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Geometry - Angles (Acute, Right, Obtuse, Straight)

Grade 4IB

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

🔑Concepts

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An angle is formed when two rays or line segments meet at a common endpoint called a vertex. Visually, think of this as the corner where two straight edges of a shape come together.

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Angles are measured in units called degrees, represented by the symbol ∘^\circ. A full rotation or a complete circle measures 360∘360^\circ, which represents a full turn back to the starting position.

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An Acute Angle is a 'sharp' angle that measures more than 0∘0^\circ but less than 90∘90^\circ. Visually, it is narrower than a perfect 'L' shape, similar to a partially opened pair of scissors or a slice of pizza.

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A Right Angle measures exactly 90∘90^\circ. It forms a perfect 'L' shape and is often referred to as a 'square corner.' In geometric drawings, a right angle is indicated by a small square drawn at the vertex instead of a curved arc.

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An Obtuse Angle is a 'blunt' or wide angle that measures more than 90∘90^\circ but less than 180∘180^\circ. Visually, it looks wider than a right angle, like a reclining chair pushed back or a hand fan opened more than halfway.

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A Straight Angle measures exactly 180∘180^\circ. It looks like a perfectly flat, straight line. It is formed by two rays pointing in opposite directions from a common vertex, representing a half-turn.

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A Protractor is the standard tool used to measure and draw angles. It usually has two scales (inner and outer) ranging from 0∘0^\circ to 180∘180^\circ to help measure angles opening from either the left or the right side.

📐Formulae

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

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

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

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

💡Examples

Problem 1:

Look at a clock showing the time 2:00. What type of angle is formed between the hour hand and the minute hand? Calculate the degree measure if each hour represents 30∘30^\circ.

Solution:

Step 1: Identify the positions of the hands. At 2:00, the minute hand is on 12 and the hour hand is on 2. Step 2: Calculate the degrees. Since there are 2 hour gaps between 12 and 2, the calculation is 2×30∘=60∘2 \times 30^\circ = 60^\circ. Step 3: Compare with 90∘90^\circ. Because 60∘60^\circ is less than 90∘90^\circ, it is an acute angle.

Explanation:

We use the benchmark of 90∘90^\circ (a right angle) to classify the angle. Since 60∘60^\circ is smaller than a square corner, it must be acute.

Problem 2:

An angle measures 135∘135^\circ. Classify this angle as acute, right, obtuse, or straight, and describe its visual appearance compared to a right angle.

Solution:

Step 1: Check if the angle is 90∘90^\circ. 135∘≠90∘135^\circ \neq 90^\circ, so it is not a right angle. Step 2: Check if it is 180∘180^\circ. 135∘≠180∘135^\circ \neq 180^\circ, so it is not a straight angle. Step 3: Compare to 90∘90^\circ and 180∘180^\circ. Since 90∘<135∘<180∘90^\circ < 135^\circ < 180^\circ, the angle is obtuse.

Explanation:

An obtuse angle is wider than a right angle (90∘90^\circ) but hasn't yet reached a flat line (180∘180^\circ). Visually, it looks like a wide-open corner.