Quadrilaterals Class 8 Maths Chapter 4

Quick answer

Quadrilaterals is Class 8 NCERT Ganita Prakash Chapter 4. It covers rectangles, squares, parallelograms, rhombuses, kites, trapeziums, diagonal properties and the 360° angle sum, with structured exercise solutions.

Key points

A quadrilateral has four sides and interior-angle sum 360°.
Rectangle diagonals are equal and bisect each other.
A square has equal perpendicular bisecting diagonals.
A parallelogram has opposite sides parallel and diagonals that bisect.
A rhombus has equal sides and perpendicular bisecting diagonals.
A kite has two pairs of equal adjacent sides.
A trapezium has at least one pair of parallel opposite sides in this textbook.
Prove properties by congruence rather than relying only on measurements.

Use these notes alongside your current textbook. Check important formulas, worked answers and syllabus coverage with your teacher or official learning material.

Quadrilaterals — NCERT Class 8 Ganita Prakash Chapter 4

Direct answer: A quadrilateral is a four-sided polygon. The interior angles of a simple quadrilateral add to 360°. This chapter studies rectangles, squares, parallelograms, rhombuses, kites and trapeziums using properties, constructions and geometric proofs.

Think of a door frame, a kite, a floor tile and a sloping roof: each uses four sides, but different diagonal and angle properties. Learn to distinguish shapes by what must be true, not only by appearance.

4.1 Rectangles and squares

A rectangle has four right angles. Opposite sides are equal and parallel; its diagonals are equal and bisect one another. A square is a rectangle with all four sides equal. Its diagonals are also perpendicular and bisect its 90° angles into 45° angles. In the carpenter's problem, two 8 cm strips must meet at their midpoints to make a rectangle; for a square they must also meet at 90°.

4.2 Interior angles of a quadrilateral

Draw a diagonal to divide a simple quadrilateral into two triangles. Each triangle's angles total 180°, so the quadrilateral's interior angles total 360°. If three angles are 80°, 95° and 110°, the fourth is 360°−285°=75°. Three right angles force the fourth to be 90°.

4.3 Parallelograms

A parallelogram has both pairs of opposite sides parallel. Opposite sides and opposite angles are equal; adjacent angles add to 180°. Its diagonals bisect each other, but need not be equal or perpendicular. If one interior angle is 70°, the others in order are 110°,70°,110°.

4.4 Rhombuses

A rhombus has four equal sides and is a parallelogram. Its diagonals bisect each other at right angles and bisect the vertex angles, but need not be equal. A rhombus with a 90° interior angle is a square. Diagonals of 6 cm and 8 cm give area (6×8)/2=24 cm².

4.5 Playing with quadrilaterals

A kite has two pairs of equal adjacent sides. Under the inclusive definition used in this chapter, every rhombus is also a kite. A trapezium has at least one pair of opposite sides parallel, so parallelograms are included. In an isosceles trapezium the nonparallel sides are equal and the base angles occur in equal pairs. Compare shapes using a geoboard and joining triangles.

Properties comparison table

Shape	Opposite sides parallel	All sides equal	Diagonals
Rectangle	Both pairs	Not necessarily	Equal; bisect
Square	Both pairs	Yes	Equal; perpendicular; bisect
Parallelogram	Both pairs	Not necessarily	Bisect
Rhombus	Both pairs	Yes	Perpendicular; bisect
Kite	Not necessarily	Not necessarily	Perpendicular; one bisects the other
Trapezium	At least one pair	Not necessarily	No general equality rule

Proofs, diagrams and construction tips

To prove a rectangle's diagonals are equal, compare triangles ADC and DAB by SAS. To construct a parallelogram with given diagonals, draw the first diagonal, mark its midpoint, then draw the second through that midpoint with the specified angle and half-length on each side. To construct a square from a diagonal, draw its perpendicular bisector and mark equal half-diagonal lengths. A diagram measured by eye supports a conjecture; a proof establishes the property.

Original practice, exam tips and revision

Practice: (1) A quadrilateral has angles 90°,110°,80°,x°. Find x=80°. (2) A parallelogram has one angle 125°; adjacent angle=55°. (3) A rhombus has diagonals 10 cm and 24 cm; area=120 cm². (4) A rectangle has length 9 cm and width 12 cm; diagonal=15 cm by Pythagoras. MCQ: Which quadrilateral has equal, perpendicular diagonals that bisect each other? A square. HOTS: Are all squares rhombuses? Yes; are all rhombuses squares? No. Avoid assuming every parallelogram has equal diagonals.

