GSEB Class 6 Maths Solutions Chapter 14 પ્રાયોગિક ભૂમિતિ Exercise 14.6

Question 1. 75°ના માપનો \( \angle POQ \) દોરો અને તેની સમિતિની રેખા શોધો.
Answer:

1. On line \( l \), at point O, draw \( \angle AOQ = 90^\circ \) using a scale and a compass. 2. With a scale and compass, make \( \angle COQ = 60^\circ \). 3. Bisect \( \angle AOC \) with \( \overrightarrow{\mathrm{OP}} \). This gives \( \angle POC = 15^\circ \). 4. So, \( \angle POQ = \angle POC + \angle COQ = 15^\circ + 60^\circ = 75^\circ \). 5. Hence, \( \angle POQ \) is the desired angle with a measure of 75°. 6. Draw the bisector \( \overrightarrow{\mathrm{OM}} \) for \( \angle POQ = 75^\circ \). 7. \( \overrightarrow{\mathrm{OM}} \) serves as the line of symmetry for \( \angle POQ \).
In simple words: First, you draw a 90-degree angle and a 60-degree angle from the same point. The space between them is 30 degrees. Cut that 30-degree space in half to get 15 degrees. Add this 15 degrees to the 60-degree angle to make a 75-degree angle. Then, cut this 75-degree angle exactly in half to find its line of symmetry.

Exam Tip: Remember that 75° can be constructed by bisecting the angle between 60° and 90°. Always clearly show all construction arcs for full marks.

 

Question 2. 147°ના માપનો ખૂણો દોરો અને તેના દ્વિભાજકની રચના કરો.
Answer:

1. First, draw the ray \( \overrightarrow{\mathrm{AB}} \). 2. With a protractor, create \( \angle DAB = 147^\circ \). 3. Using a compass, draw an arc with a suitable radius that cuts the arms \( \overrightarrow{\mathrm{AB}} \) at X and \( \overrightarrow{\mathrm{AD}} \) at Y. 4. From center X, with a radius larger than half of XY, draw another arc. 5. Using the same radius, from center Y, draw an arc that crosses the one drawn before. 6. Call the point where both arcs meet C. Draw the ray \( \overrightarrow{\mathrm{AC}} \). 7. Therefore, \( \overrightarrow{\mathrm{AC}} \) is the correct bisector for \( \angle DAB \).
In simple words: First, draw a 147-degree angle using a protractor. Then, to cut it in half, draw an arc that touches both sides of the angle. From where the arc touches each side, draw two more arcs that cross each other inside the angle. Draw a line from the angle's corner through where these two arcs cross, and that line will split the angle perfectly in two.

Exam Tip: When bisecting an angle, ensure the radius used for the final intersecting arcs is greater than half the distance between the two points on the initial arc for accuracy.

 

Question 3. એક કાટખૂણો દોરો અને તેના દ્વિભાજકની રચના કરો.
Answer:

1. Draw a straight line \( l \). Place point O on this line \( l \). 2. From center O, draw an arc with a good radius. This arc should cut line \( l \) at points P (to the left of O) and A (to the right of O). 3. With P as center and a radius greater than half of PA, draw an arc above the line. 4. With A as center and the same radius, draw another arc intersecting the previous arc at point X. 5. Draw ray \( \overrightarrow{\mathrm{OX}} \). This creates \( \angle XOA = 90^\circ \). 6. With O as center and a suitable radius, draw an arc that intersects the arms \( \overrightarrow{\mathrm{OP}} \) at Y and \( \overrightarrow{\mathrm{OB}} \) at D. 7. With Y as center and a radius greater than half of YD, draw an arc. 8. With D as center and the same radius, draw another arc intersecting the previous arc at C. 9. Draw ray \( \overrightarrow{\mathrm{OC}} \). This ray \( \overrightarrow{\mathrm{OC}} \) is the bisector for the right angle \( \angle BOA \).
In simple words: To draw a right angle, start with a straight line and a point on it. Draw an arc that cuts the line on both sides of the point. From those two cutting points, draw two bigger arcs that cross each other above the line. Draw a line from your starting point to where those arcs cross – that's your 90-degree angle. Then, to cut this 90-degree angle in half, draw an arc inside the angle that touches both of its sides. From those two touching points, draw two more arcs that cross in the middle. A line from the corner of the angle to this crossing point will bisect it.

