Measuring and Drawing Angles Using Protractors

Theory: Angles are measured in degrees (°), where one full rotation is divided into 360 equal parts. Each part is 1°.

Example:
Question: What is the measure of an angle if it includes 30 unit parts of a circle?
Solution: It is 30° since 1 unit = 1°, so 30 units = 30°.

Theory: A full circle represents a full turn, i.e., 360°. It helps quantify angles by dividing it into equal parts.

Example:
Question: What is the degree measure of one-quarter of a circle?
Solution: One-quarter of 360° is 360°4=90°\(\frac{360°}{4}\) = 90°4360°​=90°. Hence, it’s a right angle.

Theory: 360 was chosen historically for its divisibility and cultural reasons (e.g., calendars, Babylonian math). It’s divisible by many numbers.

Example:
Question: List any three reasons why 360 is used to measure angles.
Solution: 360 is divisible by 2 to 10 (except 7)
Ancient calendars had 360 days
Easier to break into equal parts (30°, 45°, 60°, etc.)

Theory: A protractor is a tool for measuring angles. It may be a half-circle (180°) or full-circle (360°). It shows degrees marked from 0° to 180°.

Example:
Question: Why does a protractor have two number scales?
Solution: To measure angles from either side (left or right), depending on which arm is aligned with 0°.

Theory:Unlabelled protractors show degree marks without numbers. You count the 10° and 5° marks to get the angle.

Example:
Question: Measure ∠KAL using tick marks.
Solution: Count from 0 to 30 using tick marks. Hence, ∠KAL = 30°.

Theory: Labelled protractors have both inner and outer scales. Align one arm with 0°, then read the angle at the second arm.

Example:
Question: If OT lies on 20° and OS lies on 55° of the outer scale, find ∠TOS.
Solution: ∠TOS = 55° − 20° = 35°.

Theory: Incorrect alignment, misreading the wrong scale, or measuring from the wrong zero can lead to wrong results.

Example:
Question: A student marks ∠U = 35° but the base arm aligns with 180° and measures from the wrong side. Is it correct?
Solution: No. The measurement should begin from 0°, not 180°. The correct reading must be taken from the aligned scale.

Theory: To draw an angle: draw a base line, place the protractor’s center on the vertex, mark the desired degree, and connect.

Example:
Question: Draw ∠TIN = 30°
Solution: Draw IN as base
Place center of protractor at I
Mark 30° from 0° side
Join I to marked point → ∠TIN = 30°

Theory: Angles can be found in clocks, doors, swings, and other objects. Estimating such angles builds understanding of rotation.

Example:
Question: What is the angle between clock hands at 4 o’clock?
Solution: Each hour = 30°. So at 4 o’clock, angle = 4 × 30 = 120°.

Theory: Paper can be folded to create a semicircle and marked to divide it into parts like 45°, 90°, 135°, etc.

Example:
Question: What angle is formed when a semicircle is folded into 4 equal parts?
Solution: 180° ÷ 4 = 45°. So, each fold forms a 45° angle.

Theory: Folding a paper to halve an angle results in an angle bisector. It creates two equal angles from a given one.

Example:
Question: Fold a 90° angle. What is the measure of each resulting angle?
Solution: 90°2=45°\(\frac{90°}{2}\) = 45°290°​=45°. Both new angles are 45° each.

Theory: Estimating angles visually helps build intuition. Drawing helps practice the use of a protractor.

Example:
Question: Estimate and draw an angle close to 110°.
Solution: Draw an angle that looks a little greater than 90°, use a protractor to correct it, and mark as 110°.

Theory: The interior angles of any triangle always add up to 180°.

Example:
Question: Triangle has angles 70°, 60°, and one unknown. Find the unknown angle.
Solution: Sum = 180°, So, unknown = 180° − (70° + 60°) = 50°.

Theory: Certain letters (M, Y, L) form specific angles. This can be used for practice in drawing and measuring angles creatively.

Example:
Question: Draw letter M with two 40° side angles and one 60° middle angle.
Solution: Use a protractor to draw two 40° angles sloping outward and a central 60° angle connecting them.

Theory: Reflex angles measure between 180° and 360°. These are angles that form more than a straight angle.

Example:
Question: Find a reflex angle if the interior angle is 120°.
Solution: Reflex angle = 360° − 120° = 240°.