Page 1 1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. Solution: Construct an angle ?BAC and draw a ray OP. Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. Now measure XY with the help of compass. Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it as point N. Now join the points O and N and extend it to the point Q. Here, ?POQ is the required angle. 2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. Solution: We know that obtuse angles are those which are greater than 90 o and less than 180 o . Construct an obtuse angle ?BAC. Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. Now join A and R and extend it to the point X. So the ray AX is the required bisector of ?BAC. By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65 o . 3. Using your protractor, draw an angle of measure 108 o . With this angle as given, drawn an angle of 54 o . Solution: Page 2 1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. Solution: Construct an angle ?BAC and draw a ray OP. Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. Now measure XY with the help of compass. Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it as point N. Now join the points O and N and extend it to the point Q. Here, ?POQ is the required angle. 2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. Solution: We know that obtuse angles are those which are greater than 90 o and less than 180 o . Construct an obtuse angle ?BAC. Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. Now join A and R and extend it to the point X. So the ray AX is the required bisector of ?BAC. By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65 o . 3. Using your protractor, draw an angle of measure 108 o . With this angle as given, drawn an angle of 54 o . Solution: Construct a ray OA. Using protractor, draw an angle ?AOB of 108 o where 108/2 = 54 o Hence, 54 o is half of 108 o . In order to get angle 54 o , we must bisect the angle of 108 o . Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc and name it as point R. Now join the points O and R and extend it to the point X. Here, ?AOX is the required angle of 54 o . 4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45 o . Solution: Construct a ray OA. Using a protractor construct ?AOB of 90 o . Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. Taking P as centre and radius which is more than half of PQ, construct an arc. Taking Q as centre and same radius, construct another arc which intersects the previous arc and name it as point R. Now join the points O and R and extend it to the point X. Here, ?AOX is the required angle of 45 o where ?AOB = 90 o and ?AOX = 45 o . 5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other. Page 3 1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. Solution: Construct an angle ?BAC and draw a ray OP. Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. Now measure XY with the help of compass. Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it as point N. Now join the points O and N and extend it to the point Q. Here, ?POQ is the required angle. 2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. Solution: We know that obtuse angles are those which are greater than 90 o and less than 180 o . Construct an obtuse angle ?BAC. Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. Now join A and R and extend it to the point X. So the ray AX is the required bisector of ?BAC. By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65 o . 3. Using your protractor, draw an angle of measure 108 o . With this angle as given, drawn an angle of 54 o . Solution: Construct a ray OA. Using protractor, draw an angle ?AOB of 108 o where 108/2 = 54 o Hence, 54 o is half of 108 o . In order to get angle 54 o , we must bisect the angle of 108 o . Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc and name it as point R. Now join the points O and R and extend it to the point X. Here, ?AOX is the required angle of 54 o . 4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45 o . Solution: Construct a ray OA. Using a protractor construct ?AOB of 90 o . Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. Taking P as centre and radius which is more than half of PQ, construct an arc. Taking Q as centre and same radius, construct another arc which intersects the previous arc and name it as point R. Now join the points O and R and extend it to the point X. Here, ?AOX is the required angle of 45 o where ?AOB = 90 o and ?AOX = 45 o . 5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other. Solution: We know that the two angles which are adjacent and supplementary are known as linear pair of angles. Construct a line AB and mark a point O on it. By constructing an angle ?AOC we get another angle ?BOC. Now bisect ?AOC using a compass and a ruler and get the ray OX. In the same way bisect ?BOC and get the ray OY. We know that ?XOY = ?XOC + ?COY It can be written as ?XOY = 1/2 ?AOC + 1/2 ?BOC So we get ?XOY = 1/2 (?AOC + ?BOC) We know that ?AOC and ?BOC are supplementary angles ?XOY = 1/2 (180) = 90 o 6. Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the same line. Solution: Construct two lines AB and CD which intersects each other at the point O Since vertically opposite angles are equal we get ?BOC = ?AOD and ?AOC = ?BOD Now bisect angle AOC and construct the bisecting ray as OX. In the same way, we bisect ?BOD and construct bisecting ray OY. We get ?XOA + ?AOD + ?DOY = 1/2 ?AOC + ?AOD + 1/2 ?BOD We know that ?AOC = ?BOD ?XOA + ?AOD + ?DOY = 1/2 ?BOD + ?AOD + 1/2 ?BOD So we get ?XOA + ?AOD + ?DOY = ?AOD + ?BOD AB is a line We know that ?AOD and ?BOD are supplementary angles whose sum is equal to 180 o . ?XOA + ?AOD + ?DOY = 180 o The angles on one side of a straight line is always 180 o and also the sum of angles is 180 o Here, XY is a straight line where OX and OY are in the same line. Page 4 1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. Solution: Construct an angle ?BAC and draw a ray OP. Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. Now measure XY with the help of compass. Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it as point N. Now join the points O and N and extend it to the point Q. Here, ?POQ is the required angle. 2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. Solution: We know that obtuse angles are those which are greater than 90 o and less than 180 o . Construct an obtuse angle ?BAC. Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. Now join A and R and extend it to the point X. So the ray AX is the required bisector of ?BAC. By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65 o . 3. Using your protractor, draw an angle of measure 108 o . With this angle as given, drawn an angle of 54 o . Solution: Construct a ray OA. Using protractor, draw an angle ?AOB of 108 o where 108/2 = 54 o Hence, 54 o is half of 108 o . In order to get angle 54 o , we must bisect the angle of 108 o . Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc and name it as point R. Now join the points O and R and extend it to the point X. Here, ?AOX is the required angle of 54 o . 4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45 o . Solution: Construct a ray OA. Using a protractor construct ?AOB of 90 o . Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. Taking P as centre and radius which is more than half of PQ, construct an arc. Taking Q as centre and same radius, construct another arc which intersects the previous arc and name it as point R. Now join the points O and R and extend it to the point X. Here, ?AOX is the required angle of 45 o where ?AOB = 90 o and ?AOX = 45 o . 5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other. Solution: We know that the two angles which are adjacent and supplementary are known as linear pair of angles. Construct a line AB and mark a point O on it. By constructing an angle ?AOC we get another angle ?BOC. Now bisect ?AOC using a compass and a ruler and get the ray OX. In the same way bisect ?BOC and get the ray OY. We know that ?XOY = ?XOC + ?COY It can be written as ?XOY = 1/2 ?AOC + 1/2 ?BOC So we get ?XOY = 1/2 (?AOC + ?BOC) We know that ?AOC and ?BOC are supplementary angles ?XOY = 1/2 (180) = 90 o 6. Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the same line. Solution: Construct two lines AB and CD which intersects each other at the point O Since vertically opposite angles are equal we get ?BOC = ?AOD and ?AOC = ?BOD Now bisect angle AOC and construct the bisecting ray as OX. In the same way, we bisect ?BOD and construct bisecting ray OY. We get ?XOA + ?AOD + ?DOY = 1/2 ?AOC + ?AOD + 1/2 ?BOD We know that ?AOC = ?BOD ?XOA + ?AOD + ?DOY = 1/2 ?BOD + ?AOD + 1/2 ?BOD So we get ?XOA + ?AOD + ?DOY = ?AOD + ?BOD AB is a line We know that ?AOD and ?BOD are supplementary angles whose sum is equal to 180 o . ?XOA + ?AOD + ?DOY = 180 o The angles on one side of a straight line is always 180 o and also the sum of angles is 180 o Here, XY is a straight line where OX and OY are in the same line. 7. Using ruler and compasses only, draw a right angle. Solution: Construct a ray OA. Taking O as centre and convenient radius construct an arc PQ using a compass intersecting the ray OA at the point Q. Taking P as centre and same radius construct another arc which intersects the arc PQ at the point R. Taking R as centre and same radius, construct an arc which cuts the arc PQ at the point C opposite to P. Using C and R as the centre construct two arcs of radius which is more than half of CR intersecting each other at the point S. Now join the points O and S and extend it to the point B. Here, ?AOB is the required angle of 90 o . 8. Using ruler and compasses only, draw an angle of measure 135 o . Solution: Construct a line AB and mark a point O on it. Taking O as centre and convenient radius, construct an arc PQ using a compass which intersects the line AB at the point P and Q. Taking P as centre and same radius, construct another arc which intersects the arc PQ at the point R. Taking Q as centre and same radius, construct another arc which intersects the arc PQ at the point S which is opposite to P. Considering S and R as centres and radius which is more than half of SR, construct two arcs which intersects each other at the point T. Page 5 1. Draw an angle and label it as ?BAC. Construct another angle, equal to ?BAC. Solution: Construct an angle ?BAC and draw a ray OP. Taking A as centre and suitable radius, construct an arc which intersects AB and AC at points X and Y. Taking O as centre and same radius, construct an arc which intersects the arc OP at the point M. Now measure XY with the help of compass. Taking M as centre and XY as radius construct an arc which intersects the arc which is drawn from O and name it as point N. Now join the points O and N and extend it to the point Q. Here, ?POQ is the required angle. 2. Draw an obtuse angle. Bisect it. Measure each of the angles so obtained. Solution: We know that obtuse angles are those which are greater than 90 o and less than 180 o . Construct an obtuse angle ?BAC. Taking A as centre with appropriate radius construct an arc which intersects AB and AC at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc at the point R. Now join A and R and extend it to the point X. So the ray AX is the required bisector of ?BAC. By measuring ?BAR and ?CAR we get ?BAR = ?CAR = 65 o . 