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NCERT Solutions Ganita Prakash (Part 1) Chapter 4 –994.3 More Quadrilaterals with Parallel Opposite Sides — In-text Questions

Book page 95 Updated on2026-09-05

Q1.
Rectangles (and therefore squares) have parallel opposite sides. Are there quadrilaterals that have parallel opposite sides that are not rectangles?
Answer

Yes, plenty. Draw two pairs of parallel lines that do not cross at right angles, and the quadrilateral they cut out has both pairs of opposite sides parallel while none of its angles is 90°.

Such quadrilaterals are called parallelograms.

Parallelogram: a quadrilateral in which opposite sides are parallel.
Why rectangles do not exhaust the list: being parallel is about direction only. Two families of parallel lines can meet at any angle at all; 90° is just one choice out of infinitely many. Every other choice gives a parallelogram that is not a rectangle.
Tip: so the set of parallelograms is larger than the set of rectangles. In the Venn diagram, the rectangle oval sits entirely inside the parallelogram oval.
Q2.
Construct such a figure by recalling how parallel lines can be constructed using a ruler and a set-square, or a compass and a ruler.
Answer

With a ruler and a set-square.

  1. Draw a line l. Slide the set-square along the ruler and draw a second line m parallel to l.
  2. Draw a third line p crossing both, at any angle other than 90°.
  3. Slide the set-square along p's direction and draw a fourth line q parallel to p.
  4. The four crossing points are the vertices of a parallelogram.

With a compass and a ruler — copy an angle instead. Draw line l and a transversal through a point P on it. Copy the angle that l makes with the transversal, at a point Q further along it. The new arm is parallel to l, because equal corresponding angles mean parallel lines.

Why the copied-angle method works: it is the converse of the corresponding-angles property. If a transversal makes equal corresponding angles with two lines, those lines can never meet — so they are parallel. The set-square method is the same idea in one motion: sliding keeps the angle with the ruler constant.
Q3.
Is a rectangle a parallelogram?
Answer

Yes. A rectangle has both pairs of opposite sides parallel (proved on page 90), which is exactly the parallelogram's definition.

More precisely, a rectangle is a special kind of parallelogram — one with all its angles equal to 90°.

StatementTrue?
Every rectangle is a parallelogramYes
Every parallelogram is a rectangleNo
Every square is a parallelogramYes
Tip: in the Venn diagram this becomes three nested regions — square inside rectangle inside parallelogram. Reading “inside” as “every ... is a ...” keeps the whole family straight.
Q4.
Draw a parallelogram with adjacent sides of lengths 4 cm and 5 cm, and an angle of 30° between them. What are the remaining angles of the parallelogram? What are the lengths of the remaining sides?
Answer

Angles: 30°, 150°, 30°, 150°. Sides: 4 cm, 5 cm, 4 cm, 5 cm.

Construction. Draw AB = 4 cm and AD = 5 cm with ∠A = 30° between them. Through D draw a line parallel to AB, and through B a line parallel to AD; call their meeting point C.

Angles (Deduction 6). AB ∥ CD with AD as transversal, so ∠A and ∠D are interior angles on the same side:

∠A + ∠D = 180°  ⇒ ∠D = 180 – 30 = 150°
∠A + ∠B = 180°  ⇒ ∠B = 150°
∠C + ∠D = 180°  ⇒ ∠C = 30°
Check: 30 + 150 + 30 + 150 = 360° ✓

Sides (Deduction 7). Compare ∆ABD and ∆CDB: the angles marked with one arc are equal (opposite angles of the parallelogram), the angles marked with two arcs are equal (alternate angles, since AD ∥ BC with BD as transversal), and BD is common.

∆ABD ≅ ∆CDB  (AAS)
⇒ AD = CB = 5 cm  and  AB = CD = 4 cm
Why adjacent angles must add to 180°: the two sides you started from are cut by a transversal that joins two parallel sides. Interior angles on the same side of a transversal always add to a straight angle — so knowing one angle of a parallelogram tells you all four.
Q5.
What about the opposite angles? Will they be equal in all parallelograms? If yes, how can we be sure? Let us take one of the angles to be x. What are the other angles?
Answer

Yes — the opposite angles of a parallelogram are always equal. Using a letter instead of a number proves it for every parallelogram at once.

In parallelogram PEAR (vertices in order P, E, A, R), take ∠P = x.

∠P + ∠R = 180°  ⇒ ∠R = 180 – x
∠A + ∠R = 180°  ⇒ ∠A = 180 – (180 – x) = 180 – 180 + x = x
So ∠P = ∠A = x
Similarly ∠R = ∠E = 180 – x

Check: x + (180 – x) + x + (180 – x) = 360° ✓

Why the two supplements cancel: ∠A is supplementary to ∠R, and ∠R is supplementary to ∠P. Taking the supplement twice brings you back where you started, so ∠A must equal ∠P. The x's cancel in the algebra for exactly that reason.
Tip: this gives Property 3 — in a parallelogram, adjacent angles add up to 180° and opposite angles are equal. One measured angle is therefore enough to fill in all four.
Q6.
Deduction 7 — What can we say about the sides of a parallelogram? Can we again use congruence to show this? Which two triangles can be considered for this?
Answer

Use ∆ABD and ∆CDB — the two halves cut off by the diagonal BD.

