NCERT Class 6 Maths: Practical Geometry Application Ex 14.6 – Constructing Angles

Why Construct Angles with Ruler and Compass?

In Practical Geometry, we learn to draw geometric figures accurately using only a ruler (for drawing straight lines) and a compass (for drawing arcs and circles). You might wonder why we don't just use a protractor to measure and draw angles. The reason is that constructions with a ruler and compass teach us fundamental geometric principles and help us understand the relationships between different parts of a figure. It's like learning to build something from scratch rather than just using a ready-made part.

This method requires precision and patience, which are valuable skills not just in maths but in many aspects of life. In Exercise 14.6, we focus on constructing specific angles such as 60 degrees, 30 degrees, 90 degrees, and 120 degrees. We will also learn how to bisect an angle, which means dividing it exactly into two equal angles. These basic constructions are the building blocks for creating more complex geometric figures later on. Every arc you draw and every line you connect follows a geometric rule that guarantees the accuracy of your final angle. Let's dive in and see how we can create these precise angles with just two simple tools!

Step-by-Step Angle Constructions

1. Constructing a 60° Angle — This is the fundamental construction from which many other angles are derived. 1. Draw a Ray: Start by drawing a ray OA with O as the initial point. 2. Draw an Arc: With O as the centre and any convenient radius, draw an arc that cuts ray OA at a point, let's call it P. 3. Second Arc: With P as the centre and the same radius (this is crucial!), draw another arc that cuts the first arc at a point, let's call it Q. 4. Draw the Ray: Join O to Q and extend it to form a ray OB. The angle ∠BOA is your 60° angle.
2. Bisecting an Angle (e.g., to get 30° from 60°) — Bisecting an angle means dividing it into two equal angles. 1. Draw an Angle: First, draw any angle, say ∠ABC. Let B be the vertex. 2. Draw an Arc: With B as the centre and any convenient radius, draw an arc that cuts ray BA at point P and ray BC at point Q. 3. Third and Fourth Arcs: With P as the centre and a radius greater than half the distance PQ, draw an arc. Then, with Q as the centre and the same radius, draw another arc that cuts the previous arc at a point, let's call it R. 4. Draw the Bisector: Join B to R and extend it to form a ray BD. Ray BD is the angle bisector of ∠ABC, meaning ∠ABD = ∠DBC = (1/2)∠ABC.
3. Constructing a 30° Angle — You can get a 30° angle by bisecting a 60° angle. 1. Construct 60°: Follow the steps to construct a 60° angle (∠BOA). 2. Bisect the 60° Angle: Now, use the angle bisector method on ∠BOA. With O as the vertex, and the arc already drawn, place the compass at P (where the first arc cuts OA) and draw an arc. Then place the compass at Q (where the second arc cut the first arc) and draw another arc to intersect the previous one. Let this intersection be R. 3. Draw the Ray: Join O to R. The angle ∠ROA will be 30°.
4. Constructing a 90° Angle — A 90° angle can be constructed by bisecting the angle formed by a 60° and a 120° mark, or by a more direct method. 1. Draw a Ray: Draw a ray OA with O as the initial point. 2. Draw a Semicircle: With O as the centre and any convenient radius, draw a semicircle that cuts OA at P. 3. Mark Arcs: With P as the centre and the same radius, draw an arc cutting the semicircle at Q (this makes a 60° angle). With Q as the centre and the same radius, draw another arc cutting the semicircle at R (this makes a 120° angle). 4. Bisect QR: Now, with Q as the centre and a radius slightly greater than half of QR, draw an arc. With R as the centre and the same radius, draw another arc that cuts the previous arc at S. 5. Draw the Ray: Join O to S and extend it to form ray OB. The angle ∠BOA is your 90° angle.
5. Constructing a 120° Angle — This is similar to the 60° angle construction, but you mark two 60° segments. 1. Draw a Ray: Draw a ray OA with O as the initial point. 2. Draw an Arc: With O as the centre and any convenient radius, draw an arc that cuts ray OA at P. 3. First 60° Mark: With P as the centre and the same radius, draw an arc cutting the first arc at Q. 4. Second 60° Mark: With Q as the centre and the same radius, draw another arc cutting the first arc at R. 5. Draw the Ray: Join O to R and extend it to form ray OB. The angle ∠BOA is your 120° angle.

Exam Tip: Achieving Precision in Constructions

Accuracy is key in practical geometry. Here are some tips to ensure your constructions are perfect for exams:

• Sharp Pencil: Always use a well-sharpened pencil for fine lines and accurate points. A blunt pencil will lead to thick lines and imprecise intersections.
• Firm Compass: Make sure your compass is firm and doesn't slip. The radius must remain constant when required, especially during arc intersections.
• Clear Arcs: Draw arcs long enough so that their intersection points are clear and easy to identify. Don't make them too faint or too short.
• Label Points: Label all points (O, A, P, Q, etc.) clearly as you draw them. This helps you follow the steps and makes your construction easy to understand for anyone checking it.
• Check with Protractor (after construction): While you shouldn't use a protractor during construction, you can use it afterwards to check if your angles are correct. This helps you identify if you made any errors in your steps.

Practice Questions with Solutions

• Q: Construct an angle of 60° using a ruler and compass. A: Step 1: Draw a ray OA with initial point O. Step 2: With O as the centre and a convenient radius, draw an arc that cuts OA at point P. Step 3: With P as the centre and the same radius, draw another arc cutting the previous arc at point Q. Step 4: Draw a ray OB passing through Q. Angle BOA is the required 60° angle. Final answer: Angle BOA = 60° constructed.
• Q: Construct an angle of 90° using a ruler and compass. A: Step 1: Draw a ray OA with initial point O. Step 2: With O as the centre and a convenient radius, draw an arc intersecting OA at P. Extend this arc to form a semicircle. Step 3: With P as the centre and the same radius, draw an arc cutting the semicircle at Q. Step 4: With Q as the centre and the same radius, draw another arc cutting the semicircle at R. Step 5: With Q and R as centres, and a radius greater than half of QR, draw two arcs that intersect each other at S. Step 6: Draw a ray OB passing through S. Angle BOA is the required 90° angle. Final answer: Angle BOA = 90° constructed.
• Q: Construct an angle of 30° using a ruler and compass. A: Step 1: First, construct a 60° angle. Draw ray OA. With O as centre, draw an arc cutting OA at P. With P as centre and same radius, cut the first arc at Q. Draw ray OB through Q. Angle BOA = 60°. Step 2: Now, bisect angle BOA. With O as centre and a convenient radius, draw an arc cutting OA at P and OB at Q (if not already done). Step 3: With P as centre and a radius greater than half of PQ, draw an arc. With Q as centre and the same radius, draw another arc intersecting the previous one at R. Step 4: Draw a ray OC passing through R. Angle COA is the required 30° angle. Final answer: Angle COA = 30° constructed.
• Q: Draw an angle of 80° using a protractor and then bisect it using a ruler and compass. A: Step 1: Draw a ray OA. Place the protractor at O, mark 80° and draw ray OB. So, ∠BOA = 80°. Step 2: Now, bisect ∠BOA. With O as the centre and a convenient radius, draw an arc that cuts OA at P and OB at Q. Step 3: With P as the centre and a radius greater than half of PQ, draw an arc. With Q as the centre and the same radius, draw another arc intersecting the previous one at R. Step 4: Draw a ray OC passing through R. Ray OC is the bisector of ∠BOA. Final answer: The 80° angle is bisected, resulting in two 40° angles, ∠COA and ∠COB.

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