Std 6 Mathematics · Chapter 2

2. Lines and Angles

Solutions to Practice Set 2.1, Practice Set 2.2 and Exercise 2, with diagrams. Maharashtra State Board (Balbharati), Grade Six Mathematics.

About the chapter

This chapter starts with the basic ideas of geometry. We learn about points, lines, line segments, rays and planes. Then we learn what an angle is. We measure angles and name their types.

Key points

A line has no end points. It extends endlessly on both sides.
A line segment has two end points. A ray has one starting point.
Points on one line are collinear points. Points not on one line are non-collinear points.
Two rays with the same starting point form an angle. The starting point is the vertex.
Angles are measured in degrees (°) with a protractor.

Formulas and terms

Zero angle – 0°
Acute angle – more than 0° and less than 90°
Right angle – 90°. Obtuse angle – more than 90° and less than 180°
Straight angle – 180°. Reflex angle – more than 180° and less than 360°
Complete angle – 360°

Practice Set 2.1

1Look at the adjacent diagram and write the names of the terms given below.

Points D, E, O and B lie on one straight line, so they are collinear.

(i) Points: point A, point B, point C, point D, point E, point O
(ii) Line segments: seg OA, seg OB, seg OC, seg OE, seg ED, seg OD, seg EB, seg DB
(iii) Rays: ray OA, ray OB, ray OC, ray OD (ray OE), ray ED, ray EB
(iv) Collinear points: D, E, O, B
(v) Line: line DB
(vi) Non-collinear points: A, O, C (any three points not on one line)

Note: This answer is based on the textbook diagram, where D, E, O and B lie on one line. Check the figure in your book once.

2Draw the following figures. (i) Line BT (ii) Ray DE (iii) Line MP (iv) Line y (v) Point M

(i) Line BT — arrows at both ends

(ii) Ray DE — starts at D, arrow at the E end

(iv) Line y — named by a small letter

(v) Point M — a dot with a capital letter

(i) Line BT: Draw a straight line. Mark points B and T on it. Put arrow heads at both ends.
(ii) Ray DE: Mark the starting point D. Draw a line through E. Put an arrow head only at the E end.
(iii) Line MP: Draw a straight line with points M and P. Put arrow heads at both ends.
(iv) Line y: Draw a line with arrow heads at both ends. Write the small letter y near it.
(v) Point M: Make a small dot with a sharp pencil. Write the capital letter M near it.

3Draw diagrams that illustrate the following points.

(i) Point P lies on line AB

(ii) E and F are on line m; D is not on line m

(iii) Ray OP and ray OQ — same starting point O

(i) Point P lies on line AB.
Draw line AB. Mark point P on the line between A and B.

(ii) Points E and F are on line m. But point D is not on line m.
Draw line m. Mark points E and F on it. Mark point D outside the line.

(iii) Draw ray OP and ray OQ.
Mark point O. Draw ray OP and ray OQ from O. Both rays have the same starting point O.

4Show the points, line segments and planes from the rangoli in the accompanying picture.

Points: Every dot of the rangoli is a point.
Line segments: Each straight line joining two dots is a line segment.
Plane: The flat floor on which the rangoli is drawn is a plane.

5Four important places are marked with points on the map of India. Draw as many line segments as possible connecting these places.

Joining 4 points in every possible way gives 4 × 3 ÷ 2 = 6 line segments.

The four places are Delhi, Mumbai, Calcutta and Chennai. We join each place with every other place. We get 6 line segments:

seg Delhi–Mumbai
seg Delhi–Calcutta
seg Delhi–Chennai
seg Mumbai–Calcutta
seg Mumbai–Chennai
seg Calcutta–Chennai

Practice Set 2.2

1Find the given letters of the English alphabet whose lines form right angles. (A, E, N, K, L, H)

The letters whose lines form right angles are E, L and H.

In A, N and K, the lines are slanting. So they do not form right angles.