∠A = ∠C  (opposite angles of a parallelogram, from Deduction 6)
∠ADB = ∠CBD  (alternate angles, since AD ∥ BC and BD is a transversal)
BD = DB  (common side)
⇒ ∆ABD ≅ ∆CDB by the AAS condition
⇒ AD = CB and AB = CD

So the opposite sides of a parallelogram are equal — Property 1.

Why a diagonal is the natural tool: the sides you want to compare are not in the same triangle until you draw one. The diagonal creates two triangles that between them contain all four sides, and the parallel lines immediately hand you a pair of equal alternate angles.
Tip: notice the logical order — Deduction 6 (angles) is used inside Deduction 7 (sides). Once a property is proved, it becomes a tool for the next proof.
Q7.
Is it wrong to write ∆ABD ≅ ∆CBD? Why?
Answer

Yes, it is wrong. The vertices are not in corresponding order.

The correct statement is ∆ABD ≅ ∆CDB, which matches A↔C, B↔D, D↔B. Writing ∆ABD ≅ ∆CBD would match A↔C, B↔B, D↔D, and so would claim:

AB = CB  and  ∠ABD = ∠CBD

Neither of these is true in a general parallelogram — AB and CB are adjacent sides of different lengths, and BD does not bisect the angle at B.

Why order is not a formality: the whole value of a congruence is the list of equal parts you can read off it (CPCT). If the letters are in the wrong order, that list is a list of false statements, and any later step built on it collapses.
Q8.
Are the diagonals of a parallelogram always equal? Check with the parallelogram that you have constructed.
Answer

No. In a general parallelogram the diagonals have different lengths.

In the 4 cm by 5 cm parallelogram with ∠A = 30°, the diagonals come out very different — the one across the 150° corners is far longer than the one across the 30° corners. Measure both in your own drawing and you will find they never agree unless the angles are 90°.

ParallelogramDiagonals equal?
General parallelogramNo
RhombusNo (unless it is a square)
RectangleYes
SquareYes
Why equal diagonals mean a rectangle: Deduction 3 showed that equal diagonals which bisect each other force all four angles to be 90°. Since the diagonals of every parallelogram bisect each other, adding “equal” is enough to turn a parallelogram into a rectangle. So the diagonals of a parallelogram are equal precisely when it is a rectangle.
Q9.
Do they bisect each other (do they intersect at their midpoints)? Reason and/or experiment to figure this out.
Answer

Yes — the diagonals of a parallelogram always bisect each other, even though they need not be equal.

Deduction 8. In parallelogram EASY the diagonals meet at O. Compare ∆AOE and ∆YOS:

AE = YS  (opposite sides of the parallelogram)
∠AEO = ∠YSO  (alternate angles, AE ∥ YS)
∠EAO = ∠SYO  (alternate angles)
⇒ ∆AOE ≅ ∆YOS by the ASA condition
⇒ OA = OY and OE = OS

So O is the midpoint of both diagonals — Property 4.

Why this works without equal diagonals: the congruence uses only one pair of equal sides and the alternate angles that parallelism guarantees. It says the two halves of the figure across O are copies of each other, so each diagonal is cut into two equal pieces — but it says nothing about comparing one diagonal with the other.
Q10.
Is it wrong to write ∆AOE ≅ ∆SOY? Why?
Answer

Yes. The correct statement is ∆AOE ≅ ∆YOS.

Correct: ∆AOE ≅ ∆YOSWrong: ∆AOE ≅ ∆SOY
A ↔ Y, O ↔ O, E ↔ SA ↔ S, O ↔ O, E ↔ Y
gives OA = OY, OE = OS ✓would give OA = OS, OE = OY ✗

OA and OS are halves of different diagonals, and there is no reason for them to be equal — in a general parallelogram they are not.

Tip: to get the order right, look at which vertex of one triangle plays the same role as which vertex of the other. A is opposite the side EO; the vertex opposite SO is Y, so A must be paired with Y.
Q11.
Do the diagonals of a parallelogram intersect at a particular angle?
Answer

No — the angle between them can be anything, and it changes as the shape of the parallelogram changes.

QuadrilateralAngle between the diagonals
General parallelogramany value; not fixed
Rectangleany value (still not fixed)
Rhombusalways 90°
Squarealways 90°

Being perpendicular is what the equal sides buy you, not what parallelism buys you. That is exactly the question section 4.4 goes on to answer, and Deduction 10 settles it: the diagonals of a rhombus intersect at 90°.

Why sides control this angle: if all four sides are equal, each vertex is the same distance from the two vertices next to it, so every vertex lies on the perpendicular bisector of a diagonal. That forces the diagonals to be perpendicular. Unequal sides put the vertices off those bisectors, and the angle is free again.
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