhat the diagonals alone tell you about a quadrilateralLay one segment across another and you have already drawn a quadrilateral. The four ends are its corners; nothing is left to choose.9 min · 12 partsहिंदी10 minWhich quadrilaterals you can build by joining two trianglesPress two identical triangles together along a matching edge and that edge stops being an edge. It becomes a diagonal.10 min · 12 partsहिंदी10 minThe kite: why one diagonal bisects the other at right angles, and the angles tooA kite assumes almost nothing: two separate pairs of neighbouring sides equal. No angle, no parallel pair, and yet a right angle comes out.10 min · 12 partsहिंदी10 minThe trapezium: what a single pair of parallel sides forces, and what "isosceles" addsA trapezium is the least parallelism you can ask for and still be given a name. Let one pair go and see what survives.10 min · 12 partsहिंदी

What you will be able to do

  • State the carpenter's three questions about the two strips — length, joining point, crossing angle — and say which of the three turns out not to matter
  • State what a conjecture is, and give the chapter's own reason why observing a property in many drawn figures does not settle it
  • State the four numbered properties of a rectangle in the chapter's order, and for each one name the argument it came from
  • State the question the section opens with: can a quadrilateral have three right angles and a fourth angle that is not a right angle?
  • Construct a parallelogram from two adjacent sides and the angle between them
  • Construct a quadrilateral with four equal sides that is not a square, starting from two equal sides at a chosen angle and cutting arcs of that same length
  • Explain why fixing the two diagonals fixes the quadrilateral, and hence why diagonal conditions can serve as definitions
  • Join two congruent cardboard triangles along a matching edge and name the quadrilateral produced, with a justification
  • State the definition of a kite, and explain why the labelling matters — the equal pairs are neighbouring sides, not opposite ones
  • State the definition of a trapezium and explain what the words "at least" do

What this chapter assumes you already know

  • The three congruence conditions for triangles by name — SAS, ASA, AAS — and the habit of concluding that matching parts of congruent triangles are equal
  • That the three angles of a triangle total 180°
  • Angles at a point: vertically opposite angles are equal, and two angles on a straight line total 180°
  • That a triangle with two equal sides has two equal angles facing them
  • Reading a labelled diagram: vertices, diagonals, the point where diagonals meet
  • The everyday definition of a rectangle as a four-sided figure with square corners and matching opposite sides (Part I, §4.1, p.83)
  • The congruence conditions by name, especially AAS, and the habit of naming matching parts in matching order
  • Constructing a quadrilateral with a ruler, a compass and a set-square, and measuring sides and angles well enough to compare them

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "You need a protractor to get a right angle."
  • "Equal diagonals are enough."
  • "Diagonals that bisect each other are enough."
  • "The crossing angle must be 90°."
  • "We proved rectangles have these diagonals, so we're done."
  • "I built five and they all came out right, so it works."
  • "If I cannot draw it, it is impossible."
  • "If I measured it and it worked, it is proved."