Std 5 Mathematics · Chapter 6

6. Angles

Solutions to Problem Sets 24 to 27, with diagrams — naming, measuring and drawing angles, and parallel and perpendicular lines. Maharashtra State Board (Balbharati), Grade Five Mathematics, Part One.

Key Ideas

Parts of an angle: the common point is the vertex; the two lines that meet there are the arms.

Naming: use three letters with the vertex in the middle. ∠ABC and ∠CBA are the same angle, with vertex B.

Measuring: put the protractor’s centre on the vertex and its baseline on one arm. Count from the 0 on that arm’s side and read where the other arm crosses the scale.

Types: right angle = 90°; acute angle < 90°; obtuse angle > 90°.

Lines: parallel lines never meet, however far they are extended. Perpendicular lines meet at a right angle (90°).

Revision — Angles on the Clocks

RLook at the clocks. Write whether the hands make a right angle, an acute angle or an obtuse angle.

Clock 1 (hands pointing to 6 and about 2): Obtuse angle
Clock 2 (hands pointing to 6 and a little below 9): Acute angle
Clock 3 (hands pointing to 12 and 3 (3 o’clock)): Right angle
Clock 4 (hands pointing to about 7 and a little above 3): Obtuse angle
Clock 5 (hands pointing to 9 and a little to the right of 6): Obtuse angle
Clock 6 (hands pointing to 12 and about 10): Acute angle

Tip: hold the corner of a notebook (a right angle) between the hands. If the opening is smaller than the corner, the angle is acute; if it is wider, the angle is obtuse. In clocks 4 and 5 the hands open slightly more than a right angle.

Problem Set 24

QComplete the following table.

Diagram	Name of the angle	Vertex	Arms of the angle
P, Q, R (vertex at Q)	∠PQR or ∠RQP	Q	QP and QR
L, N, M (vertex at M)	∠LMN or ∠NML	M	ML and MN
S, U, T (vertex at U)	∠SUT or ∠TUS	U	US and UT

The first row is the example given in the textbook; answers filled in are in bold. In the second diagram, both lines meet at M, so M is the vertex (N and L are the ends of the arms). In the third, both lines meet at U.

Problem Set 25

QMeasure the angles given below and write the measure in the given boxes.

Angle 1: 40° — acute angle
Angle 2: 115° — obtuse angle
Angle 3: 90° — right angle
Angle 4: 85° — acute angle

These measures are taken from the textbook figures. A reading 1° or 2° either way is acceptable when measuring by hand. If an arm is too short to reach the scale, extend it with a ruler first; this does not change the angle.

Problem Set 26

Step 1: Draw one arm, say BC, with a ruler. B is the vertex.

Step 2: Place the centre of the protractor on B and its baseline along BC.

Step 3: Start from the 0 on the side of C, count up to the required measure and mark a dot.

Step 4: Remove the protractor. Join B to the dot, extend the line and name its end A. Write the measure inside the angle.

Remember Rahul’s mistake: always count from the 0 that lies on the same side as the arm you drew first.

QDraw and name the following angles with the help of a protractor: (1) 60° (2) 120° (3) 90° (4) 150° (5) 30° (6) 165° (7) 45°

(1) ∠ABC = 60° — acute

(2) ∠PQR = 120° — obtuse

(3) ∠LMN = 90° — right

(4) ∠XYZ = 150° — obtuse

(5) ∠RST = 30° — acute

(6) ∠DEF = 165° — obtuse

(7) ∠GHI = 45° — acute

Any three letters may be used to name the angles, as long as the vertex letter is in the middle.

Activity: Making a Right Angle by Folding

AFold a sheet of paper, fold it again across the first fold, and unfold. Verify with a protractor that the angle between the two fold lines is 90°.

When the protractor’s centre is placed where the two fold lines cross and its baseline is laid along one fold line, the other fold line passes through the 90° mark. So the folds make a right angle, and the two fold lines are perpendicular to each other.

Problem Set 27

1Give two examples of parallel lines you can see in your environment.

The two rails of a railway track.
The ruled lines on a notebook page.

Other examples: opposite edges of a door, table or blackboard; steps of a ladder; bars of a window; stripes of a zebra crossing.

2Give two examples of perpendicular lines you can see in your environment.

Two adjacent edges of a book or notebook.
A flagpole standing straight on the ground and the ground.

Other examples: the corner edges of a door or window frame; the margin line and the ruled lines of a notebook page; a wall and the floor.

3Look at the pictures given below. Decide whether the lines given in each picture are parallel or perpendicular to each other and write the answer in the box.

Parallel lines (given)
Perpendicular lines (given)
Parallel lines — Two vertical lines that do not meet, however far they are extended.
Parallel lines — Two slanting lines leaning the same way; they never meet.
Perpendicular lines — The two lines cross each other at a right angle (90°).

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