253 S045 S1&2 Geometry&Measurement · 2019-07-13 · 1 Angles, Triangles and Polygons 1 2 Congruence and Similarity 94 3 Mensuration 161 4 Pythagoras’ Theorem 235 ... AB and CD

Contents

Chapter No Chapter Name Page No

1 Angles, Triangles and Polygons 1

2 Congruence and Similarity 94

3 Mensuration 161

4 Pythagoras’ Theorem 235

5 Statistics and Probability 260

Answers 423

253_S045_S1&2_Geometry&Measurement_5thPass 23/02/17

Chapte

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Secondary 1 & 2 Geometry and Statistics • Angles, Triangles and Polygons 1

Angles, Triangles and Polygons1

ũũ 1.1 Types of Angles

(a) Acute angle

¾ an angle that is less than 90°.

(b) Right angle

¾ an angle that is 90°.

(c) Obtuse angle

¾ an angle that is greater than 90° but less than 180°.

(d) Reflex angle

¾ an angle that is greater than 180° but less than 360°.

(e) Complementary angles

¾ ∠x and ∠y are complementary if

∠ + ∠ = °x y 90 .

(f) Supplementary angles

¾ ∠x and ∠y are supplementary if

∠ + ∠ = °x y 180 .

xx

x

x

xy

x y


© Fairfield Book Publishers Pte. Ltd.2

ũũ 1.2 Properties of Angles

(a) The sum of adjacent angles on a straight line, ∠ + ∠ = °a b 180 .

(adj. ∠s on a str. line)

(b) The sum of angles at a point is 360°, ∠ + ∠ + ∠ + ∠ = °a b c d 360 .

(∠s at a point)

(c) If the lines AB and CD are parallel, then

(i) ∠ = ∠a b

Vertically Opposite angles (vert. opp. ∠s)

(ii) ∠ = ∠a cCorresponding angles (corr. ∠s)

(iii) ∠ = ∠b cAlternate angles (alt. ∠s)

(iv) ∠ + ∠b d = 180°Interior angles (int. ∠s)

ũũ 1.3 Triangles

(a) Types of triangles

(i) Equilateral triangle (three equal sides, three equal angles)

(ii) Isosceles triangle (two equal sides, two equal angles)

a

A B C

b a b

cd

dc

ba

A

C

B

D



(iii) Scalene triangle (No equal sides, no equal angles)

(b) Angle properties

(i) The angles sum of a triangle i.e. ∠ + ∠ + ∠a b c = 180° (∠s sum of )

(ii) The exterior angles of a trianglei.e. ∠ = ∠ + ∠x a b (ext. ∠s of )

ũũ 1.4 Quadrilaterals

(a) The sum of the interior angles of a quadrilateral is 360°

(b) Types of quadrilaterals

Trapezium

¾ Only one pair of opposite sides are parallel.

Parallelogram

¾ Opposite sides are parallel.

¾ Opposite sides are equal in length.

¾ Opposite angles are equal.

¾ Diagonals bisect each other.

Rectangle


¾ Opposite sides are equal in length.

¾ All angles are right angles.

¾ Diagonals bisect each other.

¾ Diagonals are equal in length.

a

b c x



Rhombus


¾ All sides are equal in length.

¾ Opposite angles are equal.

¾ Diagonals bisect each other at right angle.

¾ Diagonals bisect the interior angles.

Square


¾ All sides are equal in length.

¾ All angles are right angles.

¾ Diagonals are equal in length.


¾ Diagonals bisect the interior angles.

Kite

¾ No parallel sides.

¾ Two pairs of equal adjacent sides.


¾ One diagonals bisect the interior angles.

ũũ 1.5 Polygons

(a) A plane figure with three or more straight lines joined together is called a polygon.

(b) A regular polygon has all its sides and all its angles equal.

