CBSE Class 6 Mathematics – 13. Angles and Their Measurement (Notes)

Revision Notes on Angle and Their Measurement

Learning Goal

1. Angle & Types of Angles

   – Zero (θ = 0⁰), Acute (0⁰<θ <90⁰), Right (θ = 90⁰), Obtuse (0⁰<θ <90⁰),

Straight/Linear (θ = 180⁰), Reflex (0⁰<θ <90⁰), Complete angle (θ = 360⁰).

2. Properties of Angles

   – Angles on as straight line add up to 180 degrees.

   – Angles around a point add up to 360 degrees.

3. Pair of Angles

   – Adjacent angles

   – Vertical opposite angles (They are equal)

   – Linear pair of angles (Sum of LPA = 180⁰)

   – Complementary and supplementary pairs (A+B=90⁰ & A+B=180⁰)

 

Tools to be used for construction

1. The Ruler

A ruler is a straight edge which we sometimes call a scale.

It is marked with the centimetres on one side and inches on the other side.

It is used to draw line segments and also to measure them.

2. The Compass

The compass has two ends –one is a pointer and the other is for the pencil. It is used to draw arcs and circles and also to measure the line segment.

3. The Divider

It is a pair of pointers on both the ends. It is used to compare the length of line and arcs.

4. Set -squares

It is a set of two triangular pieces.

One is having 45°, 45° and 90° angles at the corners. If we join two same triangular pieces, we will get a square. It is used to draw parallel and perpendicular lines.

Other is having 30°, 60° and 90° angles at the corners. If we join two same triangular pieces, we will get a rectangle. It is also used to draw parallel and perpendicular lines.

5. The Protractor

It is a semi-circular device which is divided into 180-degree parts. It starts from the 0° from the right-hand side and ends with the 180° on the left-hand side and vice versa. It is used to measure and draw angles.

Angles

Angle is a stature formed by two rays. The two rays have a common endpoint which is called the Vertex of the Angle.

1. Construction of an angle with a given measure

Draw an angle of 60 ° using a protractor.

Step 1: Draw a line BA.

Step 2: Put the protractor on the line in such a way that the centre of the protractor lays on point B and the zero edge comes on the line segment BA.

Step 3: Start from 0° and mark the point C at 60°.

Step 4: Join BC.

∠CBA is the required angle.

2. Construction of a copy of an angle of unknown measure (using a ruler and a compass).

Draw a copy of ∠A.

Step 1: P is a point on the line l.

Step 2: Take A as the centre in the given angle and draw an arc of any radius which cuts the two rays at B and C.

Step 3: In the line l, take P as the centre and draw an arc with the same radius as above which cuts line l at Q.

Step 4: Open your compass to take the length of the arc BC.

Step 5: Take Q as the centre and draw an arc with the same radius, to cut the arc drawn earlier, at point R. 

Step 6: Join PR. It will make the angle of the same measure as given.

Hence, ∠P = ∠A

3. The bisector of an angle

An angle bisector is the line segment which divides a particular angle into two equal parts. It is also called the line of symmetry of the angle.

Construction of angle bisector (using a ruler and a compass)

Draw the angle bisector of ∠O.

Step 1: Put the pointer on O and draw an arc of any radius so that it cut the rays at point A and B.

Step 2: Put the pointer on point A and draw an arc of the radius of more than half of AB.

Step 3: While taking B as the centre we will draw an arc of the same radius so that it cut the previous arc at point C.

Step 4: Join OC.OC is the required angle bisector of ∠O.

Hence, ∠BOC = ∠COA.

4. Angles of special measures

There are some angles which we can construct accurately with the help of a compass without using a protractor.

a. Construction of 60° angle.

Step 1: Draw a line m and mark a point C on it.

Step 2: Take C as the centre and draw an arc of any radius to cut the line at point D.

Step 3: While taking D as the centre we need to draw an arc of the same radius to cut the previous arc.

Step 4: Join CE. ∠C = 60°.

b. Construction of 120° angle.

It is the twice of 60° angle.

Step 1: Draw a line m and mark a point C on it.

Step 2: Put the pointer on point C and draw an arc of any radius which cuts the line m at point D.

Step 3: While taking D as the centre we need to draw an arc with the same radius to cut the previous arc at E.

Again take E as the centre and draw an arc of the same radius to cut the first arc at point F.

Step 4: Join CF. ∠FCD = 120°.

c. Construction of 30° angle.

To construct an angle of 30°, we need to draw an angle of 60° as above then bisect it with the process of an angle bisector.

d. Construction of 90° angle

It can be made by two methods-

i. Draw a perpendicular bisector of 180° i.e. a straight line.

ii. Draw a bisector of 60° and 120°.

e. Construction of 45° angle.

Draw an angle of 90° then bisect it to make an angle of 45°.