In this chapter, we use the tools of the compass box. We draw and measure line segments and angles. We draw perpendicular lines with a compass and a set square. We also learn to bisect an angle and to draw an angle equal to a given angle.
Key points
A compass box has a ruler, compass, divider, protractor and two set squares.
While measuring a segment, keep the 0 mark of the ruler on the first point.
While using a protractor, keep its centre on the vertex and its base line on one arm.
Two lines that make a right angle are perpendicular to each other.
A bisector divides an angle into two equal angles.
Formulas and terms
Perpendicular lines – lines that intersect at 90°. Written as line n ⊥ line y.
Angle bisector – the ray that divides an angle into two equal parts.
Each half of a bisected angle = angle ÷ 2
Set square – a triangular tool with a right angle. It is used to draw perpendiculars.
Practice Set 14.1
1Measure and write the lengths of all the line segments in the figure.
The figure has six sides — AB, BC, CD, DE, EF and FA.
Measure each segment in your book with a ruler. Sample readings are:
l(AB) = 4.0 cm
l(BC) = 2.0 cm
l(CD) = 2.6 cm
l(DE) = 1.6 cm
l(EF) = 6.6 cm
l(FA) = 3.6 cm
Keep the 0 mark of the ruler exactly on the first point each time.
2Draw line segments of the following measurements: (i) seg AR = 6 cm (ii) seg BT = 4 cm (iii) seg DS = 9 cm.
Step 1: Draw a line with a ruler. Mark point A on it.
Step 2: Keep the 0 mark of the ruler on point A.
Step 3: Mark point R at the 6 cm mark.
Step 4: Write l(AR) = 6 cm.
Draw seg BT = 4 cm and seg DS = 9 cm in the same way.
3Draw angles of the following measurements: (i) m∠BRT = 70° (ii) m∠XYZ = 105° (iii) m∠ABC = 140°.
Step 1: Draw ray RT. R is the vertex.
Step 2: Place the centre of the protractor on R. Keep its base line on ray RT.
Step 3: Start from the 0 on ray RT. Mark a point at 70°. Name it B.
Step 4: Draw ray RB. m∠BRT = 70°.
Draw m∠XYZ = 105° and m∠ABC = 140° in the same way. Here Y and B are the vertices.
Practice Set 14.2
1Draw a line y, take a point D on it, and using a compass draw a line n perpendicular to y through D.
Step 1: Draw line y. Mark point D on it.
Step 2: Keep the compass point on D. Draw arcs on both sides of D to cut line y. Name the points P and Q.
Step 3: Take a distance more than half of PQ in the compass. Draw arcs from P and Q. Let them meet at R.
Step 4: Draw line DR. Name it line n. Line n ⊥ line y.
2Draw line PQ, take a point T on it, and using a set square draw a perpendicular through T.
Step 1: Draw line PQ. Mark point T on it.
Step 2: Keep one arm of the right angle of the set square on line PQ.
Step 3: Slide the set square till its right-angle corner is on T.
Step 4: Draw a line along the other arm through T. This line is perpendicular to PQ.
3Draw line DF. Take two points B and M outside the line and draw perpendiculars to DF through them, using a compass and ruler.
Step 1: Draw line DF. Mark points B and M outside the line.
Step 2: Keep the compass point on B. Draw an arc that cuts line DF at two points.
Step 3: From these two points, draw two arcs on the other side of the line. Let them meet at a point.
Step 4: Join B to this point. This line is perpendicular to DF.
Step 5: Draw the perpendicular through M in the same way.
4Draw a line m. Take points P and Q on opposite sides of the line and draw perpendiculars to m through them. Measure the angles formed.
Step 1: Draw line m. Mark point P above it and point Q below it.
Step 2: Draw a perpendicular from P to line m with a compass.
Step 3: Draw a perpendicular from Q to line m in the same way.
Step 4: Measure the angles with a protractor.
All the angles formed are of 90°.
5Draw a square using the tools in the compass box and measure its sides.
Step 1: Draw seg AB = 5 cm.
Step 2: Draw perpendiculars to AB at A and at B.