(c) Name of polygons

(i) Triangles (3 sides) (ii) Quadrilateral (4 sides)

(iii) Pentagon (5 sides) (iv) Hexagon (6 sides)

(v) Heptagon (7 sides) (vi) Octagon (8 sides)

(vii) Nonagon (9 sides) (viii) Decagon (10 sides)



(d) The sum of the interior angles of an n-sided polygon = ( )n - × °2 180

Each interior angle of a regular n-sided polygon = ( )n

n- × °2 180

(e) The sum of the exterior angles of an n-sided polygon = 360°

Each exterior angle of a regular n-sided polygon = 360°

n

B

AInterior angles

C

DE

F

G

Exterior angles

G

D

E

C

BA

F



ũũ 1.6 Constructions

(a) Perpendicular bisector

The perpendicular bisector of the line AB is a line that bisects and is perpendicular to AB. Any point on the perpendicular bisector is equidistant from A and B.

To construct a perpendicular bisector of AB,

(i) place the compass point at A and make arc 1 above the line AB and make arc 2 below the line AB as shown,

(ii) using the same radius as (i) and with the compass point at B, make arc 3 to cut arc 1 at P and arc 4 to cut arc 2 at Q,

(iii) join PQ to cut AB at C, PQ is the perpendicular bisector of AB.

(b) Angle bisector

The angle bisector is a line that divides an angle into two equal parts. Any point on the angle bisector is equidistant from the two sides of the angle.

To construct the angle bisector of ∠ABC,

(i) place the compass point at B, draw an arc to cut AB at P and BC at Q,

(ii) with P as centre, draw an arc 1 as shown,

(iii) with Q as centre, draw an arc 2 to cut arc 1 at R,

(iv) joint BR, BR is the angle bisector of ∠ABC.

A B

P PA = PB

13

24Q

CA B

P

A

B C

P

1

2

Q

A

B C

R


Worked ExamplesWorked Examples


Example 1AB and CD are intersecting straight lines. Calculate angles a and b.

Example 2Find the value of x in the figure below.

Ba

b

A

C

D

54°

Solution:

∠ = ° - ° - °a 180 90 54 (adj. ∠s on a str. line)

= °36

∠ = ° + °b 90 54 (vert. opp. ∠s)

= °144

A

C

B

D

63°

122°

x°

Solution:

∠ = °a 63 (alt. ∠s)

∠ = ° -b 122 63

= °59

∠ = ∠ = °x b 59 (alt. ∠s)

A

C

B

D

63°

122°

x°

a

b

Note!

You have to draw a parallel line as shown for this question.



Example 3DCB is a straight line and AD = AC. Given that ∠ = °DAC 40 and ∠ = °ABC 46 , calculate

(a) ∠ADC,

(b) ∠BAC,

(c) Reflex ∠ABC.

Example 4Find the values of x and y in the diagram.

A

BCD

40°

46°

Solution:

(a) ∠ = ° - °ADC 180 40

2 (isosceles )

= °70

(b) ∠ = ° + °BCA 40 70 (ext. ∠s of )

= °110

∠ = ° - ° - °BAC 180 110 46 (∠s sum of )

= °24(c) Reflex ∠ = ° - °ABC 360 46 (∠s at a point)

= °314

72°66°

y° x°

Solution:

∠ = ° - °x 180 66 (int. ∠s)

= °114

∠ = °y 72

Note!

For parallelogram, opposite angles are equal.



Example 5(a) Construct

(i) the perpendicular bisector of AB,

(ii) the bisector of angle ABC.

(b) The point P is equidistant from the lines BC and AB and also equidistant from the points A and B. Mark point P clearly on your diagram.

C

BA

Solution:

P

C

BA


Exercise


1. Find the

(a) complementary angle of 75°, (b) supplementary angle of 35°.

2. Name the types of angle shown below.

(a) (b)

3. Find the value of y.

4. Find the value of x.

63°5y°4y°

x°2x°

4x°

3x°



5. Calculate the unknown angles x and y in the following figure.

State the reasons clearly.

6. In the following figure, calculate the values of x and y.

7. In the following figure, it is given that BC is parallel to DE. Find the values of x, y and z.

State the reasons clearly.

8. The figure below is not drawn to scale. ABC is an equilateral triangle and AC = CD. Find the value of x.

x

y

52°

y°

x°

105°

115°

y°

x°

z° 124°

72°

A C E

D

B

x

A

B C D


Documents

253 S045 S1&2 Geometry&Measurement · 2019-07-13 · 1 Angles, Triangles and Polygons 1 2 Congruence and Similarity 94 3 Mensuration 161 4 Pythagoras’ Theorem 235 ... AB and CD