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Geometry - Types of Angles and Measurement

Grade 5ICSE

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

🔑Concepts

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An angle is formed by two rays meeting at a common endpoint called the vertex. The rays are known as the arms of the angle.

Diagram showing an angle with vertex O and arms OA and OB.
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Angles are classified by their magnitude. An acute angle is less than 90∘90^{\circ}, a right angle is exactly 90∘90^{\circ}, and an obtuse angle is greater than 90∘90^{\circ} but less than 180∘180^{\circ}.

Diagram of a right angle measuring 90 degrees.
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A straight angle measures exactly 180∘180^{\circ} and forms a straight line. A reflex angle is greater than 180∘180^{\circ} but less than 360∘360^{\circ}.

Diagram of a straight angle forming a horizontal line.
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A protractor is the standard tool used to measure angles in degrees (∘^{\circ}). It has two scales: an inner scale and an outer scale, used depending on the direction of the angle's arm.

📐Formulae

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

Right Angle:θ=90∘\text{Right Angle}: \theta = 90^{\circ}

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

Straight Angle:θ=180∘\text{Straight Angle}: \theta = 180^{\circ}

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

Complete Angle:θ=360∘\text{Complete Angle}: \theta = 360^{\circ}

💡Examples

Problem 1:

Classify the following angles based on their measures: (a) 45∘45^{\circ}, (b) 120∘120^{\circ}, (c) 210∘210^{\circ}, (d) 90∘90^{\circ}.

Solution:

(a) 45∘45^{\circ} is an Acute Angle because 0∘<45∘<90∘0^{\circ} < 45^{\circ} < 90^{\circ}. (b) 120∘120^{\circ} is an Obtuse Angle because 90∘<120∘<180∘90^{\circ} < 120^{\circ} < 180^{\circ}. (c) 210∘210^{\circ} is a Reflex Angle because 180∘<210∘<360∘180^{\circ} < 210^{\circ} < 360^{\circ}. (d) 90∘90^{\circ} is a Right Angle because it is exactly 90∘90^{\circ}.

Explanation:

To classify angles, we compare the given degree measure against the standard definitions of acute, right, obtuse, and reflex angles.

Problem 2:

If a straight angle is divided into two parts, and one part measures 75∘75^{\circ}, what is the measure of the other part?

Solution:

Step 1: We know that a straight angle measures 180∘180^{\circ}. Step 2: Let the unknown angle be xx. Step 3: Set up the equation: 75∘+x=180∘75^{\circ} + x = 180^{\circ}. Step 4: Solve for xx: x=180∘−75∘=105∘x = 180^{\circ} - 75^{\circ} = 105^{\circ}. Final Answer: The other part measures 105∘105^{\circ}.

Explanation:

Since the total measure of a straight angle is a fixed value of 180∘180^{\circ}, we subtract the known angle from 180∘180^{\circ} to find the missing portion.

Problem 3:

Identify the type of angle shown in the figure if its measure is 145∘145^{\circ}.

An obtuse angle measuring 145 degrees.

Solution:

The angle is an Obtuse Angle.\text{The angle is an Obtuse Angle.}

Explanation:

An angle that measures more than 90∘90^{\circ} but less than 180∘180^{\circ} is classified as an obtuse angle. Since 90∘<145∘<180∘90^{\circ} < 145^{\circ} < 180^{\circ}, it is obtuse.

Problem 4:

Find the measure of the missing angle xx in the following figure, given that the total angle is a right angle.

A 90 degree angle divided into two parts: 35 degrees and x.

Solution:

x=90∘−35∘=55∘x = 90^{\circ} - 35^{\circ} = 55^{\circ}

Explanation:

A right angle measures 90∘90^{\circ}. To find the value of xx, we subtract the known angle from 90∘90^{\circ}. Thus, 90∘−35∘=55∘90^{\circ} - 35^{\circ} = 55^{\circ}.