18.1 Angular measurement
Geometry is a part of mathematics in which the properties of points, lines, surfaces and solids are investigated.
An angle is the amount of rotation between two straight lines.
Angles may be measured in either degrees or radians (see Section 23.3).
1 revolution= 360 degrees, thus 1 degree = 1
360th of one revo-lution. Also 1 minute= 1
60th of a degree and 1 second = 1 60th of a minute. 1 minute is written as 1 and 1 second is written as 1 Thus 1◦= 60and 1= 60
Problem 1. Add 14◦53 and 37◦19
14◦53 37◦19 52◦12
1◦
53 + 19 = 72 . Since 60 = 1◦, 72 = 1◦12 . Thus the 12 is placed in the minutes column and 1◦is carried in the degrees column.
Then 14◦+ 37◦+ 1◦(carried)= 52◦ Thus 14◦53+ 37◦19= 52◦12
Problem 2. Subtract 15◦47 from 28◦13
27◦
28◦13 15◦47 12◦26
13 − 47 cannot be done. Hence 1◦or 60 is ‘borrowed’ from the degrees column, which leaves 27◦ in that column. Now (60 + 13 )− 47 = 26 , which is placed in the minutes column.
27◦− 15◦= 12◦, which is placed in the degrees column.
Thus 28◦13− 15◦47= 12◦26
Problem 3. Determine (a) 13◦42 51 + 48◦22 17 (b) 37◦12 8 − 21◦17 25
(a) 13◦42 51
48◦22 17 Adding: 62◦58
1◦1
(b) 36◦11
37◦12 8 21◦17 25 Subtracting: 15◦5443
Problem 4. Convert (a) 24◦42 (b) 78◦15 26 to degrees and decimals of a degree.
(a) Since 1 minute= 1
60th of a degree,
42 =
42 60
◦
= 0.70◦
Hence 24◦42= 24.70◦ Telephone:+254728801352
Email Address: <[email protected]><[email protected]>
Property of Mr. Alphonce Kimutai Kirui
(b) Since 1 second= 1
60th of a minute,
26 =
26 60
= 0.4333
Hence 78◦15 26 = 78◦15.4˙3
15.4333 =
15.4˙3 60
◦
= 0.2572◦,
correct to 4 decimal places.
Hence 78◦1526= 78.26◦, correct to 4 significant places.
Problem 5. Convert 45.371◦ into degrees, minutes and seconds.
Since 1◦= 60 , 0.371◦= (0.371 × 60) = 22.26
Since: 1 = 60 , 0.26 = (0.26×60) = 15.6 = 16 to the nearest second.
Hence 45.371◦= 45◦2216
Now try the following exercise
Exercise 65 Further problems on angular measurement (Answers on page 277) 1. Add together the following angles:
(a) 32◦19 and 49◦52 (b) 29◦42 , 56◦37 and 63◦54 (c) 21◦33 27 and 78◦42 36 (d) 48◦11 19 , 31◦41 27 and 9◦9 37 2. Determine:
(a) 17◦− 9◦49 (b) 43◦37 − 15◦49 (c) 78◦29 41 − 59◦41 52 (d) 114◦− 47◦52 37 3. Convert the following angles to degrees and decimals of
a degree, correct to 3 decimal places:
(a) 15◦11 (b) 29◦53 (c) 49◦42 17 (d) 135◦7 19 4. Convert the following angles into degrees, minutes and
seconds:
(a) 25.4◦ (b) 36.48◦ (c) 55.724◦ (d) 231.025◦
18.2 Types and properties of angles
(a) (i) Any angle between 0◦and 90◦is called an acute angle.
(ii) An angle equal to 90◦is called a right angle.
(iii) Any angle between 90◦and 180◦is called an obtuse angle.
(iv) Any angle greater than 180◦and less than 360◦is called a reflex angle.
(b) (i) An angle of 180◦lies on a straight line.
(ii) If two angles add up to 90◦they are called complemen-tary angles.
(iii) If two angles add up to 180◦they are called supplemen-tary angles.
