Ex 12.1 Class 9 Maths Ganita Manjari Solutions | Sid Classes
Welcome to Sid Classes, your trusted destination for comprehensive academic resources! In this detailed study guide, we present complete, step-by-step solutions for the NCERT Class 9 Maths Ganita Manjari Chapter 12 Exercise 12.1 Foundations of Quadrilaterals, Sides, Angles, and Planar Classifications. Designed to reflect how top-scoring students write their solutions and reasoning in school examinations, this walkthrough covers adjacent and opposite components of quadrilaterals, internal angle definitions for convex and non-convex shapes, region analysis using plane partitioning, and spatial geometry entirely clear.
🔑 Key Concepts & Definitions:
- Quadrilateral Components: A closed 4-sided polygon with 4 vertices and 4 interior angles summing to \(360^\circ\).
- Adjacent vs. Opposite: Adjacent elements share a common vertex or side, whereas opposite elements share no common vertex or boundary segment.
- Convex vs. Non-Convex (Reflex): A convex quadrilateral has all interior angles strictly less than \(180^\circ\) and all diagonals inside; a non-convex quadrilateral contains at least one internal reflex angle (\(>180^\circ\)).
NCERT Class 9 Maths Ganita Manjari Chapter 12 Exercise 12.1 Solutions
Question 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 \(\angle A\). List all angles opposite to \(\angle A\).
(iii) Define a pair of opposite sides and a pair of opposite angles without using the names of the vertices.
Solution:
(i) Sides adjacent and opposite to \(AB\):
* Sides adjacent to \(AB\): Side \(AD\) (sharing vertex \(A\)) and side \(BC\) (sharing vertex \(B\)).
* Side opposite to \(AB\): Side \(DC\) (or \(CD\), which shares no common vertex with \(AB\)).
(ii) Angles adjacent and opposite to \(\angle A\):
* Angles adjacent to \(\angle A\): \(\angle B\) and \(\angle D\) (since they share the endpoints/vertices of side \(AB\) and side \(AD\)).
* Angle opposite to \(\angle A\): \(\angle C\).
(iii) Definition without vertex names:
* Opposite Sides: Two straight line segments forming the boundary of a quadrilateral that do not share any common endpoint.
* Opposite Angles: Two interior angles formed at vertices that do not share any common bounding edge (side).
Question 2
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.)
Solution:
At any given vertex \(V\) of a quadrilateral, the two adjacent sides form two angular regions. The internal angle at vertex \(V\) is defined as the angular region formed by the two adjacent rays meeting at \(V\) that contains the interior of the quadrilateral.
* For a convex vertex: The interior angle measures less than \(180^\circ\) and lies entirely on one side of the extended lines.
* For a non-convex (reflex) vertex: The interior angle is a reflex angle (measuring greater than \(180^\circ\) up to \(360^\circ\)) because the interior of the quadrilateral spreads outwards around that vertex, enclosing the opposite vertex within its angular sweep.
Question 3
In a quadrilateral \(ABCD\), suppose \(AB \parallel DC\). Can \(ABCD\) be non-convex? What if we instead assume \(AB = CD\)? What if we instead assume \(\angle A = \angle C\)?
Solution:
* Case 1 (\(AB \parallel DC\)): Can \(ABCD\) be non-convex?
Yes. An arrowhead (reflex) quadrilateral can have one pair of parallel sides while one of its vertices points inwards, creating a reflex interior angle.
* Case 2 (\(AB = CD\)): Can \(ABCD\) be non-convex?
Yes. A non-convex kite or dart shape can have pairs of equal-length sides without being convex.
* Case 3 (\(\angle A = \angle C\)): Can \(ABCD\) be non-convex?
Yes. A non-convex quadrilateral (such as an arrowhead where the reflex angle equals another acute or obtuse angle depending on symmetry) can have equal opposite angles.
Question 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.)
Solution:
The three lines \(AB, BC, CA\) form a triangle that divides the plane into 7 distinct regions (1 interior triangular region and 6 exterior angular/strip regions). The classification of quadrilateral \(ABCD\) depends entirely on where point \(D\) is placed:
* Region 1 (Interior of Triangle \(ABC\)): If \(D\) is inside the triangle, quadrilateral \(ABCD\) is non-convex (reflex / arrowhead).
* Regions 2, 3, 4 (Across the 3 sides / Butterfly regions): If \(D\) lies in the three exterior regions directly bordering the triangle's edges, the connecting segments cross each other, making \(ABCD\) a self-intersecting (crossed / bowtie) quadrilateral.
* Regions 5, 6, 7 (Vertically opposite angular regions / Apex regions): If \(D\) lies in the three unbounded angular regions opposite to the triangle's vertices, quadrilateral \(ABCD\) is convex.
Question 5
Can a quadrilateral be both self-intersecting and non-planar?
Solution:
No. By geometric definition, a self-intersecting polygon (such as a planar bowtie quadrilateral) requires all four vertices and intersecting line segments to lie within the same flat 2D plane so that the edges can physically cross each other.
In 3D space, a non-planar quadrilateral is known as a skew quadrilateral. The non-coplanar edges of a skew quadrilateral pass by each other in space without intersecting (they are skew lines), meaning they can never be self-intersecting in the traditional planar sense.
Frequently Asked Questions (FAQs)
Q1: What distinguishes a convex quadrilateral from a non-convex one?
A: A convex quadrilateral has all interior angles less than \(180^\circ\) and its diagonals lie entirely inside, whereas a non-convex quadrilateral has at least one interior angle greater than \(180^\circ\) (reflex angle).
Q2: Why can't skew quadrilaterals self-intersect?
A: Because their vertices do not lie on a single flat plane, lines connecting non-adjacent vertices in 3D space are skew and pass each other without crossing.
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