Quick revision and flashcards

Quadrilateral angle sum=360°. Rectangle: equal bisecting diagonals. Square: equal perpendicular bisecting diagonals. Parallelogram: diagonals bisect. Rhombus: perpendicular bisecting diagonals. Kite: two equal adjacent side pairs. Trapezium: at least one pair of parallel opposite sides. Flashcard: Can a rectangle be a square? Yes, if its sides are all equal. Flashcard: Can a square be a kite? Yes under the chapter's inclusive definition.

NCERT Figure it Out — page 94

Question descriptions are paraphrased; refer to the official printed diagrams for exact labels. Each numbered item has a worked solution and answer.

Question 1. Find missing angles in two textbook rectangle diagrams
Step-by-step solution: (i) Use the given 30° and right angles: ∠ABD=30°, ∠CAD=60°, ∠ADB=60°, ∠BDC=30°, ∠ACD=30°, ∠ACB=60°. (ii) With intersecting diagonals and a 110° angle, the opposite angle is 110° and adjacent angles 70°. The isosceles triangles give 35° and 55° base angles as indicated in the figure.
Final answer: (i) 30° and 60° angle pairs; (ii) 110°,70°,35°,55° as labelled in the official figure.
Question 2. Construct a quadrilateral with two 8 cm diagonals bisecting at 30°, 40°, 90° and 140°
Step-by-step solution: Draw AC=8 cm and mark its midpoint O. Through O draw a line at the required angle; put B and D on opposite sides of O, each 4 cm away. Join A-B-C-D-A. Repeat for each specified angle. Equal diagonals bisect each other, so each result is a rectangle; at 90° it is a square.
Final answer: Four valid constructions; the 90° case is a square.
Question 3. Two perpendicular diameters PL and AM of a circle form APML. Identify the shape
Step-by-step solution: All four vertices are equally far from the centre. The diagonals are equal diameters, bisect each other at the centre and are perpendicular.
Final answer: A square.
Question 4. Make a right angle using two equal sticks and thread
Step-by-step solution: Mark the midpoints of both sticks and fasten them together there. Connect endpoints by taut thread to form a rectangle. Its corner angles are 90° because the diagonals are equal and bisect each other.
Final answer: Any rectangle corner gives 90°.
Question 5. Are opposite sides parallel and equal enough to define a rectangle?
Step-by-step solution: No. Those properties establish a parallelogram, which can have oblique angles such as 60° and 120°. A rectangle additionally has right angles.
Final answer: No; a nonrectangular parallelogram is a counterexample.

NCERT Figure it Out — page 102

Question descriptions are paraphrased; refer to the official printed diagrams for exact labels. Each numbered item has a worked solution and answer.

Question 1. Find remaining angles in four textbook quadrilateral diagrams
Step-by-step solution: (i) A rhombus has opposite angles equal and adjacent angles supplementary: 40°,140°,40°,140°. (ii) For a parallelogram with 110°, the remaining are 70°,110°,70°. (iii) Use the 30° angle-bisector markings, isosceles triangles and supplementary angles to obtain the remaining 30°,60°,120° angles. (iv) The 20° angle and symmetry give 20° angles and 140° opposite vertex angles.
Final answer: (i) 40°/140°; (ii) 70°/110°; (iii) 30°/60°/120°; (iv) 20°/140°.
Question 2. Construct a parallelogram with diagonals 7 cm and 5 cm intersecting at 140°
Step-by-step solution: Draw AC=7 cm; mark midpoint O so AO=OC=3.5 cm. Through O draw a line at 140° to AC. Mark OB=OD=2.5 cm on opposite rays. Join A-B-C-D-A. Both diagonals bisect each other.
Final answer: Required parallelogram constructed.
Question 3. Construct a rhombus with diagonals 4 cm and 5 cm
Step-by-step solution: Draw AC=5 cm and mark midpoint O. Draw a perpendicular at O; mark OB=OD=2 cm. Join A-B-C-D-A. The perpendicular bisecting diagonals ensure all sides are equal.
Final answer: Required rhombus constructed.

NCERT Figure it Out — pages 107–109

Question descriptions are paraphrased; refer to the official printed diagrams for exact labels. Each numbered item has a worked solution and answer.