(iv) Parallel lines are straight lines which are in the same plane and never meet. (Such lines are denoted by arrows, as in Fig. 18.1).
(v) A straight line which crosses two parallel lines is called a transversal (see MN in Fig. 18.1).
P
R
Q
h e S g
M
N
f
d a c b
Fig. 18.1
(c) With reference to Fig. 18.1:
(i) a= c, b = d, e = g and f = h. Such pairs of angles are called vertically opposite angles.
(ii) a= e, b = f , c = g and d = h. Such pairs of angles are called corresponding angles.
(iii) c= e and b = h. Such pairs of angles are called alternate angles.
(iv) b+ e = 180◦and c+ h = 180◦. Such pairs of angles are called interior angles.
Problem 6. State the general name given to the following angles:
(a) 159◦ (b) 63◦ (c) 90◦ (d) 227◦
(a) 159◦lies between 90◦ and 180◦ and is therefore called an obtuse angle.
(b) 63◦lies between 0◦and 90◦and is therefore called an acute angle.
(c) 90◦is called a right angle.
Property of Mr. Alphonce Kimutai Kirui
Geometry and triangles 133
(d) 227◦is greater than 180◦and less than 360◦and is therefore called a reflex angle.
Problem 7. Find the angles complementary to (a) 41◦ (b) 58◦39
(a) The complement of 41◦is (90◦− 41◦), i.e. 49◦
(b) The complement of 58◦39 is (90◦− 58◦39 ), i.e. 31◦21
Problem 8. Find the angles supplementary to (a) 27◦ (b) 111◦11
(a) The supplement of 27◦is (180◦− 27◦), i.e. 153◦
(b) The supplement of 111◦11 is (180◦− 111◦11 ), i.e. 68◦49
Problem 9. Two straight lines AB and CD intersect at 0. If
∠AOC is 43◦, find∠AOD, ∠DOB and ∠BOC.
From Fig. 18.2, ∠AOD is supplementary to ∠AOC. Hence
∠AOD = 180◦− 43◦= 137◦. When two straight lines intersect the vertically opposite angles are equal. Hence∠DOB = 43◦and
∠BOC = 137◦
A D
C B 43° 0
Fig. 18.2
Problem 10. Determine angle β in Fig. 18.3.
b
133° a
Fig. 18.3
α= 180◦− 133◦= 47◦(i.e. supplementary angles).
α= β = 47◦(corresponding angles between parallel lines).
Problem 11. Determine the value of angle θ in Fig. 18.4.
A 23°37′ B
F u G
E
C 35°49′ D
Fig. 18.4
Let a straight line FG be drawn through E such that FG is parallel to AB and CD.∠BAE = ∠AEF (alternate angles between paral-lel lines AB and FG), hence∠AEF = 23◦37 .∠ECD = ∠FEC (alternate angles between parallel lines FG and CD), hence
∠FEC = 35◦49
Angle θ= ∠AEF + ∠FEC = 23◦37 + 35◦49 = 59◦26
Problem 12. Determine angles c and d in Fig. 18.5.
d 46°
b a
c
Fig. 18.5
b= 46◦(corresponding angles between parallel lines).
Also b+ c + 90◦= 180◦(angles on a straight line).
Hence 46◦+ c + 90◦= 180◦, from which c= 44◦. b and d are supplementary, hence d= 180◦− 46◦= 134◦. Alternatively, 90◦+ c = d (vertically opposite angles).
Now try the following exercise
Exercise 66 Further problems on types and properties of angles (Answers on page 277)
1. State the general name given to the (a) 63◦ (b) 147◦ (c) 250◦
2. Determine the angles complementary to the following:
(a) 69◦ (b) 27◦37 (c) 41◦3 43 3. Determine the angles supplementary to
(a) 78◦ (b) 15◦ (c) 169◦41 11 Telephone:+254728801352
Email Address: <[email protected]><[email protected]>
Property of Mr. Alphonce Kimutai Kirui
4. With reference to Fig. 18.6, what is the name given to the line XY . Give examples of each of the following:
(a) vertically opposite angles.
(b) supplementary angles.