Step 3: On the perpendicular at A, mark D so that AD = 5 cm. On the perpendicular at B, mark C so that BC = 5 cm.
Step 4: Join D and C. □ABCD is a square.
On measuring, each side of the square is 5 cm. Each angle is 90°.
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Practice Set 14.3
1Draw the angles below and bisect them using a compass and a ruler.
Step 1: Draw the angle with a protractor.
Step 2: Keep the compass point on the vertex. Draw an arc that cuts both arms.
Step 3: From the two points on the arms, draw two arcs of the same radius inside the angle. Let them meet.
Step 4: Join the vertex to the meeting point. This ray is the bisector.
Each half after bisecting:
(i) 110° ÷ 2 = 55°
(ii) 30° ÷ 2 = 15°
(iii) 45° ÷ 2 = 22.5°
(iv) 90° ÷ 2 = 45°
2Using compass and ruler, draw angles equal to the given angles ∠PQR and ∠GST.
Step 1: Keep the compass point on vertex Q. Draw an arc that cuts both arms of ∠PQR.
Step 2: Draw a ray in your notebook. With the same radius, draw an arc from its end point.
Step 3: Take the distance between the two points on the first arc in the compass. Mark the same distance on the new arc.
Step 4: Join the end point of the ray to this mark. The new angle is equal to ∠PQR.
Draw an angle equal to ∠GST in the same way.
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Exercise 14
1Take an acute angle and bisect it.
Draw ∠ABC of 70°.
Step 1: Keep the compass point on B. Draw an arc that cuts both arms.
Step 2: From these two points, draw arcs of the same radius. Let them meet at D.
Step 3: Draw ray BD. It is the bisector of ∠ABC.
m∠ABD = m∠DBC = 70° ÷ 2 = 35°
2Draw ∠RTG of measure 105° and bisect it.
Step 1: Draw ray TG. Draw ray TR with a protractor so that m∠RTG = 105°.
Step 2: Keep the compass point on T. Draw an arc that cuts both arms.
Step 3: From these two points, draw arcs of the same radius. Let them meet at K.
Step 4: Draw ray TK. It is the bisector of ∠RTG.
m∠RTK = m∠KTG = 105° ÷ 2 = 52.5°
3Measure and write the measures of the angles given below.
(i) about 45° — acute
(ii) about 120° — obtuse
(iii) about 290° — reflex
(i) ∠IHJ – about 45° (acute angle)
(ii) ∠IHJ – about 120° (obtuse angle)
(iii) ∠PQR – about 290° (reflex angle)
Measure the angles in your book with a protractor. For a reflex angle, measure the inner angle and subtract it from 360°.
4Draw a circle using a compass and make a 'line-drawing' (Rekha chitra) using the other tools.
Step 1: Draw a circle of radius 5 cm. Mark its centre O.
Step 2: With the same radius, keep the compass on the circle and mark arcs all around it. You get six points.
Step 3: Draw arcs from these points to make a flower design. Draw lines with the ruler and set squares.
Step 4: Colour the design.
5Measure and write the measures of ∠AXR, ∠RXT, ∠AXT and ∠RXE.
Measure each angle in your book with a protractor. Sample readings are:
m∠AXT = about 80°
m∠AXR = about 120°
m∠RXT = 120° − 80° = about 40°
m∠RXE = 180° − 120° = about 60°
Check: m∠AXR + m∠RXE = 180°, because A, X and E are on one line.
6Draw the following figures using only set squares.
Rectangle ABCD with the diagonal AC
The tilted square with right angles marked
First figure (rectangle with a diagonal):
Step 1: Draw seg AB with the edge of a set square.
Step 2: Keep the right-angle corner of the set square at A. Draw AD. Draw BC at B in the same way.
Step 3: Join D and C. Then join A and C to draw the diagonal.
Second figure (tilted square): Draw the right angles with the corner of the set square. Then join the ends with its straight edge.
7Complete the table — estimate each angle first, then measure it.
Figure of angle
Approximate measure
Exact measure
∠IHJ
about 40°
40°
∠QFE
about 120°
120°
∠DXD
about 90°
90°
Measure the angles in your book with a protractor and write your own readings.
