Lines and Angles — NCERT Solutions, AI Tutor & Practice

(a) Line segment: A part of a line with two fixed endpoints. It has a definite length. We write it as $\overline{AB}$.

(b) Ray: A part of a line that starts at one point and extends endlessly in one direction. It has one endpoint. We write it as $\overrightarrow{AB}$ (starting from $A$).

(c) Line: A straight path that extends endlessly in both directions. It has no endpoints. We write it as $\overleftrightarrow{AB}$.

Summary:

• Line segment → 2 endpoints
• Ray → 1 endpoint
• Line → 0 endpoints

Infinitely many rays can be drawn starting from a single point, because we can point in any direction from that point.

The common starting point of these rays is called the vertex (or common endpoint).

When two such rays start from the same point, the figure formed is an angle, and the common point is the vertex of that angle.

In naming an angle, the middle letter is always the vertex.

For $\angle PQR$:

• Vertex = $Q$
• Arms (rays) = $\overrightarrow{QP}$ and $\overrightarrow{QR}$

Both arms start from the vertex $Q$ and form the angle.

Recall the classification:

• Acute: $0^\circ < \theta < 90^\circ$
• Right: $\theta = 90^\circ$
• Obtuse: $90^\circ < \theta < 180^\circ$
• Straight: $\theta = 180^\circ$
• Reflex: $180^\circ < \theta < 360^\circ$

(a) $45^\circ$ → Acute angle

(b) $90^\circ$ → Right angle

(c) $130^\circ$ → Obtuse angle

(d) $180^\circ$ → Straight angle

(e) $250^\circ$ → Reflex angle

A full clock face is $360^\circ$ and is divided into $12$ equal hour-marks.

Each hour-mark = $\dfrac{360^\circ}{12} = 30^\circ$.

At 3 o'clock:

• Minute hand points to 12
• Hour hand points to 3

The gap between 12 and 3 is $3$ hour-marks.

Angle $= 3 \times 30^\circ = 90^\circ$.

So the angle between the hands is a right angle of $90^\circ$.

A complete turn of the clock hand = $360^\circ$.

Moving from 12 to 6 means covering half of the clock face.

Angle turned $= \dfrac{1}{2} \times 360^\circ = 180^\circ$.

Alternatively, from 12 to 6 there are $6$ hour-marks, each $30^\circ$: $6 \times 30^\circ = 180^\circ.$

This is a straight angle.

Steps to measure an angle with a protractor:

1. Place the centre point of the protractor exactly on the vertex of the angle.
2. Align the baseline (0° line) of the protractor along one arm of the angle.
3. Choose the correct scale (inner or outer) that starts at $0^\circ$ on that arm.
4. Read the mark on that scale where the second arm crosses the protractor.
5. That reading is the measure of the angle.

Always check: if the angle looks acute the reading should be less than $90^\circ$; if it looks obtuse, more than $90^\circ$.

Steps to draw $\angle AOB = 70^\circ$:

1. Draw a ray $\overrightarrow{OA}$ using a ruler.
2. Place the centre of the protractor on point $O$ and align the $0^\circ$ line along $\overrightarrow{OA}$.
3. Starting from $0^\circ$ on the correct scale, count up to $70^\circ$ and mark a point $B$.
4. Remove the protractor and join $O$ to $B$ with a ruler to draw ray $\overrightarrow{OB}$.

Now $\angle AOB = 70^\circ$, which is an acute angle.

(a) Right angles ($90^\circ$):

• The corner of a blackboard or notebook
• The corner where two walls meet

(b) Acute angles (less than $90^\circ$):

• The angle between the two hands of an open pair of scissors when slightly opened
• The angle at the tip of a slice of pizza

These examples show that angles are all around us in daily life.

At 2 o'clock:

• Minute hand points to 12
• Hour hand points to 2

Each hour-mark = $30^\circ$, and there are $2$ hour-marks between 12 and 2.

Angle $= 2 \times 30^\circ = 60^\circ$.

Since $60^\circ < 90^\circ$, the angle is an acute angle.

A full turn (all the way around) = $360^\circ$.

East and West are exactly opposite directions, so turning from East to West is half of a full turn.

Angle $= \dfrac{1}{2} \times 360^\circ = 180^\circ$.

This is a straight angle of $180^\circ$.

An obtuse angle must satisfy $90^\circ < \theta < 180^\circ$.

Here $\theta = 200^\circ$.

Since $200^\circ > 180^\circ$, it does not fall in the obtuse range. It cannot be a straight angle either, because that is exactly $180^\circ$.

A reflex angle satisfies $180^\circ < \theta < 360^\circ$, and $200^\circ$ lies in this range.

Therefore $200^\circ$ is a reflex angle.