(c) corresponding angles.
(d) alternate angles.
12 43 x
y 56 87
Fig. 18.6
5. In Fig. 18.7, find angle α.
a
137°29′
16°49′
Fig. 18.7
6. In Fig. 18.8, find angles a, b and c.
c
b 29°
69° a
Fig. 18.8
7. Find angle β in Fig. 18.9.
133°
98° b
Fig. 18.9
18.3 Properties of triangles
A triangle is a figure enclosed by three straight lines. The sum of the three angles of a triangle is equal to 180◦. Types of triangles:
(i) An acute-angled triangle is one in which all the angles are acute, i.e. all the angles are less than 90◦.
(ii) A right-angled triangle is one which contains a right angle.
(iii) An obtuse-angled triangle is one which contains an obtuse angle, i.e. one angle which lies between 90◦and 180◦. (iv) An equilateral triangle is one in which all the sides and all
the angles are equal (i.e. each 60◦).
(v) An isosceles triangle is one in which two angles and two sides are equal.
(vi) A scalene triangle is one with unequal angles and therefore unequal sides.
With reference to Fig. 18.10:
A
B C u
c b
a Fig. 18.10
(i) Angles A, B and C are called interior angles of the triangle.
(ii) Angle θ is called an exterior angle of the triangle and is equal to the sum of the two opposite interior angles, i.e. θ= A + C (iii) a+ b + c is called the perimeter of the triangle.
Problem 13. Name the types of triangles shown in Fig. 18.11.
2
(a) (b)
2
2.6
2.1 2.8
107°
39° 51°
2.5
2.5 2.1 2
(d) (e)
(c)
Fig. 18.11 Property of Mr. Alphonce Kimutai Kirui
Geometry and triangles 135
(a) Equilateral triangle.
(b) Acute-angled scalene triangle.
(c) Right-angled triangle.
(d) Obtuse-angled scalene triangle.
(e) Isosceles triangle.
Problem 14. Determine the value of θ and α in Fig. 18.12.
A
B C
D
E u
a 62°
15°
Fig. 18.12
In triangle ABC, ∠A + ∠B + ∠C = 180◦ (angles in a trian-gle add up to 180◦), hence∠C = 180◦− 90◦− 62◦= 28◦. Thus
∠DCE = 28◦(vertically opposite angles).
θ= ∠DCE + ∠DEC (exterior angle of a triangle is equal to the sum of the two opposite interior angles). Hence
∠θ = 28◦+ 15◦= 43◦
∠α and ∠DEC are supplementary, thus α= 180◦− 15◦= 165◦
Problem 15. ABC is an isosceles triangle in which the unequal angle BAC is 56◦. AB is extended to D as shown in Fig. 18.13. Determine the angle DBC.
56° A
B
D
C
Fig. 18.13
Since the three interior angles of a triangle add up to 180◦then 56◦+ ∠B + ∠C = 180◦, i.e.∠B + ∠C = 180◦− 56◦= 124◦. Triangle ABC is isosceles hence∠B = ∠C =124◦
2 = 62◦.
∠DBC = ∠A + ∠C (exterior angle equals sum of two interior opposite angles), i.e.∠DBC = 56◦+ 62◦= 118◦[Alternatively,
∠DBC + ∠ABC = 180◦(i.e. supplementary angles)].
Problem 16. Find angles a, b, c, d and e in Fig. 18.14.
e 55°
62° d c
b a
Fig. 18.14
a= 62◦ and c= 55◦ (alternate angles between parallel lines) 55◦+ b + 62◦= 180◦(angles in a triangle add up to 180◦), hence b= 180◦− 55◦− 62◦= 63◦
b= d = 63◦(alternate angles between parallel lines).
e+ 55◦+ 63◦= 180◦(angles in a triangle add up to 180◦), hence e= 180◦− 55◦− 63◦= 62◦
[Check: e= a = 62◦ (corresponding angles between parallel lines)].
Now try the following exercise
Exercise 67 Further problems on properties of triangles (Answers on page 278)
1. In Fig. 18.15, (i) and (ii), find angles w, x, y and z.
What is the name given to the types of triangle shown in (i) and (ii)?