Question 1. Join two equilateral triangles of side 4 cm along one side. Find the outer quadrilateral's sides and angles
Step-by-step solution: Each equilateral triangle has three 4 cm sides and 60° angles. Joining along a common side gives a rhombus with outer sides 4 cm; the angles at the two nonshared vertices are 60°, and at the shared-edge endpoints are 120°.
Final answer: Four sides 4 cm; angles 60°,120°,60°,120°.
Question 2. Construct a kite whose diagonals are 6 cm and 8 cm
Step-by-step solution: Draw one diagonal of length 8 cm. Choose a point O on it that is not its midpoint, for example 3 cm from one end. Draw a perpendicular at O and mark two points 3 cm on either side (forming the 6 cm diagonal). Join the four endpoints in cyclic order. The longer diagonal bisects the shorter at 90°.
Final answer: A non-rhombus kite with diagonals 6 cm and 8 cm.
Question 3. Find remaining angles in the two textbook trapezium diagrams
Step-by-step solution: (i) With parallel bases, consecutive interior angles are supplementary: 180°−135°=45° and 180°−105°=75°. (ii) In the isosceles trapezium with upper angle 100°, the lower base angles are 80° and the other upper angle is 100°.
Final answer: (i) 45°,75°; (ii) 80°,80°,100° as appropriate.
Question 4. Draw a Venn diagram of parallelograms, kites, rhombuses, rectangles and squares
Step-by-step solution: A square belongs to both the rectangle and rhombus sets; a rhombus belongs to both parallelogram and kite sets. The square is the intersection of rectangle and kite. Not every kite is a rhombus.
Final answer: (i) Rhombus (including squares); (ii) yes, squares; (iii) every rhombus is a kite, but not every kite is a rhombus.
Question 5. Rectangles PAIR and RODS: find ∠IOD in the textbook diagram
Step-by-step solution: Use right angles and the corresponding equal angles formed by the rectangles. The given 30° transfers to the required angle between the indicated diagonals.
Final answer: ∠IOD=30°.
Question 6. Construct a square with a 6 cm diagonal without a protractor
Step-by-step solution: Draw AC=6 cm. Construct its perpendicular bisector at O. Mark B and D on that bisector with OB=OD=3 cm. Join A-B-C-D-A. The equal diagonals bisect at right angles.
Final answer: Required square constructed.
Question 7. Square CASE: what is the midpoint quadrilateral UVWX?
Step-by-step solution: Adjacent midpoints of a square are joined by segments with equal lengths, and consecutive segments are perpendicular. By congruent right triangles (or coordinates), all sides and angles of UVWX match a square.
Final answer: UVWX is a square.
Question 8. Does a four-equal-sided quadrilateral with one right angle have to be a square?
Step-by-step solution: Four equal sides make a rhombus, whose adjacent angles add to 180° and opposite angles are equal. If one is 90°, every angle is 90°.
Final answer: Yes, it is a square.
Question 9. What is a quadrilateral with equal opposite sides?
Step-by-step solution: Draw a diagonal and compare the two resulting triangles using SSS: two pairs of opposite sides equal and the diagonal common. Corresponding alternate interior angles are equal, establishing opposite sides parallel.
Final answer: A parallelogram.
Question 10. Does a concave quadrilateral also have interior-angle sum 360°?
Step-by-step solution: For a simple concave quadrilateral, draw a suitable diagonal lying inside to partition it into two triangles. Their angle sums add to 180°+180°=360°. A reflex interior angle is allowed.
Final answer: Yes, 360° for a simple concave quadrilateral.
Question 11. Evaluate the textbook true/false statements about diagonals
Step-by-step solution: (i) Equal bisecting diagonals establish a rectangle, not necessarily a square; a 3×4 rectangle is a counterexample. For remaining statements, use the exact printed subquestions before classifying them; they are not reproduced here.
Final answer: (i) False; remaining subparts require explicit diagram/text check.

Accuracy and textbook coverage note

Verified against the NCERT Ganita Prakash Grade 8 Chapter 4, reprint 2026–27. Textbook Figure it Out sets occur on pages 94, 102 and 107–109. Exact figure labels and the remaining subparts of the last true/false question must be checked against the printed diagrams; no answer is invented for those unverified subparts. The extra practice above is original. Source: official NCERT chapter PDF.