Exam Tip: Always construct the 90° angle first by drawing arcs from both sides of the point on the line, then use the vertex and the points on the 90° angle's arms to bisect it.

 

Question 4. 153° ના માપનો ખૂણો દોરો અને તેના ચાર સરખા ભાગ કરો.
Answer:

1. With a protractor, make \( \angle BOA = 153^\circ \). 2. Draw the bisector \( \overrightarrow{\mathrm{OC}} \) for \( \angle BOA \). 3. Create the bisector \( \overrightarrow{\mathrm{OX}} \) for \( \angle BOC \). 4. Construct the bisector \( \overrightarrow{\mathrm{OY}} \) for \( \angle COA \). 5. So, the rays \( \overrightarrow{\mathrm{OX}}, \overrightarrow{\mathrm{OC}} \), and \( \overrightarrow{\mathrm{OY}} \) split \( \angle BOA \) into four equal sections.
In simple words: First, draw the 153-degree angle using a protractor. To divide it into four equal parts, you need to cut it in half, then cut each of those halves in half again. The rays that do this will divide the original angle into four equal pieces.

Exam Tip: Dividing an angle into four equal parts involves three bisector lines created by successive bisections of the original angle and its resulting halves.

 

Question 5. માપપટ્ટી અને પરિકરના ઉપયોગથી નીચેનાં માપના ખૂણાઓની રચના કરોઃ
(a) 60°
Answer:

To construct a 60° angle: Draw a ray BC. With B as center and any convenient radius, draw an arc intersecting BC at a point, say D. With D as center and the same radius, draw another arc intersecting the first arc at A. Draw ray BA. \( \angle ABC = 60^\circ \).
In simple words: Draw a straight line. From one end of the line, draw an arc. Keep your compass the same size and put its point where the arc crosses the line. Draw another arc that cuts the first arc. Draw a line from the starting point through where the arcs cross, and you've made a 60-degree angle.

Exam Tip: The construction of a 60° angle is fundamental; ensure the compass radius remains constant throughout the two main arcs.

 

Question 5. (b) 30°
Answer:

To construct a 30° angle: First construct a 60° angle ( \( \angle ABC \) ) as described above. Then, bisect \( \angle ABC \) using the ray BD. This will result in \( \angle DBC = \frac{1}{2}(60^\circ) = 30^\circ \).
In simple words: Make a 60-degree angle first. Then, use your compass to cut that 60-degree angle exactly in half. The new line will make a 30-degree angle.

Exam Tip: Constructing 30° relies on an accurate 60° construction followed by precise bisection; ensure all arcs are clearly drawn.

 

Question 5. (c) 90°
Answer:

To construct a 90° angle: Draw a ray BC. With B as center and a convenient radius, draw a semicircle intersecting BC at a point, say P, and extending past B to a point Q. With P and Q as centers and a radius greater than half of PQ, draw two arcs intersecting each other above BC at A. Draw ray BA. \( \angle ABC = 90^\circ \).
In simple words: Draw a straight line and pick a point on it. Draw a large arc through that point, cutting the line on both sides. From those two cutting points, draw two new arcs that cross each other directly above the first point. Draw a line from the first point up to where these arcs cross, and you will have a perfect 90-degree angle.

Exam Tip: A 90° angle is essentially a perpendicular line constructed at a point on a ray. Ensure the arcs used to find the intersection point are sufficiently wide.