2 cm 2 cm 70° 110°
110° x
(i) (ii)
y v
z
Fig. 18.15
2. Find the values of angles a to g in Fig. 18.16 (i) and (ii).
56°29′
14°41′ b
(i) (ii)
a 131°
68°
f g d e c
Fig. 18.16 Telephone:+254728801352
Email Address: <[email protected]><[email protected]>
Property of Mr. Alphonce Kimutai Kirui
3. Find the unknown angles a to k in Fig. 18.17. is produced to D. If the bisectors of∠ABC and ∠ACD meet at E, determine∠BEC.
5. If in Fig. 18.18, triangle BCD is equilateral, find the interior angles of triangle ABE.
E
Two triangles are said to be congruent if they are equal in all respects, i.e. three angles and three sides in one triangle are equal to three angles and three sides in the other triangle. Two triangles are congruent if:
(i) the three sides of one are equal to the three sides of the other (SSS),
(ii) they have two sides of the one equal to two sides of the other, and if the angles included by these sides are equal (SAS),
(iii) two angles of the one are equal to two angles of the other and any side of the first is equal to the corresponding side of the other (ASA), or
(iv) their hypotenuses are equal and if one other side of one is equal to the corresponding side of the other (RHS).
Problem 17. State which of the pairs of triangles shown in Fig. 18.19 are congruent and name their sequence.
B
(a) Congruent ABC, FDE (Angle, side, angle, i.e. ASA).
(b) Congruent GIH , JLK (Side, angle, side, i.e. SAS).
(c) Congruent MNO, RQP (Right-angle, hypotenuse, side, i.e.
RHS).
(d) Not necessarily congruent. It is not indicated that any side coincides.
(e) Congruent ABC, FED (Side, side, side, i.e. SSS).
Problem 18. In Fig. 18.20, triangle PQR is isosceles with Z the mid-point of PQ. Prove that triangle PXZ and QYZ are congruent, and that triangles RXZ and RYZ are congruent.
Determine the values of angles RPZ and RXZ.
P Q
Since triangle PQR is isosceles PR= RQ and thus
∠QPR = ∠RQP
∠RXZ = ∠QPR + 28◦ and ∠RYZ = ∠RQP + 28◦ (exterior angles of a triangle equal the sum of the two interior opposite angles). Hence∠RXZ = ∠RYZ.
∠PXZ = 180◦− ∠RXZ and ∠QYZ = 180◦− ∠RYZ. Thus
∠PXZ = ∠QYZ.
Triangles PXZ and QYZ are congruent since
∠XPZ = ∠YQZ, PZ = ZQ and ∠XZP = ∠YZQ (ASA) Property of Mr. Alphonce Kimutai Kirui
Geometry and triangles 137
Hence XZ= YZ
Triangles PRZ and QRZ are congruent since PR= RQ,
∠RPZ = ∠RQZ and PZ = ZQ (SAS). Hence ∠RZX = ∠RZY Triangles RXZ and RYZ are congruent since ∠RXZ = ∠RYZ, XZ= YZ and ∠RZX = ∠RZY (ASA). ∠QRZ = 67◦ and thus
∠PRQ = 67◦+ 67◦= 134◦. Hence
∠RPZ = ∠RQZ =180◦− 134◦ 2 = 23◦
∠RXZ = 23◦+ 28◦= 51◦(external angle of a triangle equals the sum of the two interior opposite angles).
Now try the following exercise
Exercise 68 Further problems on congruent triangles (Answers on page 278)
1. State which of the pairs of triangles in Fig. 18.21 are congruent and name their sequence.
A AB and BC, respectively, such that AD= CE. Show that triangles AEB and CDB are congruent.
18.5 Similar triangles
Two triangles are said to be similar if the angles of one triangle are equal to the angles of the other triangle. With reference
to Fig. 18.22: Triangles ABC and PQR are similar and the corresponding sides are in proportion to each other, i.e.
p
Problem 19. In Fig. 18.23, find the length of side a.
