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USP IX Quadrilaterals Test Paper with Solutions

Class IX Maths 📋 Sample Paper Ch 12: Quadrilaterals NCERT book: Ganita Manjari

Updated 22 Oct 2026

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Class 9 Maths Ganita Manjari Chapter 12: Quadrilaterals – Test Paper with Solutions

Part II • CBSE 2026-27 • 40 Marks • 60 Minutes

18 QuestionsStep-by-step solutionsFree

This free practice test paper with solutions for Class 9 Maths Chapter 12, Quadrilaterals (NCERT Ganita Manjari, Part II, session 2026-27) has 18 questions for 40 marks and is made to be finished in about 60 minutes. The paper has MCQs, short answer questions, a case study and long answer questions. Try every question on your own first. Then tap Show Answer to see the full working, with the marks for each step, and compare it with your own solution.

What this test paper covers

Quadrilaterals: angle sum, naming and convex quadrilaterals
Parallelograms: properties and the tests that prove a quadrilateral is a parallelogram
Midpoint Theorem: the theorem and its converse
Medians and the centroid, and the Varignon parallelogram
Tiling the plane: why the angles at a vertex must add up to 360 degrees

How the paper is arranged

Section	Type of questions	Questions	Marks each	Total
A	Multiple choice questions	8	1	8
B	Short answer questions	3	2	6
C	Short answer questions	4	3	12
D	Case study	1	4	4
E	Long answer questions	2	5	10
Total		18		40

Try each question first, then tap Show Answer to check your working.

SECTION AMultiple Choice Questions (1 mark each)8 marks

The angles of a quadrilateral are 95°, 85°, 110° and x°. The value of x is:

1 mark

A60

B65

C70

D80

Two adjacent angles of a parallelogram are in the ratio 4 : 5. The smaller angle is:

1 mark

A70°

B80°

C90°

D100°

The diagonals of parallelogram PQRS meet at O. If QS = 26 cm, then OQ is:

1 mark

A13 cm

B26 cm

C52 cm

D6.5 cm

Which of these is NOT enough to prove that a quadrilateral is a parallelogram?

1 mark

Aboth pairs of opposite sides are equal

Bthe diagonals bisect each other

Cone pair of opposite sides is equal and parallel

Done pair of opposite sides is parallel

D and E are the midpoints of the sides AB and AC of triangle ABC. If DE = 9 cm, then BC is:

1 mark

A4.5 cm

B18 cm

C27 cm

D9 cm

The centroid G of triangle ABC lies on the median BE, and BG = 10 cm. Then GE is:

1 mark

A10 cm

B20 cm

C5 cm

D15 cm

The midpoints of the sides of a quadrilateral whose diagonals are equal and perpendicular form a:

1 mark

Arectangle

Brhombus

Ctrapezium

Dsquare

The plane can be tiled with copies of any quadrilateral because:

1 mark

Aits four angles add up to 360°

Ball its sides are equal

Cits diagonals are equal

Dits opposite sides are parallel

SECTION BShort Answer Questions (2 marks each)6 marks

The angles of a quadrilateral are (x + 20)°, (2x − 10)°, (x + 50)° and 2x°. Find x and the four angles.

2 marks

The diagonals of parallelogram ABCD meet at O. AO = (3x − 2) cm and OC = (x + 8) cm. Find x and the length of AC.

2 marks

The quadrilateral ABCD can also be named BCDA. (a) Write two other names for it. (b) Why can it not be named ACBD?

2 marks

SECTION CShort Answer Questions (3 marks each)12 marks

Three angles of a quadrilateral are 75°, 105° and 95°.

3 marks

(a)Find the fourth angle.[2]

(b)Is the quadrilateral convex? Give a reason.[1]

In parallelogram ABCD, the bisector of ∠A meets the side DC at E. Show that DE = DA.

3 marks

In triangle ABC, M is the midpoint of AB. The line through M parallel to BC meets AC at N. If BC = 11 cm and AC = 14 cm, find MN and AN. Name the theorems you use.

3 marks

In quadrilateral PQRS, PQ = RS = 6 cm and QR = SP = 4 cm.

3 marks

(a)Name the figure PQRS and the test you use.[1]

(b)If ∠P = 65°, find ∠Q, ∠R and ∠S.[2]

SECTION DCase Study (4 marks)4 marks

A farmer’s field is the quadrilateral ABCD, with diagonals AC = 80 m and BD = 60 m. He puts poles P, Q, R and S at the midpoints of the sides AB, BC, CD and DA, and joins the poles with a rope to make a smaller field PQRS.