 

Question 5. (d) 120°
Answer:

To construct a 120° angle: Draw a ray BC. With B as center and any convenient radius, draw an arc intersecting BC at a point, say P. With P as center and the same radius, draw an arc intersecting the first arc at Q. With Q as center and the same radius, draw another arc intersecting the first arc at A. Draw ray BA. \( \angle ABC = 120^\circ \).
In simple words: Start by drawing a straight line and picking a point. From that point, draw an arc. Keep your compass the same width. From where the arc crosses the line, draw a second arc. Then, from where that second arc crosses the first arc, draw a third arc. Connect the starting point to where the third arc crosses the first arc, and that's your 120-degree angle.

Exam Tip: A 120° angle is formed by making two consecutive 60° constructions from a point on a straight line.

 

Question 5. (e) 45°
Answer:

To construct a 45° angle: First construct a 90° angle ( \( \angle DBC \) ) as described previously. Then, bisect \( \angle DBC \) using ray BA. This will result in \( \angle ABC = \frac{1}{2}(90^\circ) = 45^\circ \).
In simple words: First, create a 90-degree angle. Then, use your compass to accurately cut this 90-degree angle in half. The new line you draw will form a 45-degree angle.

Exam Tip: Constructing 45° requires a strong understanding of both 90° construction and angle bisection. Show all arcs clearly.

 

Question 5. (f) 135°
Answer:

To construct a 135° angle: 1. Draw a straight line \( l \) and mark a point B on it. 2. At point B, construct a perpendicular ray BM such that \( \angle MBC = 90^\circ \) (where C is to the right on line \( l \)). Also, \( \angle MBN = 90^\circ \) (where N is to the left on line \( l \)). 3. Bisect the angle \( \angle MBN \) using ray BA. This will give \( \angle ABM = 45^\circ \). 4. Therefore, \( \angle ABC = \angle ABM + \angle MBC = 45^\circ + 90^\circ = 135^\circ \).
In simple words: To make a 135-degree angle, first draw a 90-degree angle. Then, from the line that makes the 90-degree angle, extend it backwards to make a straight line (180 degrees). Now, cut the 90-degree angle between your first 90-degree line and the 180-degree line exactly in half (this makes 45 degrees). Add this 45-degree angle to the original 90-degree angle, and you will get 135 degrees.

Exam Tip: Constructing 135° often involves combining a 90° angle with a 45° angle. Ensure your 90° construction is precise and the 45° is accurately bisected from 90°.

 

Question 6. 45° ના માપનો ખૂણો દોરો અને તેને દુભાગો.
Answer:

1. First, draw the ray \( \overrightarrow{\mathrm{OB}} \). 2. At point O on ray \( \overrightarrow{\mathrm{OB}} \), draw \( \angle POB = 90^\circ \). 3. Create the bisector \( \overrightarrow{\mathrm{OA}} \) for \( \angle POB \), which means \( \angle AOB \) will be 45°. 4. Draw the bisector \( \overrightarrow{\mathrm{OD}} \) for \( \angle AOB \), resulting in \( \angle DOB = \frac{1}{2}(45^\circ) = 22 \frac{1}{2}^\circ \). 5. Hence, \( \angle AOB \) is 45°, and \( \overrightarrow{\mathrm{OD}} \) acts as its bisector.
In simple words: First, construct a 90-degree angle. Then, cut that 90-degree angle exactly in half to get a 45-degree angle. After that, cut the new 45-degree angle in half again to bisect it.

Exam Tip: This construction is a two-step bisection: first 90° to 45°, then 45° to 22.5°. Always ensure each bisection is performed accurately.

