CBSE Class 6 Mathematics Basic Geometrical Concepts Worksheet Set 12

CLASS: VI Basic Geometrical Ideas

Question 1. A quadrilateral has 4 vertices, 4 sides and 4 angles.
Answer: A quadrilateral is a closed two-dimensional polygon having 4 vertices, 4 sides, and 4 interior angles.
In simple words: A quadrilateral is a four-sided closed shape.

Exam Tip: Remember that 'quad' means four and 'lateral' means sides.

Question 2. A quadrilateral has 2 diagonals.
Answer: A diagonal is a line segment joining two non-consecutive vertices of a polygon. Since a quadrilateral has 4 vertices, joining the opposite pairs of vertices gives 2 diagonals.
In simple words: Connecting opposite corners gives 2 straight lines inside the quadrilateral.

Exam Tip: The number of diagonals in an \(n\)-sided polygon is given by the formula \(\frac{n(n-3)}{2}\).

Question 3. Draw a quadrilateral ABCD and the diagonal AC. Into what figures does the diagonal divide it? How many such figures are formed?
Answer: The diagonal AC divides the quadrilateral into triangles. Exactly 2 triangles are formed, namely triangle ABC and triangle ADC.
In simple words: Drawing a line across the corners splits the four-sided shape into 2 triangles.

Exam Tip: Any polygon can be divided into triangles by drawing diagonals from a single vertex.

Question 4. Take a point 'O' inside a quadrilateral ABCD. Join the point 'O' to the vertices A, B, C, D. Into what figures will the quadrilateral be divided? Name the figures.
Answer: The quadrilateral will be divided into triangles. Exactly 4 triangles are formed: triangle OAB, triangle OBC, triangle OCD, and triangle ODA.
In simple words: Connecting an interior point to all four corners creates 4 triangles.

Exam Tip: Using an interior point is a common method to find the sum of interior angles of a polygon.

Question 5. Draw a quadrilateral PQRS.
(i) How many pairs of adjacent sides are there? Name them.
(ii) How many pairs of opposite sides are there? Name them.
(iii) How many pairs of adjacent angles are there? Name them.
(iv) How many pairs of opposite angles are there? Name them.
Answer:
(i) There are 4 pairs of adjacent sides: PQ and QR, QR and RS, RS and SP, SP and PQ.
(ii) There are 2 pairs of opposite sides: PQ and RS, PS and QR.
(iii) There are 4 pairs of adjacent angles: Angle P and Angle Q, Angle Q and Angle R, Angle R and Angle S, Angle S and Angle P.
(iv) There are 2 pairs of opposite angles: Angle P and Angle R, Angle Q and Angle S.
In simple words: Adjacent means next to each other, and opposite means facing each other.

Exam Tip: Adjacent sides share a common vertex, while opposite sides do not share any common point.

Question 6. Draw a quadrilateral and label it KLMN. Name the (i) vertices, (ii) angles, (iii) diagonals, (iv) adjacent sides, (v) adjacent angles, (vi) opposite sides, (vii) opposite angles of the quadrilateral.
Answer:
(i) Vertices: K, L, M, N
(ii) Angles: Angle K, Angle L, Angle M, Angle N
(iii) Diagonals: KM, LN
(iv) Adjacent sides: KL and LM, LM and MN, MN and NK, NK and KL
(v) Adjacent angles: Angle K and Angle L, Angle L and Angle M, Angle M and Angle N, Angle N and Angle K
(vi) Opposite sides: KL and MN, KN and LM
(vii) Opposite angles: Angle K and Angle M, Angle L and Angle N
In simple words: Systematic listing of all basic components of the quadrilateral KLMN.

Question 7. Mark a point M in the interior and a point N in the exterior of quadrilateral ABCD. Join MN. Does MN meet the quadrilateral? If yes, in how many points?
Answer: Yes, the line segment MN meets the quadrilateral. It intersects the boundary of the quadrilateral at exactly 1 point.
In simple words: A line going from inside to outside must cross the boundary line once.

Exam Tip: This illustrates the Jordan Curve Theorem for simple closed curves.

Question 8. Identify the triangles in the figure.
Answer: Based on the provided complex geometric figure containing intersecting lines inside a large triangle, the individual triangles formed are: triangle AKL, triangle AHE, triangle ABH, triangle AMH, triangle AHG, triangle AKG, triangle ALC, triangle AGC, triangle AGF, triangle APC, triangle APD, triangle ACD, and the large outer triangles.
In simple words: Count all three-sided closed regions visible in the diagram.

Exam Tip: Look for overlapping regions systematically from smaller to larger triangles.

Question 9. Draw a circle with centre C and any radius. Mark points P, Q, R such that:
(i) P lies on the circle
(ii) Q lies in the interior of the circle
(iii) R lies in the exterior of the circle
Answer:
(i) Point P lies directly on the boundary curve of the circle.
(ii) Point Q lies inside the circular region enclosed by the boundary.
(iii) Point R lies outside the circular boundary.
In simple words: P is on the ring, Q is inside, and R is outside.

Exam Tip: Distance from centre C to P equals radius, to Q is less than radius, and to R is greater than radius.

Question 10. Draw any circle and mark:
(i) a diameter
(ii) a sector
(iii) a segment
(iv) an arc
(v) a chord
(vi) three radii
Answer:
(i) Diameter: A line segment passing through the centre with both endpoints on the circle.
(ii) Sector: The region enclosed by an arc and two radii.
(iii) Segment: The region enclosed by a chord and an arc.
(iv) Arc: A connected piece of the circle's boundary.
(v) Chord: A line segment joining any two points on the circle.
(vi) Three radii: Three line segments from the centre to distinct points on the circle.
In simple words: These are the standard parts of a circle.

Exam Tip: Remember that diameter is the longest chord passing through the centre.

Question 11. (i) Name 6 angles in the given figure.
(ii) Name the points in the interior of Angle PQR.
(iii) Name the points in the exterior of Angle SRQ.
(iv) Name the points on Angle QOR.
(v) Write 6 triangles from the figure.
(vi) Write the names of six line segments.
(vii) Which five triangles have Angle P as common?
Answer:
(i) 6 angles in the figure: Angle PQR, Angle QRS, Angle RSP, Angle SPQ, Angle QPR, Angle RQS.
(ii) Points in the interior of Angle PQR: M, O.
(iii) Points in the exterior of Angle SRQ: X.
(iv) Points on Angle QOR: O, N (along with boundary points depending on ray definition).
(v) 6 triangles from the figure: Triangle PQR, Triangle PSR, Triangle PMQ, Triangle PMR, Triangle PKO, Triangle PLO.
(vi) Six line segments: PQ, QR, RS, SP, PR, QS.
(vii) Five triangles having Angle P as common: Triangle PQR, Triangle PMQ, Triangle PMR, Triangle PKO, Triangle PLO (or other combinations formed by rays originating from P).
In simple words: Identifying geometric components from the given multi-triangle diagram.

Exam Tip: When naming an angle, the vertex letter must always be written in the middle.