4 marks

(not to scale)

(i)What kind of figure is PQRS?[1]

(ii)Find the lengths PQ and QR.[1]

(iii)Find the length of rope needed to go once round PQRS.[1]

(iv)If the diagonals AC and BD are perpendicular, what special figure is PQRS?[1]

SECTION ELong Answer Questions (5 marks each)10 marks

The diagonals of a rhombus ABCD are AC = 16 cm and BD = 12 cm. They meet at O.

5 marks

(a)What is ∠AOB, and why?[1]

(b)Find AO and BO.[1]

(c)Find the side AB.[2]

(d)Find the perimeter of the rhombus.[1]

Answer and working

(a) ∠AOB = 90°, because the diagonals of a rhombus bisect each other at right angles.

1 mark

(b) AO = 12 × 16 = 8 cm and BO = 12 × 12 = 6 cm.

1 mark

AB = 10 cm.

1 mark

(d) All four sides are equal, so the perimeter = 4 × 10 = 40 cm.

1 mark

Answer: (a) 90° (b) 8 cm and 6 cm (c) 10 cm (d) 40 cm

Answer these questions on tiling the plane.

5 marks

(a)The angles of a quadrilateral are 85°, 95°, 75° and 105°. Show that four copies of it can fit around a point with no gap and no overlap.[2]

(b)Name one method used to tile the plane with any quadrilateral.[1]

(c)Each angle of a regular polygon is 144°. Can copies of it tile the plane? Give a reason using the angles at a vertex.[2]

Answer and working

(a) 85 + 95 + 75 + 105 = 360. Place one angle of each of four copies around the point, using each angle once. They add up to 360°, so there is no gap and no overlap.

2 marks

(b) Rotate a copy by 180° about the midpoint of a side (it fits exactly along that side), or place copies on alternate parallelograms of the Varignon grid.

1 mark

(c) No. The angles at a vertex must add up to 360°, but 360 ÷ 144 = 2.5 is not a whole number. Two angles give 288° (a gap) and three give 432° (an overlap).

2 marks

Answer: (a) shown (b) rotate about a side’s midpoint (c) no

Key ideas from Chapter 12

Quadrilaterals. The four angles of every quadrilateral add up to 360°. A quadrilateral is convex when every angle is less than 180°. Name it by going round the figure, for example ABCD or BCDA, never ACBD.

Parallelograms. Opposite sides are equal, opposite angles are equal, adjacent angles add up to 180° and the diagonals bisect each other. A quadrilateral is a parallelogram if both pairs of opposite sides are equal, both pairs of opposite angles are equal, the diagonals bisect each other, or one pair of opposite sides is equal and parallel. One pair of parallel sides alone is not enough.

Midpoint Theorem. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. Its converse: a line through the midpoint of one side, parallel to another side, bisects the third side.

Centroid and Varignon parallelogram. The three medians meet at the centroid, which divides each median 2 : 1 from the vertex. The midpoints of the sides of any quadrilateral form a parallelogram whose perimeter is AC + BD. It is a rhombus if AC = BD, a rectangle if AC ⊥ BD, and a square if both.

Tiling. The angles at each vertex of a tiling add up to 360°. Any quadrilateral tiles the plane because its four angles add up to 360°.

Common mistakes to avoid

Treating one pair of parallel sides as enough to prove a parallelogram.
Taking the centroid ratio as 1 : 2 from the vertex. The longer part is towards the vertex.
Thinking the midpoint figure of a rectangle is a rectangle. It is a rhombus.

Frequently asked questions

What is a convex quadrilateral?

A quadrilateral is convex when every internal angle is less than 180 degrees. Its two diagonals intersect. A quadrilateral with a dent, where one angle is more than 180 degrees, is non-convex.

How do I prove that a quadrilateral is a parallelogram?

Show any one of these: both pairs of opposite sides are equal, both pairs of opposite angles are equal, the diagonals bisect each other, or one pair of opposite sides is both equal and parallel.

What does the Midpoint Theorem say?

If P and Q are the midpoints of two sides of a triangle, then PQ is parallel to the third side and equal to half of it.

What is the centroid of a triangle?

The centroid is the point where the three medians meet. It divides each median in the ratio 2 : 1, with the longer part towards the vertex.

Why can any quadrilateral tile the plane?

The four angles of any quadrilateral add up to 360 degrees. So four copies can be placed around a point using each angle once, with no gap and no overlap.

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