CLASS 6 · GANITA PRAKASH · CHAPTER 2
Understand points, lines, rays and angles, compare turns, measure degrees with a protractor and solve angle problems.
73 worked answers · short steps first, explanations when you need them
Textbook pages 14–16
Question 1
One point does not fix a direction, so a line can turn through it in any direction.
Two distinct points fix one straight path.
Infinitely many lines through one point; exactly one line through two distinct points.
Question 2
The segments are LM, MP, PQ and QR.
The two end points each touch one segment; each interior point touches two.
Segments: LM, MP, PQ, QR. L and R lie on exactly one; M, P and Q lie on two.
Question 3
The pictured rays are TA, TB, TN and NB.
A ray starts at its first named point.
TA, TB, TN and NB; T starts TA, TB and TN, but not NB.
Question 4(a)
Place and label the points as stated.
A correct labelled sketch must show rays op and oq meeting at o.
Question 4(b)
Place and label the points as stated.
A correct labelled sketch must show lines xy and pq intersecting at m.
Question 4(c)
Place and label the points as stated.
A correct labelled sketch must show line l through e and f, with d off l.
Question 4(d)
Place and label the points as stated.
A correct labelled sketch must show segment ab with p on it.
Question 5
Points: D, E, O, B, C.
Line DE also passes through O and B.
Rays from O include OC, OB, OE and OD.
Points D, E, O, B, C; line DE (also DO, DB, EO, EB or OB); rays OC, OB, OE, OD; segments DE, DO, DB, EO and EB.
Question 6(a)
O is still the starting point and B lies in the same direction.
Yes. OA and OB name the same ray when A and B lie along the same direction from O.
Question 6(b)
OA starts at O; AO starts at A.
No. They have different starting points.
Textbook pages 19–21
Question 1
For example, the two drawn rays DB and DC meet at D.
One example is ∠BDC; vertex D; arms DB and DC. Other visible angles may also be named.
Question 2
The common endpoint is S, so it must be the middle letter.
For example, ∠TSR (or ∠RST), with vertex S.
Question 3
More than one angle may meet at P.
Because other rays through P can form different angles; ∠APB specifies the two arms.
Question 4
Read each curved mark and name one point on both arms.
∠RTQ and ∠RTP.
Question 5
Join each pair: AB, BC and CA.
Each point is a vertex for one angle.
Three lines; three angles: ∠ABC, ∠BCA and ∠CAB (each may also be named in reverse order).
Question 6
Choose pairs for lines: 4×3÷2=6.
At each vertex the other three points make 3 angle pairs; 4×3=12.
Six lines and 12 smaller angles: three at each vertex. Name each with the vertex in the middle.
Textbook pages 20–24
Question 1
Trace or fold one angle over another with vertices aligned.
Compare how far the arms turn.
Superimpose them with vertices together; the angle with the wider turn is larger. The examples depend on the folds.
Question 2(a)
∠AOB includes ∠XOY plus two extra angle parts.
∠AOB.
Question 2(b)
The arms enclose one part of ∠AOB beyond ∠XOB.
∠AOB.
Question 2(c)
The two marked angles occupy the same turn in the figure.
They are equal.
Question 3
The drawing does not reliably distinguish their sizes.
Place them together or measure them.
No; superimpose or measure the angles.
Question Angles without tracing
Place its centre at the vertex and mark where each arm meets the circle.
Keep one arm aligned while comparing the other.
The angle whose second arm turns farther from the aligned arm is larger; matching arms mean equal angles.
Question Slit activity
Try matching the vertex and both arms.
Arm lengths may differ, but the opening must match.
Only arms with the same angle as the slit pass through fully.
Textbook pages 27–32
Question Straight angle bisector
Fold OA over OB; the crease from O lies halfway between them.
Yes. OC is an angle bisector and makes two equal right angles.
Question Right angle fraction
A straight angle is half a turn; a right angle is half of that.
One quarter of a full turn.
Question Classroom corners
Each rectangular window has four square corners.
Four per rectangular window; multiply by the number of windows.
Question Grid straight angle
Choose two grid points on opposite sides of A in one straight line.
Any opposite pair of collinear grid points through A works.
Question Grid right angle
Draw two perpendicular directions from A, such as horizontal and vertical.
Join A to points on perpendicular grid lines.
Question Perpendicular fold
The crossing creates four equal quarter turns.
Four right angles.
Question Angle words
Acute means sharp; obtuse means blunt.
An acute opening is smaller than a right angle; an obtuse one is wider.
Acute angles look sharper and obtuse angles more open or blunt.
Question Acute-angle pattern
The counts are 3, 12 and 21.
Each new figure adds 9.
3, 12, 21, then 30 acute angles; the count increases by 9 each step.
Question Classify angle examples
Compare each with 90°, 180° and one full turn.
Acute: 0°<a<90°; right: 90°; obtuse: 90°<a<180°; straight: 180°; reflex: 180°<a<360°.
Textbook pages 33–49
Question Circle divisions
Divide 360° by each number of parts.
360°, 180°, 120°, 90°, 72°, 60°, 45°, 40°, 36° and 30°, respectively.
Question Protractor scale
The base ray may point either left or right.
Choose the scale that starts at 0° on the base ray.
The two scales let you measure from either direction; start at 0° on the aligned arm.
Question 1(a)
Align the protractor centre at A and count from AK to AL.
30°.
Question 1(b)
Align at A and read the scale from the starting arm.
50°.
