GSEB Class 6 Maths Solutions Chapter 14 Practical Geometry Exercise 14.6

Question 1. Draw \( \angle POQ \) of measure \( 75^\circ \) and find its line of symmetry.
Answer:

Steps of construction:
Step I: We construct \( \angle BOA = 90^\circ \), where \( \angle AOQ = 60^\circ \) and \( \angle BOQ = 30^\circ \).
Step II: Draw \( \overrightarrow{OP} \), which is the angle bisector of \( \angle BOQ \), so that \( \angle QOP = \frac{1}{2} \angle BOQ \).

\( \angle QOP = \frac{1}{2} (30^\circ) = 15^\circ \)
Step III: Since \( 60^\circ + 15^\circ = 75^\circ \).
So, \( \angle AOQ + \angle QOP = \angle POA \). Thus, \( \angle POA = 75^\circ \).
In simple words: First, we create a 90-degree angle, which also gives us a 60-degree angle and a 30-degree angle. Then, we cut the 30-degree angle exactly in half to get 15 degrees. When we add the 60-degree angle and this new 15-degree angle together, we get our desired 75-degree angle.

Exam Tip: Remember the basic angle constructions (60°, 90°, 120°) as they are often used as building blocks for other angles. Bisecting an angle precisely is key for getting angles like 30°, 45°, and 75°.

Question 2. Draw an angle of measure \( 147^\circ \) and construct its bisector.
Answer:

Steps of construction:
Step I: First, draw a ray \( \overrightarrow{AB} \).
Step II: Then, use a protractor to construct \( \angle BAC = 147^\circ \).
Step III: With point 'A' as the center and a suitable radius, draw an arc that cuts the arms \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) at points P and Q, respectively.
Step IV: With point P as the center and a radius more than half of PQ, draw an arc.
Step V: With point Q as the center and keeping the same radius, draw another arc that cuts the previous arc at point R.
Step VI: Join points A and R. Extend the line AR.
Thus, \( \overrightarrow{AR} \) is the required angle bisector of \( \angle BAC \).
In simple words: First, draw the 147-degree angle using a protractor. Then, use a compass to find the exact middle of that angle. Draw a line from the corner of the angle through this middle point, and that line will split the 147-degree angle into two equal parts.

Exam Tip: When bisecting large angles, make sure your compass arcs are drawn carefully to find the precise intersection point, as accuracy is crucial for getting full marks.

Question 3. Draw a right angle and construct its bisector.
Answer:

Steps of construction:
Step I: First, draw a line segment (or ray) `l` and mark a point 'O' on it.
Step II: With 'O' as the center and a suitable radius, draw an arc that cuts the line `l` at points A and B.
Step III: With A and B as centers, and a radius greater than half of AB, draw two arcs that intersect each other at point C.
Step IV: Join OC, so that \( \angle AOC \) is a right angle.
Step V: With A and D as centers (where D is the intersection of the arc from O and OC), and a radius more than half of AD, draw two arcs that cut each other at E.
Step VI: Join OE and extend it.
Thus, \( \overrightarrow{OE} \) is the required angle bisector of the right angle \( \angle AOC \), which means \( \overrightarrow{OE} \) bisects the right angle.
In simple words: First, create a perfect 90-degree angle. Then, use your compass to find the middle line that splits this 90-degree angle into two exactly equal parts. Draw that line, and it will be the bisector.

Exam Tip: Constructing a 90-degree angle correctly is the foundation. Ensure your compass openings are consistent when drawing arcs for the bisector to ensure accuracy.

Question 4. Draw an angle of measure \( 153^\circ \) and divide it into four equal parts.
Answer:

Steps of construction:
Step I: Begin by drawing a ray \( \overrightarrow{AB} \).
Step II: Using a protractor, construct \( \angle BAC = 153^\circ \).
Step III: Draw \( \overrightarrow{AD} \), which serves as the bisector of \( \angle BAC \).
Step IV: Next, draw \( \overrightarrow{AE} \), the bisector of \( \angle DAC \).
Step V: Also, draw \( \overrightarrow{AF} \), the bisector of \( \angle BAD \).
Thus, the rays \( \overrightarrow{AE} \), \( \overrightarrow{AD} \), and \( \overrightarrow{AF} \) successfully divide the original \( \angle BAC \) into four equal parts.
In simple words: First, draw a 153-degree angle. Then, cut that angle in half. After that, cut each of those halves in half again. This will give you four small angles that are all the same size.

Exam Tip: To divide an angle into multiple equal parts, repeatedly bisect the existing angles. Always ensure precision with your compass and ruler, especially for the initial angle construction.

Question 5. Construct with ruler and compasses, angles of following measures:
(a) \( 60^\circ \)
Answer:

Steps of construction:
Step I: Draw a ray OA.
Step II: With O as the center and any convenient radius, draw an arc that cuts OA at point P.
Step III: With P as the center and the same radius, draw another arc that cuts the first arc at point Q.
Step IV: Join OQ. The angle \( \angle AOQ \) will be \( 60^\circ \).
In simple words: Draw a line, then use your compass to mark an arc from the end of the line. From that arc, mark another arc using the same compass size. Connect the start of your line to the second mark, and you've made a 60-degree angle.

