Class 9 Maths Ganita Manjari Chapter 12 Exercise 12.1 Solutions Quadrilaterals

The Class 9 Maths Ganita Manjari Chapter 12 Exercise 12.1 Solutions – Quadrilaterals help students understand the basic properties, terminology and structure of quadrilaterals. This exercise focuses on concepts such as adjacent and opposite sides, adjacent and opposite angles, internal angles, convex and non-convex quadrilaterals, and conditions that determine the shape of a quadrilateral.

The solutions are explained in a clear, step-by-step manner to help students build strong geometric reasoning and write complete mathematical answers with confidence. Students can also use the Class 9 Maths Ganita Manjari Chapter 12 Exercise 12.1 Solutions PDF for quick revision and offline study.

Class 9 Maths Ganita Manjari Chapter 12 Exercise 12.1 Solutions Quadrilaterals

NCERT Class 9 Maths Ganita Manjari Chapter 12 Exercise Set 12.1

1. Let ABCD be a quadrilateral.

(i) List all sides of ABCD adjacent to side AB. List all sides opposite to AB.
(ii) List all angles of ABCD adjacent to ∠A. List all angles opposite to ∠A.
(iii) Define a pair of opposite sides and a pair of opposite angles without using the names of the vertices.

Answer: 

1. Let ABCD be a quadrilateral.

(i) List all sides of ABCD adjacent to side AB. List all sides opposite to AB.

The sides adjacent to AB are AD and BC.

The side opposite to AB is CD.

(ii) List all angles of ABCD adjacent to ∠A. List all angles opposite to ∠A.

The angles adjacent to ∠A are ∠B and ∠D.

The angle opposite to ∠A is ∠C.

(iii) Define a pair of opposite sides and a pair of opposite angles without using the names of the vertices.

A pair of opposite sides of a quadrilateral are two sides that do not share a common vertex.

A pair of opposite angles are two angles that are not adjacent, that is, they do not share a common side.

2. You have used internal angles of quadrilaterals, but they too require an exact definition, just like how we gave one for a quadrilateral. Precisely define the internal angle of a quadrilateral at a given vertex. Your answer should work for a non-convex quadrilateral too.

(Hint: use the opposite vertex as well.)

Answer:

Consider a quadrilateral ABCD and focus on vertex A. Let θ be the smaller angle formed by the two sides AB and AD.

  • If A lies outside triangle BCD, then the internal angle at A is simply θ.
  • If A lies inside triangle BCD, then A is the concave vertex, so the internal angle is the reflex angle 360° − θ.

The same method can be used at any other vertex of the quadrilateral.

Thus, the internal angle is the angle between the two sides meeting at a vertex that lies inside the region of the quadrilateral. It is less than 180° at a convex vertex and greater than 180° at a concave vertex.

3. In a quadrilateral ABCD, suppose AB ∥ DC. Can ABCD be non-convex? What if we instead assume AB = CD? What if we instead assume ∠A = ∠C?

Answer:

(i) If AB ∥ DC

No. A simple quadrilateral ABCD cannot be non-convex if the opposite sides AB and DC are parallel.

Since AB and DC lie on parallel lines, the other two sides AD and BC must join the endpoints without crossing. This gives a convex quadrilateral, so no vertex can form a reflex angle.

Therefore, AB ∥ DC implies that ABCD is convex.

(ii) If AB = CD

Yes. A quadrilateral can still be non-convex even if the opposite sides AB and CD are equal.

For example, take:

A(0, 0), B(2, 0), C(1, 1), D(1, 3).

Here, AB = CD = 2 units, but C is an inward-pointing vertex, so ABCD is non-convex.

Therefore, AB = CD does not guarantee that ABCD is convex.

(iii) If ∠A = ∠C

Yes. A quadrilateral can have equal opposite angles and still be non-convex.

For example, take:

A(−2, 0), B(0, 1), C(2, 0), D(0, 3).

The figure is symmetric, so ∠A = ∠C, but B is an inward-pointing vertex and forms a reflex angle.

Therefore, ∠A = ∠C does not guarantee that ABCD is convex.

Hence:

  • AB ∥ DC: ABCD must be convex.
  • AB = CD: ABCD may be non-convex.
  • ∠A = ∠C: ABCD may be non-convex.

4. Consider three non-collinear points A, B, C and draw the lines AB, BC, CA. For every possible location of point D in the plane outside these lines, decide if ABCD is self-intersecting, non-convex, or convex.

(Hint: the three lines divide the plane into 7 regions.)
Answer: 

The three lines AB, BC and CA divide the plane into 7 regions. The type of quadrilateral ABCD depends on which region contains D.

Position of D Nature of ABCD Reason
Inside triangle ABC Non-convex D becomes the inward-pointing vertex, so ∠D is reflex.
Across side AC, on the side opposite B Convex The vertices A, B, C, D occur around the boundary without any inward angle or crossing.
Across side AB, on the side opposite C Self-intersecting Side CD crosses side AB.
Across side BC, on the side opposite A Self-intersecting Side AD crosses side BC.
Exterior corner region near A Non-convex ∠A becomes a reflex angle.
Exterior corner region near B Non-convex ∠B becomes a reflex angle.
Exterior corner region near C Non-convex ∠C becomes a reflex angle.

Thus, among the seven regions:

  • 1 region gives a convex quadrilateral.
  • 2 regions give a self-intersecting quadrilateral.
  • 4 regions give a non-convex quadrilateral.

The asymmetry occurs because the vertices are joined specifically in the order A → B → C → D → A. Since D is adjacent to both A and C, placing D across AC produces the convex case, whereas placing it across AB or BC causes two sides to intersect.

5. Can a quadrilateral be both self-intersecting and non-planar?

Answer: 

No. A quadrilateral cannot be both self-intersecting and non-planar.

If a quadrilateral is self-intersecting, a pair of its non-adjacent sides must cross each other. Two intersecting lines always lie in the same plane. Since all four vertices lie on those two intersecting sides, all four vertices must lie in that same plane.

Therefore, a self-intersecting quadrilateral is necessarily planar.

Hence, a quadrilateral cannot be both self-intersecting and non-planar.

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