Examinable Gr11 & 12 Maths Proofs: page 1 of 16 pages
Name: _________________ Surname: _____________________ Class ______
1) Number patterns: PAPER 1
a) Sum to 𝑛 terms in an arithmetic series
b) Sum to 𝑛 terms in a geometric series
2) Trigonometry: PAPER 2
a) Sine rule
b) Cos rule (V1 + V2)
c) Area rule
d) Compound Angles
3) Euclidean Geometry: PAPER 2
a) Theorem 1: The line segment joining the centre of a circle to the midpoint
of a circle is perpendicular to the chord.
b) Theorem 2: The angle which the arc of a circle subtends at the centre is
double the angle it subtends at any given point on the
circumference.
c) Theorem 5: The opposite angles of a cyclic quadrilateral are
supplementary.
d) Theorem 8: The angle between the tangent to a circle and the chord
drawn from the point of contact is equal to the angle
subtended by the chord in the alternate segment.
e) Theorem 9: A line drawn parallel to one side of a triangle divides the
other two sides proportionally.
f) Theorem 10: Equiangular triangles are similar.
,Sum to 𝑛 terms of an arithmetic series. page 2 of 16 pages
Consider an arithmetic sequence of 𝑛 terms with 1st term 𝑎, 𝑛𝑡ℎ term 𝑙 and
common difference 𝑑. Prove that in any arithmetic series the sum of the first
𝑛 𝑛
𝑛 terms is 𝑆𝑛 = (𝑎 + 𝑙) or 𝑆𝑛 = [2𝑎 + (𝑛 − 1)𝑑].
2 2
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,Sum to 𝑛 terms of a geometric series. page 3 of 16 pages
Consider a geometric sequence of 𝑛 terms, 1st term 𝑎 and common ratio 𝑟.
Prove that in any geometric series the sum of the first 𝑛 terms is expressed by
𝑎(1−𝑟 𝑛 ) 𝑎(𝑟 𝑛 −1)
𝑆𝑛 = OR 𝑆𝑛 = .
1−𝑟 𝑟−1
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,Proof of the Sine Rule: page 4 of 16 pages
𝐶
𝐴 𝐵
sin 𝐴̂ sin 𝐵̂
Prove that for any acute-angled ∆𝐴𝐵𝐶; =
𝑎 𝑏
Solution:
𝐶
𝐴 𝐵
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, Proof of the Cos Rule: (ONE VERSION) page 5 of 16 pages
Using the sketch provided, prove that in any acute-angled
∆𝐴𝐵𝐶; 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝐵( ; ) Note (not part of cos rule)
sin 𝐴 =
∴ 𝐵𝐷 =
cos 𝐴 =
𝐶 𝐴
∴ 𝐴𝐷 =
COORDINATES: _______________
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Similarly: __________________________________________________
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Name: _________________ Surname: _____________________ Class ______
1) Number patterns: PAPER 1
a) Sum to 𝑛 terms in an arithmetic series
b) Sum to 𝑛 terms in a geometric series
2) Trigonometry: PAPER 2
a) Sine rule
b) Cos rule (V1 + V2)
c) Area rule
d) Compound Angles
3) Euclidean Geometry: PAPER 2
a) Theorem 1: The line segment joining the centre of a circle to the midpoint
of a circle is perpendicular to the chord.
b) Theorem 2: The angle which the arc of a circle subtends at the centre is
double the angle it subtends at any given point on the
circumference.
c) Theorem 5: The opposite angles of a cyclic quadrilateral are
supplementary.
d) Theorem 8: The angle between the tangent to a circle and the chord
drawn from the point of contact is equal to the angle
subtended by the chord in the alternate segment.
e) Theorem 9: A line drawn parallel to one side of a triangle divides the
other two sides proportionally.
f) Theorem 10: Equiangular triangles are similar.
,Sum to 𝑛 terms of an arithmetic series. page 2 of 16 pages
Consider an arithmetic sequence of 𝑛 terms with 1st term 𝑎, 𝑛𝑡ℎ term 𝑙 and
common difference 𝑑. Prove that in any arithmetic series the sum of the first
𝑛 𝑛
𝑛 terms is 𝑆𝑛 = (𝑎 + 𝑙) or 𝑆𝑛 = [2𝑎 + (𝑛 − 1)𝑑].
2 2
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,Sum to 𝑛 terms of a geometric series. page 3 of 16 pages
Consider a geometric sequence of 𝑛 terms, 1st term 𝑎 and common ratio 𝑟.
Prove that in any geometric series the sum of the first 𝑛 terms is expressed by
𝑎(1−𝑟 𝑛 ) 𝑎(𝑟 𝑛 −1)
𝑆𝑛 = OR 𝑆𝑛 = .
1−𝑟 𝑟−1
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,Proof of the Sine Rule: page 4 of 16 pages
𝐶
𝐴 𝐵
sin 𝐴̂ sin 𝐵̂
Prove that for any acute-angled ∆𝐴𝐵𝐶; =
𝑎 𝑏
Solution:
𝐶
𝐴 𝐵
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, Proof of the Cos Rule: (ONE VERSION) page 5 of 16 pages
Using the sketch provided, prove that in any acute-angled
∆𝐴𝐵𝐶; 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝐵( ; ) Note (not part of cos rule)
sin 𝐴 =
∴ 𝐵𝐷 =
cos 𝐴 =
𝐶 𝐴
∴ 𝐴𝐷 =
COORDINATES: _______________
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Similarly: __________________________________________________
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