Question 1(c)
Read the turn from AK to AT.
120°.
Question 3
Centre the protractor on each vertex and align its zero line with one arm.
The printed answers are 42° and 116°, respectively.
Question 4
One full turn is 360°.
360°−100°=260°.
260°.
Question 5(a–f)
Read each marked angle from its aligned protractor scale.
(a) 80°; (b) 120°; (c) 60°; (d) 130°; (e) 130°; (f) 60°.
Question 6
Read each arm’s direction from the protractor.
Subtract the directions that bound each angle.
∠BXE=115°; ∠CXE=85°; ∠AXB=65°; ∠BXC=30°.
Question 7
Read each ray from the protractor baseline.
∠PQR=45°; ∠PQS=100°; ∠PQT=150°.
Question 8
Fold the numbered steps in order, open the paper, and trace each crease.
Measure each angle between adjacent creases with a protractor.
Activity answer; angles depend on how accurately the folds are made.
Question 9
Measure each of the three angles and add them.
About 180° for each triangle; small measurement errors may change the total slightly.
Question 10
Place its centre exactly at the angle vertex.
Align the baseline with one arm and read from that arm’s zero.
Correct the centre, baseline and scale alignment before taking the reading.
Question Ray angles
Subtract their directions from the common baseline.
For each pair, use the smaller difference unless the reflex angle is requested.
∠POQ=35°, POR=95°, POS=125°, POT=160°, QOR=60°, QOS=90°, QOT=125°, QOU=145°, ROS=30°, ROT=65°, ROU=85°, SOT=35°, SOU=55°, TOU=20°.
Question Angle bisectors
A straight angle is 180°.
180÷8=22.5°.
22.5° each.
Question Measuring activities
Put the protractor centre on the vertex.
Align one arm with 0° and read where the other crosses.
Activity answer; report each measured angle in degrees.
Question Triangle angles
Measure all three interior angles and add them.
Repeat for several triangle shapes.
Each triangle’s three interior angles add to about 180°; small differences may come from measurement.
Question Protractor mistake
Check that the centre sits on the vertex.
Check that one arm follows the baseline and that the reading starts at its zero.
Correct any centre, baseline or wrong-scale placement before reading the angle.
Question Clock 1(a)
360°÷12=30°.
The full 360° turn is split into 12 equal spaces, so each is 30°.
Question Clock 1(b)
Multiply the 30° gap per hour by the number of hour spaces.
2 o’clock: 60°; 4 o’clock: 120°; 6 o’clock: 180°.
Question Door angle
The door rotates about its hinge.
Yes; the hinge is the vertex, and the door edge and its closed-position direction are the arms.
Question Swing angle
Compare the starting direction of the swing with its later direction.
The pivot is the vertex; the starting and current swing positions show the two directions.
Question Slab slope
Use the horizontal top or ground direction as one arm.
Use the slab edge as the second arm.
The angle is between the horizontal reference and the slab edge; the reference may be extended if it is not drawn.
Question Drawing an angle
Draw base ray IN.
Place the protractor centre at I, align its zero with IN, mark 30° and draw ray IT through the mark.
Draw rays IN and IT with vertex I and a 30° turn between them.
Question Angle games
Measure the true angle with a protractor.
Find the absolute difference between the guess and true measure.
Score=|guess−actual| degrees; the lower total score wins.
Question Figure 2.23
Use the middle letter for each vertex and choose one point on each arm.
Examples include ∠CAP, ∠ACD, ∠APL, ∠DLP, ∠RPL, ∠SLP, ∠PRS, ∠LSR, ∠BRS and ∠CLP; other valid angles may be named too.
Question 2(a)
Draw a base ray.
Place the centre on its endpoint, mark 110° from zero and draw the second ray.
Construct an obtuse angle of 110°.
Question 2(b)
Draw a base ray and mark 40° from its endpoint.
Construct a 40° acute angle.
Question 2(c)
Draw a base ray and mark 75°.
Construct a 75° acute angle.
Question 2(d)
Draw a base ray and mark 112° from zero.
Construct a 112° obtuse angle.
Question 2(e)
Draw a base ray and mark 134° from zero.
Construct a 134° obtuse angle.
Textbook pages 50–53
Question 2(a)
30° is greater than 0° and less than 90°.
Acute.
Question 2(b)
60° is between 0° and 90°.
Acute.
Question 2(c)
102° is more than 90° and less than 180°.
Obtuse.
Question 2(d)
258° is greater than 180° and less than 360°.
Reflex.
Question Supplementary angles
∠BET=180°−80°=100°.
The right-angle marker gives the remaining relation shown: ∠SET=90°−80°=10°.
∠BET=100° and ∠SET=10°.
Question 3
Choose example measures: 30°, 45°, 60°, 90°, 110° and 130°.
Any labelled sketch with three angles below 90°, one exactly 90°, and two between 90° and 180° works.
Question 4
At each vertex, construct rays with the requested opening.
A labelled M-shaped sketch with 40°, 60° and 40° at its three joins.
Question 5
The three angles around the central point total 360°.
150+60+150=360.
Draw three arms with adjacent angles 150°, 60° and 150°.
Question 6
360°÷24=15° per gap.
The largest multiple of 15° less than 90° is 75°.
15° between adjacent spokes; 75° is the largest acute angle.
Question 7
For x, require 0<2x,3x,4x<90, so x<22.5.
Also require 90<5x<180, so 18<x<36.
19°, 20°, 21° or 22°.