Exam Tip: The 60° angle is fundamental. Ensure your compass opening remains exactly the same for both arcs to guarantee a perfect 60° angle.

(b) \( 30^\circ \)
Answer:

Steps of construction:
Step I: Construct a \( 60^\circ \) angle, for example, \( \angle AOC \), using the method described for \( 60^\circ \).
Step II: Now, draw the angle bisector of \( \angle AOC \). To do this, with P and Q (the points where the arc intersects the rays of the \( 60^\circ \) angle) as centers and a suitable radius, draw two arcs that intersect.
Step III: Join the vertex O to this intersection point, let's call it B. The ray OB will be the bisector.
The resulting angle, for example, \( \angle AOB \), will measure \( \frac{1}{2} (60^\circ) = 30^\circ \).
In simple words: First, make a 60-degree angle. Then, use your compass to split that 60-degree angle exactly in half. The new line you draw will create two 30-degree angles.

Exam Tip: To construct 30°, always start by constructing a 60° angle. The accuracy of your 30° angle depends entirely on the precision of your initial 60° angle and its bisection.

(c) \( 90^\circ \)
Answer:

Steps of construction:
Step I: First, draw a ray OP.
Step II: With O as the center and any convenient radius, draw an arc that cuts OP at point X.
Step III: With X as the center and the same radius, draw an arc that cuts the first arc at point Y (this marks \( 60^\circ \)).
Step IV: With Y as the center and the same radius, draw another arc that cuts the first arc at point Z (this marks \( 120^\circ \)).
Step V: With Y and Z as centers and a radius greater than half of YZ, draw two arcs that intersect at point Q.
Step VI: Join OQ. The angle \( \angle POQ \) will be \( 90^\circ \).
In simple words: Draw a straight line. Use your compass to make a semi-circle from one end. From where the semi-circle touches the line, make two more marks on the semi-circle (these are 60 and 120 degrees). From these two marks, draw two arcs that cross above the semi-circle. Connect the starting point to where those arcs cross, and you'll have a 90-degree angle.

Exam Tip: Constructing 90° involves creating 60° and 120° first, then bisecting the angle between them. Ensure arcs are clear and intersections are precise for accurate results.

(d) \( 120^\circ \)
Answer:

Steps of construction:
Step I: Draw a ray OA.
Step II: With O as the center and any convenient radius, draw an arc that cuts OA at point X.
Step III: With X as the center and the same radius, draw an arc that cuts the first arc at point Y (this marks \( 60^\circ \)).
Step IV: With Y as the center and the same radius, draw another arc that cuts the first arc at point Z (this marks \( 120^\circ \)).
Step V: Join OZ. The angle \( \angle AOZ \) will measure \( 120^\circ \).
In simple words: Draw a line and an arc from one end. From where the arc meets the line, make two more marks on the arc using the same compass opening. The second mark from the start will give you 120 degrees when connected to the origin.

Exam Tip: Constructing 120° is an extension of constructing 60°. Just make two consecutive 60° arcs along the initial arc, and connect the origin to the second intersection.

(e) \( 45^\circ \)
Answer:

Steps of construction:
Step I: Begin by constructing a \( 90^\circ \) angle, for instance, \( \angle POQ \), using the compass and ruler method.
Step II: Now, draw the angle bisector of \( \angle POQ \). To do this, with X (intersection of arc and OP) and Q (the point on the vertical ray) as centers and a suitable radius, draw two arcs that intersect.
Step III: Join the vertex O to this intersection point, let's call it R. The ray OR will be the bisector.
The resulting angle, for example, \( \angle POR \), will measure \( \frac{1}{2} (90^\circ) = 45^\circ \).
In simple words: First, create a perfect 90-degree angle. Then, use your compass to find the exact middle line that splits this 90-degree angle in half. Draw that line, and it will give you a 45-degree angle.

Exam Tip: Constructing 45° always relies on a precise 90° angle first. Make sure your bisection arcs are accurately drawn to get the exact 45° measure.

(f) \( 135^\circ \)
Answer:

Steps of construction:
Step I: Draw a line segment AB and mark a point O on it.
Step II: Construct a \( 90^\circ \) angle at O, extending upwards, let's call it OC, such that \( \angle AOC = 90^\circ \).
Step III: Extend AO to the left to form a straight line, creating an angle \( \angle BOC = 90^\circ \).
Step IV: Bisect \( \angle BOC \). To do this, with O as the center and a suitable radius, draw an arc. From where it intersects OB and OC, draw two arcs that intersect each other. Join O to this intersection, let's call it D.
Step V: The ray OD will bisect \( \angle BOC \), creating an angle of \( 45^\circ \).
Therefore, \( \angle AOD = \angle AOC + \angle COD = 90^\circ + 45^\circ = 135^\circ \).
In simple words: First, make a 90-degree angle. Then, extend the base line in the opposite direction. Now, split the 90-degree angle that's on the straight line (making it 180 degrees) in half. Adding that half (45 degrees) to your original 90-degree angle will give you 135 degrees.

