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Angles (protractor, unknown angles) Knowledge-Based Study

Math · Geometry · Math · Angles & Lines

Math topic explanations on Angles (protractor, unknown angles).

Study notes

  1. An angle is formed when two rays or line segments meet at a common endpoint called the vertex. The two rays are called the arms of the angle.

    Classifying Angles by Size

    • Acute angle: Measures greater than 0° and less than 90°.
    • Right angle: Measures exactly 90°. It is typically marked with a small square symbol at the vertex.
    • Obtuse angle: Measures greater than 90° and less than 180°.
    • Straight angle: Measures exactly 180° and forms a continuous straight line.
    • Reflex angle: Measures greater than 180° and less than 360°.
    • Full turn (complete angle): Measures exactly 360°.

    A protractor is a semi-circular tool used to measure angles in degrees (°). Standard protractors feature two scales: an inner scale and an outer scale, each running from 0° to 180° in opposite directions.

    Steps to Measure an Angle

    1. Position the origin: Place the centre mark (crosshair) of the protractor directly over the vertex of the angle.
    2. Align the baseline: Ensure the bottom baseline (0° line) of the protractor lies exactly on top of one arm of the angle.
    3. Choose the correct scale: Identify which scale starts at 0° along the aligned arm. If the arm points to the right, follow the scale that reads 0° on the right (typically the inner scale). If the arm points to the left, use the scale that starts at 0° on the left (typically the outer scale).
    4. Read the measurement: Follow the second arm of the angle to where it crosses the curved edge of the protractor and read the number on the chosen scale.

    Geometric reasoning allows us to determine unknown angle measures without direct measurement by applying key angle rules.

    1. Angles on a Straight Line

    Adjacent angles lying on a straight line always add up to 180° (supplementary angles).

    If two angles aa and bb form a straight line, then:

    a+b=180∘a + b = 180^{\circ}

    2. Angles Around a Point

    The sum of all angles meeting at a single point equals one complete revolution, which is 360°.

    ∑angles=360∘\sum \text{angles} = 360^{\circ}

    3. Vertically Opposite Angles

    When two straight lines intersect, the angles directly opposite each other across the vertex are called vertically opposite angles. Vertically opposite angles are always equal.

    To find the value of an unknown angle, identify the geometric property that connects the given angles and set up an equation.

    Worked Example 1: Straight Line

    Three angles sit on a straight line: 42°, xx, and 68°.

    • Sum rule: 42∘+x+68∘=180∘42^{\circ} + x + 68^{\circ} = 180^{\circ}
    • Combine known values: 110∘+x=180∘110^{\circ} + x = 180^{\circ}
    • Subtract from 180°: x=180∘−110∘=70∘x = 180^{\circ} - 110^{\circ} = 70^{\circ}

    Worked Example 2: Angles Around a Point

    Four angles meet at a point: 125°, 85°, 90° (a right angle), and yy.

    • Sum rule: 125∘+85∘+90∘+y=360∘125^{\circ} + 85^{\circ} + 90^{\circ} + y = 360^{\circ}
    • Combine known values: 300∘+y=360∘300^{\circ} + y = 360^{\circ}
    • Subtract from 360°: y=360∘−300∘=60∘y = 360^{\circ} - 300^{\circ} = 60^{\circ}