A ABC and DEF are similar, since their angles are the same. Since corresponding sides are in proportion to each other then:
a
Problem 20. In Fig. 18.24, find the dimensions marked r and p.
Property of Mr. Alphonce Kimutai Kirui
In triangle PQR,∠Q = 180◦− 90◦− 35◦= 55◦ In triangle XYZ,∠X = 180◦− 90◦− 55◦= 35◦
Hence triangles PQR and ZYX are similar since their angles are the same. The triangles may be redrawn as shown in Fig. 18.25.
Problem 21. In Fig. 18.26, show that triangles CBD and CAE are similar and hence find the length of CD and BD.
A
Since BD is parallel to AE then ∠CBD = ∠CAE and
∠CDB = ∠CEA (corresponding angles between parallel lines).
Also∠C is common to triangles CBD and CAE. Since the angles in triangle CBD are the same as in triangle CAE the triangles are similar. Hence, by proportion:
CB
Problem 22. A rectangular shed 2 m wide and 3 m high stands against a perpendicular building of height 5.5 m. A ladder is used to gain access to the roof of the building.
Determine the minimum distance between the bottom of the ladder and the shed.
A side view is shown in Fig. 18.27, where AF is the minimum length of ladder. Since BD and CF are parallel,∠ADB = ∠DFE (corresponding angles between parallel lines). Hence triangles BAD and EDF are similar since their angles are the same.
AB= AC − BC = AC − DE = 5.5 − 3 = 2.5 m. distance from bottom of ladder to the shed
A
Property of Mr. Alphonce Kimutai Kirui
Geometry and triangles 139
Now try the following exercise
Exercise 69 Further problems on similar triangles (Answers on page 278)
1. In Fig. 18.28, find the lengths x and y.
14.58 mm
25.69 mm
4.74 mm
7.36 mm 111°
32°
32° 37° x
y
Fig. 18.28
2. PQR is an equilateral triangle of side 4 cm. When PQ and PR are produced to S and T , respectively, ST is found to be parallel with QR. If PS is 9 cm, find the length of ST . X is a point on ST between S and T such that the line PX is the bisector of ∠SPT. Find the length of PX .
3. In Fig. 18.29, find (a) the length of BC when AB= 6 cm, DE = 8 cm and DC = 3 cm, (b) the length of DE when EC= 2 cm, AC = 5 cm and AB= 10 cm.
B
C A
D E
Fig. 18.29
4. In Fig. 18.30, AF= 8 m, AB = 5 m and BC = 3 m. Find the length of BD.
C
B
A F
E D
Fig. 18.30
18.6 Construction of triangles
To construct any triangle the following drawing instruments are needed:
(i) ruler and/or straight edge, (ii) compass, (iii) protractor, (iv) pencil. For actual constructions, see Problems 23 to 26 which follow.
Problem 23. Construct a triangle whose sides are 6 cm, 5 cm and 3 cm.
With reference to Fig. 18.31:
F E
B
A 6 cm
C G
D
Fig. 18.31
(i) Draw a straight line of any length, and with a pair of compasses, mark out 6 cm length and label it AB.
(ii) Set compass to 5 cm and with centre at A describe arc DE.
(iii) Set compass to 3 cm and with centre at B describe arc FG.
(iv) The intersection of the two curves at C is the vertex of the required triangle. Join AC and BC by straight lines.
It may be proved by measurement that the ratio of the angles of a triangle is not equal to the ratio of the sides (i.e. in this problem, the angle opposite the 3 cm side is not equal to half the angle opposite the 6 cm side).
Problem 24. Construct a triangle ABC such that a= 6 cm, b= 3 cm and ∠C = 60◦.
Telephone:+254728801352
Email Address: <[email protected]><[email protected]>
Property of Mr. Alphonce Kimutai Kirui
A
C B a⫽6 cm
b⫽3 cm 60°
Fig. 18.32
With reference to Fig. 18.32:
(i) Draw a line BC, 6 cm long.
(ii) Using a protractor centred at C make an angle of 60◦to BC.
(iii) From C measure a length of 3 cm and label A.
