Grade 10 Maths · Euclidean Geometry

Prove a theorem or formula

Write out the standard proof of a theorem or formula step by step with reasons, for example proving that the angles of a triangle add up to 180°.

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Euclidean geometry proofs in less than 10 minutes - Grade 11 and12 NSC & IEBMathew's Academy Of Sciences

Angle at Centre Theorem: ProofKevinmathscience

Grade 12 Proportionality Theorem ProofKevinmathscience

Proof of Sine Rule - Grade 11Mlungisi Nkosi Maths & Science

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Question 10.1 · May/June 2025 · Paper 2 · National (DBE) · 6 marks

10.1 In the diagram,

Δ A BC

and

Δ P K L

are drawn such that

\hat{A} = \hat{P}

\hat{B} = \hat{K}

and

\hat{C} = \hat{L}

.

Use the diagram in the ANSWER BOOK to prove the theorem which states that if two triangles are equiangular, then the corresponding sides are in proportion, i.e. that

\frac{A B}{P K} = \frac{A C}{P L}

(6)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: prove a theorem or formula

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3. Question 8.1 · November 2025 · Paper 2 · Provincial · 5 marks
In the diagram below, O is the centre of the circle. AC is a chord. OB is drawn perpendicular to the chord. $Diagram$
8.1 Use the diagram above to prove the theorem which states that the line drawn from the centre of a circle perpendicular to a chord bisects the chord, that is, prove that AB $=$ BC.(5)
Show the memo (5 ticks)Memo
1. ✓
  ConstructionConstruction: Join OA and OC.
  1 mark · State the construction
2. ✓
  S/R: radiiIn $Δ A OB$ and $Δ COB$ :
  1. AO $=$ CO [radii]
  1 mark · Prove a theorem or formula
3. ✓
  S/R: right angles2. $A \hat{B} O = C \hat{B} O = 90°$ [ $∠$ s on a str. line]
  1 mark · Prove a theorem or formula
4. ✓
  S: common side3. OB $=$ OB [common]
  1 mark · Prove a theorem or formula
5. ✓
  R: RHS congruence $∴ Δ A OB \equiv Δ COB$ [RHS]
  $∴$ AB $=$ BC
  1 mark · Prove a theorem or formula
4. Question 9.1 · November 2024 · Paper 2 · Provincial · 5 marks
9.1 In the diagram, AB is a chord of a circle with center O. Prove the theorem that states: If M is the midpoint of AB, then OM $⊥$ AB.(5) $Diagram for question 9.1$
Show the memo (5 ticks)Memo
1. ✓
  ConstructionConstruction: Connect AO and OB $Memo diagram$
  1 mark · State the construction
2. ✓
  S: three pairs of equal sidesIn $Δ A OM$ and $Δ BOM$ :
  1. $A O = BO$
  2. $OM = OM$
  3. $A M = MB$
  1 mark · Prove a theorem or formula
3. ✓
  R: reasons for the sides(Radii), (Common side), (Given)
  1 mark · Prove a theorem or formula
4. ✓
  S/R: congruence $∴ Δ A OM \equiv Δ BOM$ (S S S)
  $∴ \hat{M}_{1} = \hat{M}_{2}$
  1 mark · Prove a theorem or formula
5. ✓
  S/R: conclusionBut $\hat{M}_{1} + \hat{M}_{2} = 180°$ (AMB is a straight line)
  $∴ \hat{M}_{1} = \hat{M}_{2} = 90°$
  $∴ OM ⊥ A B$
  1 mark · Prove a theorem or formula
5. Question 9.1 · November 2024 · Paper 2 · Provincial · 5 marks
In the diagram, ABCD is a cyclic quadrilateral. O is the centre of the circle. $Diagram$
9.1 Use the diagram above to prove the THEOREM which states that, the opposite angles of a cyclic quadrilateral add up to 180°, then prove that, $\hat{B} + \hat{D} = 180°$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  ConstructionConstruction: Draw AO and OC. $Memo diagram$
  1 mark · State the construction
2. ✓
  S/R $\hat{O}_{1} = 2 \hat{B}$ [ $∠$ at centre $= 2 \times ∠$ at circumf]
  1 mark · Prove a theorem or formula
3. ✓
  S $\hat{O}_{2} = 2 \hat{D}$ [ $∠$ at centre $= 2 \times ∠$ at circumf]
  1 mark · Prove a theorem or formula
4. ✓
  S/R $\hat{O}_{1} + \hat{O}_{2} = 360°$ [ $∠$ s around a point]
  1 mark · Prove a theorem or formula
5. ✓
  S $2 \hat{B} + 2 \hat{D} = 360°$
  $\hat{B} + \hat{D} = 180°$
  1 mark · Prove a theorem or formula
6. Question 6.1 · March 2022 · Provincial · 5 marks
Given $△ A BC$ with $D E ∥ BC$ as shown in the figure.
6.1 Prove that: $\frac{A D}{D B} = \frac{A E}{EC}$ (5)
Show the memo (5 ticks)Memo
1. ✓
  ConstructionConstruct an altitude $h$ from $E$ to $A D$ (height $h_{1}$ ) and from $D$ to $A E$ (height $h_{2}$ ). Join $D C$ and $BE$ .
  1 mark · State the construction
2. ✓
  Area ratios on LHS $\frac{Area Δ A D E}{Area Δ D BE} = \frac{\frac{1}{2} A D \cdot h _{1}}{\frac{1}{2} D B \cdot h _{1}} = \frac{A D}{D B}$
  1 mark · Prove a theorem or formula
3. ✓
  Area ratios on RHS $\frac{Area Δ A D E}{Area Δ D EC} = \frac{\frac{1}{2} A E \cdot h _{2}}{\frac{1}{2} EC \cdot h _{2}} = \frac{A E}{EC}$
  1 mark · Prove a theorem or formula
4. ✓
  Equate areasArea $Δ D BE$ = Area $Δ D EC$ (Same base $D E$ , same height between parallel lines $D E ∥ BC$ )
  1 mark · Prove a theorem or formula
5. ✓
  Conclusion $∴ \frac{A D}{D B} = \frac{A E}{EC}$
  1 mark · Prove a theorem or formula

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