CBSE Class 7 Maths: Practical Geometry Ex 10.1 - Constructing Parallel Lines

Understanding Parallel Lines and Their Construction

Parallel lines are lines that never meet, no matter how far they are extended. Think of railway tracks – they run side by side without ever crossing. In practical geometry, constructing a line parallel to another through a specific point is a fundamental skill. The key to this construction lies in understanding the properties of angles formed when a transversal line intersects two parallel lines.

There are several methods, but the most common one we'll use in Class 7 involves the property of corresponding angles or alternate interior angles. When a transversal cuts two parallel lines, corresponding angles are equal, and alternate interior angles are also equal. We use the converse of these properties: if we can make corresponding or alternate interior angles equal, then the lines must be parallel. We'll use a ruler and compass to transfer an angle, effectively creating equal corresponding angles, thereby ensuring our new line is perfectly parallel to the original.

Step-by-Step: Constructing a Parallel Line

1. Step 1: Draw the Base Line and Point — Draw a line 'l' using your ruler. Mark a point 'A' anywhere outside this line 'l'. This point 'A' is where our new parallel line will pass through.
2. Step 2: Draw a Transversal Line — Choose any point 'B' on line 'l'. Join 'A' to 'B' using your ruler. The line segment 'AB' acts as a transversal, intersecting line 'l' at 'B'.
3. Step 3: Draw an Arc at B — With 'B' as the center, draw an arc of any convenient radius that intersects line 'l' at point 'C' and the transversal 'AB' at point 'D'. This arc helps us 'measure' the angle.
4. Step 4: Draw a Similar Arc at A — Now, with 'A' as the center and the same radius as in Step 3, draw another arc that intersects the transversal 'AB' at point 'E'.
5. Step 5: Measure the Angle CD — Using your compass, measure the distance between points 'C' and 'D' from the first arc. This is essentially measuring the angle formed at 'B'.
6. Step 6: Mark Intersection Point F — Keeping the compass opening (distance CD) as it is, place the compass needle at point 'E' (where the second arc intersects AB) and draw an arc to intersect the arc drawn in Step 4 at point 'F'.
7. Step 7: Draw the Parallel Line — Draw a line 'm' passing through points 'A' and 'F' using your ruler. This line 'm' will be parallel to line 'l'. We have successfully created equal corresponding angles (or alternate interior angles, depending on how you visualize).

Exam Tip: Precision is Key in Geometry

In practical geometry, even a small error in drawing can lead to an inaccurate construction. Always use a sharp pencil and a well-calibrated ruler and compass. Ensure your arcs are drawn lightly but clearly, and always double-check your measurements, especially when transferring arc lengths. A common mistake is to change the compass radius accidentally between steps. Take your time, draw neatly, and label your points clearly for full marks!

Practice Questions with Solutions

• Q: Draw a line 'p'. Take a point 'X' outside it. Through 'X', draw a line parallel to 'p' using a ruler and compass. A: Step 1: Draw a line 'p'. Mark a point 'X' not on 'p'. Step 2: Take any point 'Y' on 'p'. Join 'X' to 'Y'. Step 3: With 'Y' as center, draw an arc intersecting 'p' at 'A' and 'XY' at 'B'. Step 4: With 'X' as center and the same radius, draw an arc intersecting 'XY' at 'C'. Step 5: With 'C' as center and radius equal to the length of 'AB', draw an arc to cut the previous arc at 'D'. Step 6: Draw a line 'q' passing through 'X' and 'D'. Final answer: Line 'q' is parallel to line 'p'.
• Q: Draw a line segment of length 6 cm. Mark a point 'R' 4 cm above its one endpoint. Construct a line through 'R' parallel to the line segment. A: Step 1: Draw a line segment AB = 6 cm. Mark point 'A'. Step 2: Draw a perpendicular line from 'A' and measure 4 cm upwards to mark point 'R'. (Alternatively, just mark point R visually 4cm above A and proceed as general parallel line construction). Step 3: Join 'A' to 'R'. This acts as our transversal if we choose to make 'A' the intersection point on the original line segment. Step 4: Using the method of equal corresponding angles (as described above), construct an angle at 'R' equal to the angle at 'A' (formed by AR and AB). Step 5: Extend the line through 'R' to form the parallel line 's'. Final answer: Line 's' is parallel to AB.
• Q: Draw a line 'm'. Take a point 'N' at a distance of 3 cm from 'm'. Construct a line parallel to 'm' passing through 'N'. A: Step 1: Draw a line 'm'. Step 2: Draw a perpendicular from any point on 'm' and measure 3 cm along this perpendicular to mark point 'N'. (This ensures 'N' is 3 cm away). Step 3: Now, use the standard construction method: draw a transversal from 'N' intersecting 'm' at a point, say 'P'. Step 4: Construct an angle at 'N' equal to the corresponding angle formed at 'P'. Step 5: Draw the line through 'N' using this constructed angle. Final answer: The new line passes through 'N' and is parallel to 'm'.
• Q: Construct a line 'k'. Take two points 'P' and 'Q' on opposite sides of 'k'. Draw a line through 'P' parallel to 'k' and another line through 'Q' parallel to 'k'. What can you say about these two new lines? A: Step 1: Draw line 'k'. Mark point 'P' on one side and 'Q' on the other side. Step 2: Construct a line 'p'' through 'P' parallel to 'k' using the corresponding angles method (as taught). Step 3: Construct a line 'q'' through 'Q' parallel to 'k' using the same method. Step 4: Observe the relationship between 'p'' and 'q''. Final answer: Both 'p'' and 'q'' are parallel to 'k'. Since two lines parallel to the same line are parallel to each other, lines 'p'' and 'q'' will also be parallel to each other.

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