From O-Level to Additional Mathematics You met every angle fact below at O-Level. In Additional Mathematics they become the reasons of a proof: each statement you write is followed by its reason in brackets, quoted by the standard name given here. Basic angle facts FactResultReason you write Angles at a point add up to \(360^\circ\).\(\angle a+\angle b+\angle c+\angle d=360^\circ\)(\(\angle\)s at a point) Adjacent angles on a straight line add up to \(180^\circ\).\(\angle a+\angle b=180^\circ\)(adj. \(\angle\)s on a str. line) Vertically opposite angles are equal.\(\angle a=\angle b\)(vert. opp. \(\angle\)s) Angles formed with parallel lines \(AB//CD\) Alternate angles are equalCorresponding angles are equalInterior angles add up to \(180^\circ\) (alt. \(\angle\)s, \(AB//CD\))(corr. \(\angle\)s, \(AB//CD\))\(\angle a+\angle b=180^\circ\) (int. \(\angle\)s, \(AB//CD\)) Triangle facts FactResultReason you write Interior angles add up to \(180^\circ\).\(\angle a+\angle b+\angle c=180^\circ\)(\(\angle\) sum of \(\triangle\)) An exterior angle equals the sum of the two interior opposite angles.\(\angle x=\angle a+\angle b\)(ext. \(\angle\) of \(\triangle\)) Isosceles triangle: two equal sides face equal base angles.\(AB=AC\ \Rightarrow\ \angle ABC=\angle ACB\)(base \(\angle\)s of isos. \(\triangle\)) Equilateral triangle: all sides equal.\(\angle a=\angle b=\angle c=60^\circ\)(equilateral \(\triangle\)) Exterior angle of a triangle: \(\angle x=\angle a+\angle b\) Quadrilaterals and their diagonals ShapeSides and anglesDiagonals SquareFour equal sides; opposite sides parallel; four right angles.Equal; bisect one another at \(90^\circ\); bisect the corner angles. RectangleOpposite sides equal and parallel; four right angles.Equal; bisect one another. RhombusFour equal sides; opposite sides parallel; opposite angles equal.Bisect one another at \(90^\circ\); bisect the corner angles. ParallelogramOpposite sides equal and parallel; opposite angles equal.Bisect one another. KiteTwo pairs of equal adjacent sides; one pair of equal opposite angles.Cut at \(90^\circ\); the diagonal joining the vertices where equal adjacent sides meet bisects the other. TrapeziumAt least one pair of opposite sides is parallel (inclusive convention).No special property. Exam check Quote reasons by their standard names, naming the lines or triangle involved — write (alt. \(\angle\)s, \(AB//CD\)), not "Z-angles". Do not assume facts from a diagram’s appearance. State the reason supporting each step of a proof.

Angles, Triangles and Quadrilaterals

Secondary 4

TGM Original Questions

From O-Level to Additional Mathematics

You met every angle fact below at O-Level. In Additional Mathematics they become the reasons of a proof: each statement you write is followed by its reason in brackets, quoted by the standard name given here.

Basic angle facts

Fact	Result	Reason you write
Angles at a point add up to \(360^\circ\).	\(\angle a+\angle b+\angle c+\angle d=360^\circ\)	(\(\angle\)s at a point)
Adjacent angles on a straight line add up to \(180^\circ\).	\(\angle a+\angle b=180^\circ\)	(adj. \(\angle\)s on a str. line)
Vertically opposite angles are equal.	\(\angle a=\angle b\)	(vert. opp. \(\angle\)s)

Angles formed with parallel lines \(AB//CD\)

Alternate angles are equal	Corresponding angles are equal	Interior angles add up to \(180^\circ\)

(alt. \(\angle\)s, \(AB//CD\))	(corr. \(\angle\)s, \(AB//CD\))	\(\angle a+\angle b=180^\circ\) (int. \(\angle\)s, \(AB//CD\))

Triangle facts

Fact	Result	Reason you write
Interior angles add up to \(180^\circ\).	\(\angle a+\angle b+\angle c=180^\circ\)	(\(\angle\) sum of \(\triangle\))
An exterior angle equals the sum of the two interior opposite angles.	\(\angle x=\angle a+\angle b\)	(ext. \(\angle\) of \(\triangle\))
Isosceles triangle: two equal sides face equal base angles.	\(AB=AC\ \Rightarrow\ \angle ABC=\angle ACB\)	(base \(\angle\)s of isos. \(\triangle\))
Equilateral triangle: all sides equal.	\(\angle a=\angle b=\angle c=60^\circ\)	(equilateral \(\triangle\))

Exterior angle of a triangle: \(\angle x=\angle a+\angle b\)

Quadrilaterals and their diagonals

Shape	Sides and angles	Diagonals
Square	Four equal sides; opposite sides parallel; four right angles.	Equal; bisect one another at \(90^\circ\); bisect the corner angles.
Rectangle	Opposite sides equal and parallel; four right angles.	Equal; bisect one another.
Rhombus	Four equal sides; opposite sides parallel; opposite angles equal.	Bisect one another at \(90^\circ\); bisect the corner angles.
Parallelogram	Opposite sides equal and parallel; opposite angles equal.	Bisect one another.
Kite	Two pairs of equal adjacent sides; one pair of equal opposite angles.	Cut at \(90^\circ\); the diagonal joining the vertices where equal adjacent sides meet bisects the other.
Trapezium	At least one pair of opposite sides is parallel (inclusive convention).	No special property.

Exam check

Quote reasons by their standard names, naming the lines or triangle involved — write (alt. \(\angle\)s, \(AB//CD\)), not "Z-angles". Do not assume facts from a diagram’s appearance. State the reason supporting each step of a proof.

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