2Write the type of each angle based on the measures given below.

Zero angle — 0°

Acute — 40°

Right angle — 90°

Obtuse — 155°

Straight angle — 180°

Reflex — 245°

(i) 40° – Acute angle
(ii) 0° – Zero angle
(iii) 90° – Right angle
(iv) 155° – Obtuse angle
(v) 180° – Straight angle
(vi) 245° – Reflex angle

3Draw the reflex angles of different measures.

Draw 110° first, then mark the arc on the outer side — that outer angle is 250°.

An angle of more than 180° and less than 360° is a reflex angle. For example, draw angles of 200°, 250° and 310°.

To draw 250°: 360° − 250° = 110°. Draw an angle of 110° with a protractor. Mark the outer side of it. The outer angle is 250°.

4Answer the following questions.

(i) How many right angles are included in a straight angle?
A straight angle is 180°. A right angle is 90°.
180° ÷ 90° = 2. So, a straight angle includes 2 right angles.

(ii) What is the type of angle formed between two adjacent fingers of a hand?
The angle between two adjacent fingers is less than 90°. So, it is an acute angle.

(iii) What kind of angle is formed between the hour hand and the minute hand of a clock when it is 12 o'clock?
At 12 o'clock, both hands are on 12. One hand lies on the other. So, a zero angle is formed.

Exercise 2

1Choose the correct option.

(i) A complete angle contains ………. right angles. — (d) 4
360° ÷ 90° = 4
(ii) If a straight angle is divided into three equal parts, each resulting angle will be … — (b) acute angle
180° ÷ 3 = 60°. It is less than 90°.
(iii) The starting point of ray OP and ray OQ is ………. — (a) O
(iv) ….. is infinite on both sides. — (a) Line

2Match the pairs.

Group A	Group B
i) 91°	e) Obtuse angle
ii) 180°	a) Straight angle
iii) 360°	b) Complete angle
iv) 215°	c) Reflex angle
v) 90°	d) Right angle

3Write the answers by looking at the figure.

Line PQ carries the points D and E; line CF and ray EB also pass through E.

i) Names of all the rays: ray DA, ray DP, ray DQ, ray EB, ray EC, ray EF, ray EP, ray EQ
ii) Lines on which point E lies: line PQ and line CF
iii) Names of the line segments: seg AD, seg DE, seg BE, seg CE, seg EF, seg CF
iv) Rays whose starting point is E: ray EB, ray EC, ray EF, ray EP, ray EQ
v) Type of ∠BEQ and its sides: ∠BEQ is an acute angle. Its sides are ray EB and ray EQ.

Note: This answer is based on the textbook diagram. Check the figure in your book once.

4Write the names and types of the angles in the diagram.

Four rays start at O, so 4 × 3 ÷ 2 = 6 angles are formed.

Four rays start from point O. So, 6 angles are formed.

∠POQ – Right angle
∠QOR – Acute angle
∠POR – Obtuse angle
∠POS – Obtuse angle
∠QOS – Obtuse angle
∠ROS – Obtuse angle

Note: Measure each angle with a protractor in your book to confirm its type.

5Find the number of acute angles in each of the following figures. Draw the figure that comes in place (d). What pattern is found?

(a) — 3 acute angles

(b) — 12 acute angles

(d) — 30 acute angles

Figure (a) – 3 acute angles
Figure (b) – 12 acute angles
Figure (c) – 21 acute angles
Figure (d) – 30 acute angles

Figure (d): In figure (c), join the midpoints of the sides of the smallest triangle once more.

Pattern: 3, 12, 21, 30. Each time, the number of acute angles increases by 9.

6Draw a freehand picture with an acute angle, a right angle and an obtuse angle and fill it with colours.

This is a drawing activity. For example, draw a house. The roof top shows an acute angle. The walls and the ground show right angles. An open door shows an obtuse angle. Colour the picture.

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