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

Grade 4IB

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

πŸ”‘Concepts

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An acute angle is a 'sharp' angle that measures less than 90∘90^\circ. It is smaller than a right angle (the corner of a square). Examples include 15∘15^\circ, 45∘45^\circ, and 75∘75^\circ.

A diagram showing an acute angle, which is narrower than a right angle.
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A right angle measures exactly 90∘90^\circ. It looks like the corner of a sheet of paper or the letter 'L'. We often mark it with a small square at the vertex.

A diagram of a 90-degree right angle forming an L-shape.
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An obtuse angle is a 'blunt' angle that is greater than 90∘90^\circ but less than 180∘180^\circ. It is wider than a right angle.

A diagram of an obtuse angle, which is wider than a right angle but not a flat line.
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A straight angle measures exactly 180∘180^\circ. It looks like a perfectly straight horizontal or vertical line and is equal to two right angles put together.

A diagram of a straight angle forming a flat line.

πŸ“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

πŸ’‘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 2Γ—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. 135βˆ˜β‰ 90∘135^\circ \neq 90^\circ, so it is not a right angle. Step 2: Check if it is 180∘180^\circ. 135βˆ˜β‰ 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.

Problem 3:

Identify the type of angle formed by the hands of a clock when it is exactly 4:00. Given that each hour mark on the clock represents 30∘30^\circ, calculate the exact angle measure.

A clock face showing 4:00 with the hands forming an obtuse angle of 120 degrees.

Solution:

  1. At 4:00, the minute hand is on 12 and the hour hand is on 4.
  2. The number of hour gaps between 12 and 4 is 4βˆ’0=44 - 0 = 4 units.
  3. Since each unit is 30∘30^\circ, the total angle is 4Γ—30∘=120∘4 \times 30^\circ = 120^\circ.
  4. Because 120∘120^\circ is greater than 90∘90^\circ but less than 180∘180^\circ, it is an obtuse angle.

Explanation:

We count the intervals between the clock hands and multiply by the degrees per interval to find the total rotation, then classify it based on the four main categories.

Problem 4:

Consider a triangle where one angle is 35∘35^\circ and another is 55∘55^\circ. Find the third angle and classify it. (Note: The sum of angles in a triangle is 180∘180^\circ)

A right-angled triangle with angles labeled 35, 55, and 90 degrees.

Solution:

  1. Sum of known angles: 35∘+55∘=90∘35^\circ + 55^\circ = 90^\circ.
  2. Third angle: 180βˆ˜βˆ’90∘=90∘180^\circ - 90^\circ = 90^\circ.
  3. An angle that measures 90∘90^\circ is a right angle.

Explanation:

By using the property that all internal angles of a triangle add up to 180 degrees, we find the missing value and recognize it as the standard measure for a right angle.