NCERT Solutions Class 9 Mathematics Ganita Manjari 2 Chapter 12 Quadrilaterals Exercise 12.1

Exercise Set 12.1

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 ∠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:
(i) Two sides are adjacent when they share an endpoint. AB has endpoints A and B.
DA meets AB at A and BC meets AB at B, so the sides adjacent to AB are BC and DA.
CD has neither A nor B as an endpoint, so the side opposite to AB is CD.
(ii) The vertices joined to A by a side are B and D, so the angles adjacent to ∠A are ∠B and ∠D.
The remaining vertex C is not joined to A by a side, so the angle opposite to ∠A is ∠C.
(iii) Opposite sides: two sides of a quadrilateral that have no endpoint (vertex) in common.
Opposite angles: two angles of a quadrilateral whose vertices are not the two ends of any side, that is, whose vertices are the two ends of a diagonal.

Question 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:
Take quadrilateral ABCD and look at vertex A. Its neighbours are B and D; the opposite vertex is C.
The two sides at A lie along rays AB and AD, which form the ordinary angle ∠BAD (less than 180°). The internal angle is either ∠BAD or the reflex angle 360° − ∠BAD on the other side. The opposite vertex C decides which side is inside.
Definition. The internal angle of ABCD at A is
1. ∠BAD, if A does not lie inside △BCD;
2. 360° − ∠BAD (a reflex angle), if A lies inside △BCD.

Why it works: if A is outside △BCD, the quadrilateral near A lies within ∠BAD. If A is inside △BCD, the quadrilateral is △BCD with △BAD cut away; this is the dent at A, and the region near A lies outside ∠BAD, so the internal angle is the reflex one.
For example, in DART (Fig. 12.2) D lies inside △ART, so the angle at D is 360° − ∠ADT, which is more than 180°. With this definition the internal angles of every planar, non-self-intersecting quadrilateral add up to 360°.

Teacher's Note:
1. Students often forget that a vertex angle can be reflex; the opposite vertex settles which side is 'inside'.
2. Ask them to test the definition on DART and on a square.

Question 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:
Case 1: AB ∥ DC. ABCD cannot be non-convex.
A→B and D→C must point the same way; otherwise BC and AD would cross and ABCD would be self-intersecting, which is not allowed.
So AD is a transversal of the parallel lines AB and DC, and ∠A, ∠D are co-interior angles: ∠A + ∠D = 180°. In the same way, with BC as transversal, ∠B + ∠C = 180°.
Hence every angle is less than 180°, and ABCD is convex (so its diagonals meet).
Case 2: AB = CD. Yes, it can be non-convex.
Take A(0, 0), B(4, 0), C(1, 1), D(1, 5), figure (b). Then AB = 4 and CD = 5 − 1 = 4. The sides do not cross, and C lies inside △ABD, so the angle at C is reflex. So ABCD is non-convex with AB = CD.
Case 3: ∠A = ∠C. Yes, it can be non-convex.
Take the arrowhead A(−2, 0), B(0, 4), C(2, 0), D(0, 1), figure (c). It is symmetric about line BD, which swaps A and C, so ∠A = ∠C. D lies inside △ABC, so the angle at D is reflex and ABCD is non-convex.
(The dent cannot be at A or C here: then both ∠A and ∠C would be above 180° and the sum would exceed 360°.)

Teacher's Note:
1. Parallel opposite sides force convexity; equal sides or equal angles do not.
2. Encourage students to test claims by plotting small coordinate examples.

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.)
Answer:
The sides of ABCD are AB, BC, CD and DA, so D is joined to C and to A. The three lines cut the plane into 7 regions: the inside of the triangle, three regions across a side and three regions beyond a vertex.

1. Inside △ABC: D is inside the triangle of the other three vertices, so the angle at D is reflex: non-convex (dent at D).
2. Across AC (opposite B): B and D are on opposite sides of AC and the diagonals AC and BD cross: convex.
3. Across AB (opposite C): C and D are on opposite sides of AB, so CD cuts side AB: self-intersecting.
4. Across BC (opposite A): A and D are on opposite sides of BC, so DA cuts side BC: self-intersecting.
5. Beyond vertex A: A lies inside △BCD, so the angle at A is reflex: non-convex.
6. Beyond vertex B: B lies inside △CDA: non-convex (dent at B).
7. Beyond vertex C: C lies inside △DAB: non-convex (dent at C).
Summary: convex in 1 region, self-intersecting in 2 regions, non-convex in 4 regions.

Teacher's Note:
1. Remember that D is joined to A and C, not to B; that is why the region across AC gives a convex shape.
2. Let students sketch one D in each region to check the table.

Question 5. Can a quadrilateral be both self-intersecting and non-planar?
Answer:
No. A self-intersecting quadrilateral is always planar.
In a self-intersecting 4-gon ABCD, two sides meet at a point that is not a vertex. They cannot be adjacent sides (AB and BC meet only at B unless A, B, C are collinear, and then all four points already lie in one plane). So two opposite sides meet, say AB and CD at E.
The lines AB and CD pass through the common point E, and two intersecting lines always lie in one plane.
That plane contains both lines, and so it contains A, B, C and D. Hence ABCD is planar, and it cannot be both self-intersecting and non-planar.