Exam Tip: Constructing 135° involves combining a 90° angle with a 45° angle. Ensure both base constructions (90° and bisection) are precise for an accurate final angle.

Question 6. Draw an angle of measure \( 45^\circ \) and bisect it.
Answer:

Steps of construction:
Step I: First, draw a ray OA.
Step II: Construct an angle \( \angle AOQ = 45^\circ \) using a compass and ruler (as done in Q5e).
Step III: With C (where the arc cuts OA) and D (where the arc cuts OQ) as centers and a radius more than half of CD, draw two arcs that intersect.
Step IV: Join OB, where B is the intersection point of the arcs, and extend it.
Thus, \( \overrightarrow{OB} \) bisects \( \angle AOQ \) into two equal parts.
This means \( \angle AOB = \frac{1}{2} (45^\circ) = 22\frac{1}{2}^\circ \).
In simple words: First, create a 45-degree angle. Then, use your compass to find the exact middle line that splits this 45-degree angle in half. Draw that line, and it will divide the angle into two smaller angles of \( 22\frac{1}{2} \) degrees each.

Exam Tip: Bisecting an angle requires careful attention to the compass radius. Make sure the arcs you draw from the intersection points of the initial arc are of the same radius for accurate bisection.

Question 7. Draw an angle of measure \( 135^\circ \) and bisect it.
Answer:

Steps of construction:
Step I: First, draw a line segment AB and mark a point O on it.
Step II: Construct \( \angle AOC = 135^\circ \) using a compass and ruler (as done in Q5f).
Step III: Draw \( \overrightarrow{OD} \), which is the bisector of \( \angle AOC \). To do this, with X (where the arc cuts OA) and Y (where the arc cuts OC) as centers and a suitable radius, draw two arcs that intersect.
Step IV: Join O to this intersection point, let's call it D. The ray OD will be the bisector.
Thus, \( \angle AOD = \frac{1}{2} (135^\circ) = 67\frac{1}{2}^\circ \).
In simple words: First, create a 135-degree angle. Then, use your compass to find the middle line that splits this 135-degree angle exactly in half. Draw that line, and it will divide the angle into two smaller angles of \( 67\frac{1}{2} \) degrees each.

Exam Tip: When bisecting large or unconventional angles, always perform the bisection in a fresh set of arcs, originating from the points where the initial angle's arc intersects its arms, to ensure accuracy.

Question 8. Draw an angle of \( 70^\circ \). Make a copy of it using only a straight edge and compasses.
Answer:

Steps of construction:
Step I: Draw a line `l` and mark a point O on it.
Step II: Using a protractor, construct \( \angle AOB = 70^\circ \).
Step III: With O as the center and a suitable radius, draw an arc that cuts \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) at points E and F, respectively.
Step IV: Draw a ray \( \overrightarrow{PQ} \).
Step V: Keeping the same radius as in Step III and with P as the center, draw an arc that cuts \( \overrightarrow{PQ} \) at point R.
Step VI: With R as the center and a radius equal to EF (the distance between E and F from the original angle), draw an arc that cuts the previous arc at point S.
Step VII: Join PS and extend it.
Thus, \( \angle QPS \) is a copy of \( \angle AOB = 70^\circ \).
In simple words: First, draw a 70-degree angle. Then, use your compass to measure the width of that angle. Draw a new line and use the same compass width to create an identical angle on the new line.

Exam Tip: Copying an angle relies on accurately transferring the arc length subtended by the angle. Make sure your compass opening remains constant when transferring the measurement from the original angle to the new one.

Question 9. Draw an angle of \( 40^\circ \). Copy its supplementary angle.
Answer:

Steps of construction:
Step I: Using a protractor, draw \( \angle AOB = 40^\circ \). Extend the line AO to the left to form a straight line, let the point on the extended line be C. The angle \( \angle COB \) is the supplementary angle of \( \angle AOB \), so \( \angle COB = 180^\circ - 40^\circ = 140^\circ \).
Step II: With O as the center and a suitable radius, draw an arc that cuts \( \overrightarrow{OC} \) and \( \overrightarrow{OB} \) at points E and F, respectively.
Step III: Draw a new ray \( \overrightarrow{QR} \).
Step IV: With Q as the center and the same radius used in Step II, draw an arc that cuts \( \overrightarrow{QR} \) at point L.
Step V: With L as the center and a radius equal to EF (the distance between E and F from the original supplementary angle), draw an arc that cuts the previous arc at point S.
Step VI: Join QS and extend it.
Thus, \( \angle PQS \) is a copy of the supplementary angle \( \angle COB \).
In simple words: First, draw a 40-degree angle. Extend one side of this angle into a straight line to find its supplementary angle (which is 180 - 40 = 140 degrees). Then, using only a compass and straightedge, make an exact copy of this 140-degree supplementary angle.

Exam Tip: Understanding supplementary angles (angles that add up to 180°) is key. When copying, ensure you measure the arc of the *supplementary* angle, not the initial angle, to transfer the correct size.