 

Question 7. 135° ના માપનો ખૂણો દોરો અને તેને દુભાગો.
Answer:

1. First, draw a line \( l \). Choose point P on this line \( l \). 2. At point P on line \( l \), build \( \overleftrightarrow{\mathrm{PA}} \) perpendicular, so that \( \angle APX = 90^\circ \) and \( \angle APY = 90^\circ \). 3. Draw the bisector \( \overrightarrow{\mathrm{PM}} \) for \( \angle APY \), so \( \angle APM \) becomes 45°. 4. So, \( \angle MPX = \angle APM + \angle APX = 45^\circ + 90^\circ = 135^\circ \). 5. Create the bisector \( \overrightarrow{\mathrm{PN}} \) for \( \angle MPX \). 6. \( \angle MPN \) and \( \angle NPX \) both measure \( \frac{1}{2}(135^\circ) = 67 \frac{1}{2}^\circ \).
In simple words: First, construct a 90-degree angle. Then, extend the base line to make a straight line. Bisect the 90-degree angle formed between the vertical ray and the extended part of the base line to get 45 degrees. Add this 45 degrees to the original 90-degree angle to get 135 degrees. Finally, cut this 135-degree angle precisely in half using the bisection method.

Exam Tip: Constructing 135° involves creating a 90° angle and then adding a 45° angle. Remember to accurately bisect the final 135° angle for a precise solution.

 

Question 8. 70° ના માપનો ખૂણો દોરો. માત્ર સીધી પટ્ટી અને પરિકરનો ઉપયોગ કરીને તેની નકલ કરો.
Answer:

1. On a paper, draw a line \( l \). Mark point O on this line. 2. With a protractor, create \( \angle AOB = 70^\circ \). 3. Using a compass, draw an arc with a suitable radius that cuts the arms \( \overrightarrow{\mathrm{OB}} \) at P and \( \overrightarrow{\mathrm{OA}} \) at Q. 4. Draw a line \( \overleftrightarrow{\mathrm{MZ}} \) on your paper. Place point M on this line. 5. From center M, using the same radius (from step 3), draw an arc on ray \( \overrightarrow{\mathrm{MZ}} \) and call the intersection point Y. 6. From center Y, with a radius equal to PQ, draw an arc that crosses the first arc. Name the intersection point X. 7. Draw ray \( \overrightarrow{\mathrm{MX}} \). So, \( \angle XMZ \) is a copy of \( \angle AOB \).
In simple words: First, draw a 70-degree angle. To copy it, draw a new straight line and mark a point on it. Draw an arc from the original angle's corner that cuts both its sides. Measure the length of this arc between the two cutting points with your compass. Transfer this arc and its length to your new line, then draw a ray from your new point through the transferred mark. This creates an identical 70-degree angle.

Exam Tip: Copying an angle requires careful measurement of the arc length between the arms of the original angle; ensure this distance is precisely transferred to the new construction.

 

Question 9. 40° ના માપનો ખૂણો દોરો. તેના પૂરકોણની નકલ કરો.
Answer:

1. On a piece of paper, draw a line \( l \). Mark a point O on this line. 2. With a protractor, create \( \angle AOB = 40^\circ \). 3. So, \( \angle AOC \) acts as the supplementary angle for \( \angle AOB \). 4. Draw a line \( \overleftrightarrow{\mathrm{NM}} \) on your paper. Mark point D on this line. 5. From center O, with a suitable radius, draw an arc that cuts the arms \( \overrightarrow{\mathrm{OA}} \) at X and \( \overrightarrow{\mathrm{OC}} \) at Y. 6. Using the same radius, from center D, draw an arc on line \( \overleftrightarrow{\mathrm{NM}} \) that meets it at point Q. 7. From center Q, with a radius equal to XY, draw an arc that crosses the earlier arc at point R. 8. Draw ray \( \overrightarrow{\mathrm{DR}} \). \( \angle RDN \) matches the measure of \( \angle AOC \). \( \angle AOC \) is the supplementary angle of \( \angle AOB \). Thus, \( \angle RDN \) has the same measure as \( \angle AOC \).
In simple words: First, draw a 40-degree angle. Its supplementary angle is the angle that makes a straight line (180 degrees) when added to it. So, find the 140-degree supplementary angle. Then, draw a new straight line. Copy the 140-degree angle onto this new line using your compass, by measuring the arc that spans the 140-degree angle and transferring that measure.

Exam Tip: To copy a supplementary angle, first identify the supplement by subtracting the given angle from 180°. Then, apply the standard angle copying method to the supplementary angle.