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Lines and Angles - Types of Angles and their Measures

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An angle is formed by two rays originating from a common endpoint called the vertex. We measure the rotation between these rays in degrees, where a full revolution is 360∘360^\circ. Angles are classified into several types based on their measure: Acute (<90∘< 90^\circ), Right (=90∘= 90^\circ), Obtuse (>90∘> 90^\circ and <180∘< 180^\circ), Straight (=180∘= 180^\circ), Reflex (>180∘> 180^\circ and <360∘< 360^\circ), and Complete (=360∘= 360^\circ).

An acute angle of 60 degrees showing the vertex and two rays.
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A right angle is exactly 90∘90^\circ and represents one-fourth of a full revolution. It is often represented by a small square symbol at the vertex.

A right angle showing a square corner at the vertex.
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A reflex angle is an angle that is greater than 180∘180^\circ but less than 360∘360^\circ. It represents more than a semi-circle of rotation.

A reflex angle measured from the outer side of two rays.
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A straight angle is exactly 180∘180^\circ and forms a straight line. It is equivalent to half a revolution.

📐Formulae

0∘<Acute Angle<90∘0^\circ < \text{Acute Angle} < 90^\circ

Right Angle=90∘\text{Right Angle} = 90^\circ

90∘<Obtuse Angle<180∘90^\circ < \text{Obtuse Angle} < 180^\circ

Straight Angle=180∘\text{Straight Angle} = 180^\circ

180∘<Reflex Angle<360∘180^\circ < \text{Reflex Angle} < 360^\circ

Complete Angle=360∘\text{Complete Angle} = 360^\circ

Measure in degrees=Fraction of revolution×360∘\text{Measure in degrees} = \text{Fraction of revolution} \times 360^\circ

💡Examples

Problem 1:

Classify the following angles based on their measures: 35∘35^\circ, 145∘145^\circ, 90∘90^\circ, 180∘180^\circ, and 275∘275^\circ.

Solution:

35∘35^\circ: Acute Angle, 145∘145^\circ: Obtuse Angle, 90∘90^\circ: Right Angle, 180∘180^\circ: Straight Angle, 275∘275^\circ: Reflex Angle.

Explanation:

We compare each value against the standard definitions: angles <90∘< 90^\circ are acute, 90∘90^\circ is right, 90∘<θ<180∘90^\circ < \theta < 180^\circ is obtuse, 180∘180^\circ is straight, and >180∘> 180^\circ is reflex.

Problem 2:

What fraction of a revolution does the hour hand of a clock turn through when it goes from 1212 to 33?

Solution:

312=14 revolution\frac{3}{12} = \frac{1}{4} \text{ revolution}

Explanation:

A full clock face represents 1212 hours and 360∘360^\circ. Moving from 1212 to 33 covers 33 hours. The fraction is 312\frac{3}{12}, which simplifies to 14\frac{1}{4}. In degrees, this is 14×360∘=90∘\frac{1}{4} \times 360^\circ = 90^\circ (a right angle).

Problem 3:

If an interior angle of a shape is 60∘60^\circ, what is the measure of the corresponding reflex angle at that vertex?

Solution:

360∘−60∘=300∘360^\circ - 60^\circ = 300^\circ

Explanation:

The sum of an angle and its reflex angle at a point is 360∘360^\circ. Therefore, to find the reflex angle, subtract the given angle from 360∘360^\circ.

Problem 4:

Find the sum of a right angle and a straight angle.

Solution:

90∘+180∘270∘\begin{array}{r} 90^\circ \\ + 180^\circ \\ \hline 270^\circ \end{array}

Explanation:

A right angle is 90∘90^\circ and a straight angle is 180∘180^\circ. Adding them gives 270∘270^\circ, which is a reflex angle.

Problem 5:

Identify the type of angle shown in the figure which measures 135∘135^\circ. If this angle is subtracted from a complete angle, what is the measure of the remaining angle?

An obtuse angle measuring 135 degrees.

Solution:

135∘135^\circ is an Obtuse angle. Remaining angle = 360∘−135∘=225∘360^\circ - 135^\circ = 225^\circ.

Explanation:

Since 90∘<135∘<180∘90^\circ < 135^\circ < 180^\circ, it is classified as an obtuse angle. A complete angle is 360∘360^\circ. Subtracting the given angle from 360∘360^\circ gives 225∘225^\circ, which is a reflex angle.

Problem 6:

Calculate the measure of the angle formed between the hands of a clock at 4 o'clock. What fraction of a complete revolution is this?

Clock showing 4 o'clock with an angle of 120 degrees.

Solution:

Angle at 4 o'clock = 4×30∘=120∘4 \times 30^\circ = 120^\circ Fraction of revolution = 120360=13\frac{120}{360} = \frac{1}{3}

Explanation:

A clock is divided into 12 equal segments. Each segment between two numbers represents 360∘/12=30∘360^\circ / 12 = 30^\circ. At 4 o'clock, the minute hand is at 12 and the hour hand is at 4, spanning 4 segments. The angle is 4×30∘=120∘4 \times 30^\circ = 120^\circ. This is 1/31/3 of a full 360∘360^\circ revolution.