CBSE Class 6 Mathematics Basic Geometrical Concepts Worksheet Set 04

VI - Mathematics [Basic Geometrical Ideas]

Question 1. (a) Name all the different angles shown in the figures:
Answer: Figure (i): \(\angle PRQ\), \(\angle PQR\), \(\angle RPQ\)
Figure (ii): \(\angle DAB\), \(\angle ABC\), \(\angle BCD\), \(\angle CDA\), \(\angle DAC\), \(\angle CAB\), \(\angle BCA\), \(\angle ACD\)
Figure (iii): \(\angle PMQ\), \(\angle QMO\), \(\angle PMO\), \(\angle MON\), \(\angle MNO\), \(\angle OMN\), \(\angle MQP\), \(\angle QPO\), \(\angle QOP\), \(\angle OPQ\), \(\angle PON\), \(\angle OPN\)
Figure (iv): Various angles formed at the centre and circumference involving intersecting chords, tangents, and radii.
In simple words: An angle is formed where two rays meet at a common endpoint. We name each angle using three letters, with the vertex in the middle.

Exam Tip: Always place the middle letter of the angle name at the vertex where the two arms meet.

Question 1. (b) Count the number of angles.
Answer: Figure (i): 3 angles
Figure (ii): 8 angles
Figure (iii): 12 distinct angles
Figure (iv): Multiple angles formed by intersecting lines and a circle.
In simple words: Count every distinct corner where two lines or line segments intersect to form an opening.

Exam Tip: Be systematic when counting angles in complex figures by going vertex by vertex.

Question 1. (c) List the acute angles.
Answer: Acute angles are angles whose measure is less than \(90^\circ\). Examples include \(\angle PRQ\), \(\angle RPQ\), \(\angle DAC\), \(\angle CAB\), \(\angle BCA\), and \(\angle ACD\).
In simple words: An acute angle is smaller than a right angle (less than \(90^\circ\)).

Exam Tip: Look for sharp, narrow corners to identify acute angles quickly.

Question 1. (d) List the obtuse angles.
Answer: Obtuse angles are angles whose measure is greater than \(90^\circ\) and less than \(180^\circ\). Examples include \(\angle PQR\), \(\angle DAB\), and \(\angle BCD\).
In simple words: An obtuse angle is wider than a right angle (between \(90^\circ\) and \(180^\circ\)).

Exam Tip: Look for wide-open corners that are larger than a square corner.

Question 1. (e) Identify the right angles and straight angles.
Answer: Right angles (\(90^\circ\)): Found in perpendicular intersections, such as in figure (iv) where a radius meets a tangent line.
Straight angles (\(180^\circ\)): Formed by straight lines passing through a point, such as straight line segments in figure (iv).
In simple words: Right angles look like the corner of a square (\(90^\circ\)), and straight angles form a straight line (\(180^\circ\)).

Exam Tip: A right angle is denoted by a small square symbol at the vertex in geometric diagrams.

Question 2. List all the points which are in the exterior and interior of the given angle.
Answer: For the first diagram:
Interior points: \(m, r\)
Exterior points: \(s, q, k, p, x\)
For the second diagram:
Interior points: Points lying strictly inside the region spanned by the arms of angle 1.
Exterior points: Points lying outside the arms of angle 1.
In simple words: Interior points are inside the opening of the angle, while exterior points are outside.

Exam Tip: Imagine extending the arms of the angle infinitely to correctly classify boundary and exterior points.

Question 3. Draw a rough sketch of a quadrilateral PQRS and state:
(a) two pairs of opposite sides.
(b) two pairs of opposite angles.
(c) four pairs of adjacent angles.
(d) four pairs of adjacent sides.
(e) Draw the diagonals and name them.
Answer: (a) Opposite sides: Side PQ and Side SR; Side PS and Side QR.
(b) Opposite angles: \(\angle P\) and \(\angle R\); \(\angle Q\) and \(\angle S\).
(c) Adjacent angles: \((\angle P, \angle Q)\), \((\angle Q, \angle R)\), \((\angle R, \angle S)\), \((\angle S, \angle P)\).
(d) Adjacent sides: \((PQ, QR)\), \((QR, RS)\), \((RS, SP)\), \((SP, PQ)\).
(e) Diagonals: PR and QS.
In simple words: A quadrilateral is a four-sided polygon. Opposite sides and angles do not share a common vertex or side, while adjacent ones do.

Exam Tip: Draw a convex quadrilateral to clearly show interior diagonals intersecting inside the figure.

Question 4. Name the interior and exterior points of the angle marked \(x^\circ\) in the following:
Answer: First figure:
Interior points: Points lying between ray BE and ray BC, such as points in the immediate interior of angle \(x^\circ\).
Exterior points: Points outside ray BE and ray BC, such as point A and point D.
Second figure:
Interior points: Points lying inside the angle formed at vertex O containing \(x^\circ\), including point N.
Exterior points: Points outside the angle arms, such as points D, M, P.
In simple words: Interior points are trapped inside the angle's V-shape, and exterior points are outside.

Exam Tip: Trace the two rays forming the angle \(x^\circ\) to clearly separate the inside region from the outside region.

Question 5. (a) Identify the triangles in the figure:
(b) Write the names of angles
(c) Write the names of line segments.
Answer: (a) Triangles: In figure (i): \(\triangle EFO\), \(\triangle FGO\), \(\triangle GHO\), \(\triangle EHO\), and larger combined triangles like \(\triangle EFH\), \(\triangle EGH\), etc.
(b) Angles: All individual interior and exterior angles formed by intersecting lines.
(c) Line segments: EF, FG, GH, HE, EO, FO, GO, HO, FH, EG.
In simple words: Triangles are three-sided closed figures. Line segments are straight paths with two endpoints.

Question 6. Draw any circle and mark:
(a) its centre
(b) three radii
(c) a diameter
(d) shade a minor sector
(e) colour a minor arc
(f) a chord
(g) two points in its interior
(h) two points in its exterior
(j) three points on the circle
Answer: (a) Centre: Point O inside the circle.
(b) Radii: OA, OB, OC (line segments from centre to the circle).
(c) Diameter: Line segment passing through the centre connecting two points on the circle (e.g., AB).
(d) Minor sector: Region bounded by two radii and an arc, smaller than a semicircle.
(e) Minor arc: A curved part of the circle's boundary measuring less than half the circumference.
(f) Chord: Line segment joining any two points on the circle.
(g) Interior points: Two points inside the circle (e.g., P, Q).
(h) Exterior points: Two points outside the circle (e.g., X, Y).
(j) Points on the circle: Three points lying directly on the circumference (e.g., A, B, C).
In simple words: A circle is a set of all points equidistant from a fixed centre point.

Exam Tip: Always use a sharp compass to draw a clean, accurate circle and label all requested parts clearly.