(iv) Join B to A by a straight line.
Problem 25. Construct a triangle PQR given that QR= 5 cm, ∠Q = 70◦and∠R = 44◦.
With reference to Fig. 18.33:
Q P R′
Q′
5 cm
70° 44°
R Fig. 18.33
(i) Draw a straight line 5 cm long and label it QR.
(ii) Use a protractor centred at Q and make an angle of 70◦. Draw QQ .
(iii) Use a protractor centred at R and make an angle of 44◦. Draw RR .
(iv) The intersection of QQ and RR forms the vertex P of the triangle.
Problem 26. Construct a triangle XYZ given that XY= 5 cm, the hypotenuse YZ = 6.5 cm and ∠X = 90◦.
V S
Y
X A′
A B
U Z P
C
R Q
Fig. 18.34
With reference to Fig. 18.34:
(i) Draw a straight line 5 cm long and label it XY .
(ii) Produce XY any distance to B. With compass centred at X make an arc at A and A . (The length XA and XA is arbitrary.) With compass centred at A draw the arc PQ. With the same compass setting and centred at A , draw the arc RS. Join the intersection of the arcs, C, to X , and a right angle to XY is produced at X . (Alternatively, a protractor can be used to construct a 90◦angle).
(iii) The hypotenuse is always opposite the right angle. Thus YZ is opposite∠X . Using a compass centred at Y and set to 6.5 cm, describe the arc UV .
(iv) The intersection of the arc UV with XC produced, forms the vertex Z of the required triangle. Join YZ by a straight line.
Now try the following exercise
Exercise 70 Further problems on the construction of triangles (Answers on page 278)
In problems 1 to 5, construct the triangles ABC for the given sides/angles.
1. a= 8 cm, b = 6 cm and c = 5 cm 2. a= 40 mm, b = 60 mm and C = 60◦ 3. a= 6 cm, C = 45◦and B= 75◦ 4. c= 4 cm, A = 130◦and C= 15◦
5. a= 90 mm, B = 90◦, hypotenuse= 105 mm
Property of Mr. Alphonce Kimutai Kirui
Geometry and triangles 141
Assignment 8
This assignment covers the material contained in chap-ters 16 to 18. The marks for each question are shown in brackets at the end of each question.
1. In the following equations, x and y are two related variables and k and t are constants. For the stated equa-tions to be verified it is necessary to plot graphs of the variables in modified form. State for each (a) what should be plotted on the horizontal axis, (b) what should be plotted on the vertical axis, (c) the gradient, and (d) the vertical axis intercept.
(i) y−k
x= t (ii) y
k= xt (8)
2. The following experimental values of x and y are believed to be related by the law y= ax2+ b, where a and b are constants. By plotting a suitable graph verify this law and find the approximate values of a and b.
x 2.5 4.2 6.0 8.4 9.8 11.4
y 15.4 32.5 60.2 111.8 150.1 200.9 (8) 3. State the minimum number of cycles on logarithmic graph paper needed to plot a set of values ranging from
0.065 to 480. (2)
4. Determine the law of the form y= aekx which relates the following values:
y 0.0306 0.285 0.841 5.21 173.2 1181 x −4.0 5.3 9.8 17.4 32.0 40.0
(8) 5. Evaluate: 29◦17 + 75◦51 − 47◦49 (3) 6. Convert 47.319◦ to degrees, minutes and seconds (2) 7. State the angle (a) supplementary to 49◦
(b) complementary to 49◦ (2)
8. In Fig. A8.1, determine angles x, y and z (3)
59°37°
y z
x
Fig. A8.1
9. In Fig. A8.2, determine angles a to e (5)
c
b d
a 60°
125° e
Fig. A8.2
10. In Fig. A8.3, determine the length of AC (4)
10 m
3 m
8 m A
C D
B
Fig. A8.3
11. Construct a triangle PQR given PQ= 5 cm,
∠QPR = 120◦and∠PRQ = 35◦ (5)
Telephone:+254728801352
Email Address: <[email protected]><[email protected]>
Property of Mr. Alphonce Kimutai Kirui