3. Using your protractor, draw an angle of measure 108 o . With this angle as given, drawn an angle of 54 o . Solution: Construct a ray OA. Using protractor, draw an angle ?AOB of 108 o where 108/2 = 54 o Hence, 54 o is half of 108 o . In order to get angle 54 o , we must bisect the angle of 108 o . Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. Taking P as centre and radius which is more than half of PQ construct an arc. Taking Q as centre and same radius construct another arc which intersects the previous arc and name it as point R. Now join the points O and R and extend it to the point X. Here, ?AOX is the required angle of 54 o . 4. Using protractor, draw a right angle. Bisect it to get an angle of measure 45 o . Solution: Construct a ray OA. Using a protractor construct ?AOB of 90 o . Taking O as centre and convenient radius, construct an arc which cuts the sides OA and OB at the points P and Q. Taking P as centre and radius which is more than half of PQ, construct an arc. Taking Q as centre and same radius, construct another arc which intersects the previous arc and name it as point R. Now join the points O and R and extend it to the point X. Here, ?AOX is the required angle of 45 o where ?AOB = 90 o and ?AOX = 45 o . 5. Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other. Solution: We know that the two angles which are adjacent and supplementary are known as linear pair of angles. Construct a line AB and mark a point O on it. By constructing an angle ?AOC we get another angle ?BOC. Now bisect ?AOC using a compass and a ruler and get the ray OX. In the same way bisect ?BOC and get the ray OY. We know that ?XOY = ?XOC + ?COY It can be written as ?XOY = 1/2 ?AOC + 1/2 ?BOC So we get ?XOY = 1/2 (?AOC + ?BOC) We know that ?AOC and ?BOC are supplementary angles ?XOY = 1/2 (180) = 90 o 6. Draw a pair of vertically opposite angles. Bisect each of the two angles. Verify that the bisecting rays are in the same line. Solution: Construct two lines AB and CD which intersects each other at the point O Since vertically opposite angles are equal we get ?BOC = ?AOD and ?AOC = ?BOD Now bisect angle AOC and construct the bisecting ray as OX. In the same way, we bisect ?BOD and construct bisecting ray OY. We get ?XOA + ?AOD + ?DOY = 1/2 ?AOC + ?AOD + 1/2 ?BOD We know that ?AOC = ?BOD ?XOA + ?AOD + ?DOY = 1/2 ?BOD + ?AOD + 1/2 ?BOD So we get ?XOA + ?AOD + ?DOY = ?AOD + ?BOD AB is a line We know that ?AOD and ?BOD are supplementary angles whose sum is equal to 180 o . ?XOA + ?AOD + ?DOY = 180 o The angles on one side of a straight line is always 180 o and also the sum of angles is 180 o Here, XY is a straight line where OX and OY are in the same line. 7. Using ruler and compasses only, draw a right angle. Solution: Construct a ray OA. Taking O as centre and convenient radius construct an arc PQ using a compass intersecting the ray OA at the point Q. Taking P as centre and same radius construct another arc which intersects the arc PQ at the point R. Taking R as centre and same radius, construct an arc which cuts the arc PQ at the point C opposite to P. Using C and R as the centre construct two arcs of radius which is more than half of CR intersecting each other at the point S. Now join the points O and S and extend it to the point B. Here, ?AOB is the required angle of 90 o . 8. Using ruler and compasses only, draw an angle of measure 135 o . Solution: Construct a line AB and mark a point O on it. Taking O as centre and convenient radius, construct an arc PQ using a compass which intersects the line AB at the point P and Q. Taking P as centre and same radius, construct another arc which intersects the arc PQ at the point R. Taking Q as centre and same radius, construct another arc which intersects the arc PQ at the point S which is opposite to P. Considering S and R as centres and radius which is more than half of SR, construct two arcs which intersects each other at the point T. Now join the points O and T which intersects the arc PQ at the point C. Considering C and Q as centres and radius which is more than half of CQ, construct two arcs which intersects each other at the point D. Now join the points O and D and extend it to point X to form the ray OX. Here, ?AOX is the required angle of 135 o . 9. Using a protractor, draw an angle of measure 72 o . With this angle as given, draw angles of measure 36 o and 54 o . Solution: Construct a ray OA. Using protractor construct ?AOB of 72 o Taking O as centre and convenient radius, construct an arc which cut sides OA and OB at the point P and Q. Taking P and Q as centres and radius which is more than half of PQ, construct two arcs which cuts each other at the point R. Now join the points O and R and extend it to the point X. Here, OR intersects the arc PQ at the point C. Taking C and Q as centres and radius which is more than half of CQ, construct two arcs which cuts each other at point T. Now join the points O and T and extend it to the point Y. OX bisects ?AOB It can be written as ?AOX = ?BOX = 72/2 = 36 o OY bisects ?BOX It can be written as ?XOY = ?BOY = 36/2 = 18 o We know that ?AOY = ?AOX + ?XOY = 36 o + 18 o = 54 o Here, ?AOX is the required angle of 36 o and ?AOY is the required angle of 54 o .Read More

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