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Geometry - Measuring and drawing angles with a protractor

Grade 4Cambridge (IGCSE)

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

🔑Concepts

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The vertex of the angle must be perfectly aligned with the center point of the protractor. The base line of the angle must align with the 0∘0^{\circ} line on the protractor scale.

A protractor layout showing the alignment of the center point and the zero line.
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A protractor has two scales: an inner scale and an outer scale. Always start counting from 0∘0^{\circ} on the side where the base arm of your angle lies. If the angle opens to the right, use the inner scale; if it opens to the left, use the outer scale.

An angle of 45 degrees being measured from the right-hand zero line.
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To draw a reflex angle (greater than 180∘180^{\circ} but less than 360∘360^{\circ}), subtract the angle from 360∘360^{\circ}. Draw the resulting small angle and the reflex angle is the space on the outside.

Diagram showing a reflex angle as the exterior rotation between two lines.
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Estimating before measuring prevents scale errors. If an angle looks wide (obtuse), the measurement must be between 90∘90^{\circ} and 180∘180^{\circ}.

📐Formulae

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

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

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

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

💡Examples

Problem 1:

You are measuring an angle. You place the center of the protractor on the vertex and align the right arm of the angle with the 0° mark on the inner scale. The other arm points between 50 and 60, specifically at the 55 mark on the inner scale. What is the measurement?

Solution:

55°

Explanation:

Because the baseline arm aligned with the 0° on the inner scale, we must follow the inner scale to find the measurement. The arm points to 55°, which is an acute angle.

Problem 2:

Describe the steps to draw an angle of 130°.

Solution:

  1. Draw a straight horizontal line. 2. Mark a point (vertex) at the left end. 3. Place the protractor center on the vertex and the 0° line on the ray. 4. Using the scale starting at 0°, find 130° and make a dot. 5. Remove the protractor and join the vertex to the dot.

Explanation:

Since 130° is greater than 90°, the resulting angle should be obtuse (wider than a square corner).

Problem 3:

If an angle is measured using the wrong scale and the student reads 150°, but the angle is clearly acute, what is the correct measurement?

Solution:

30°

Explanation:

Standard protractors have scales that add up to 180° at any given point (180−150=30180 - 150 = 30). If the student read 150° (obtuse) but the angle is acute, they likely looked at the wrong scale. The corresponding measurement on the opposite scale is 30°.

Problem 4:

Identify the measurement of the angle ABCABC shown in the diagram, where point BB is the vertex and the base BCBC lies on the right-side zero line.

An obtuse angle of 120 degrees with vertices A, B, and C.

Solution:

The angle measures 120∘120^{\circ}.

Explanation:

  1. Place the protractor center on point BB.
  2. Align line BCBC with the 0∘0^{\circ} mark on the inner scale.
  3. Follow the inner scale around to where line ABAB crosses it.
  4. The line points exactly at 120120 on the inner scale. Since the angle is obtuse, 120∘120^{\circ} is the correct reading.

Problem 5:

Draw an acute angle of 35∘35^{\circ} using a ruler and a protractor.

A construction of a 35 degree acute angle.

Solution:

The resulting drawing shows two lines meeting at a point with a gap of 35∘35^{\circ}.

Explanation:

  1. Use a ruler to draw a horizontal baseline.
  2. Place the protractor center at the left end of the line.
  3. Mark a dot at the 35∘35^{\circ} mark on the outer scale (starting from 0 on the left).
  4. Remove the protractor and use the ruler to connect the endpoint of the baseline to the dot.