Based directly on the official DBE 2015 Examination Guidelines. Covers all examinable proofs, theorems, corollaries, and the exact acceptable reasons you must use in Paper 2.
You must know the formal proofs for these six theorems. They routinely account for roughly 12 marks in Paper 2.
The line drawn from the centre of a circle perpendicular to a chord bisects the chord.
Proof Strategy: Construct radii to form two triangles. Prove the triangles are congruent using RHS (Right angle, Hypotenuse, Side).
The angle subtended by an arc at the centre of a circle is double the size of the angle subtended by the same arc at the circle.
Proof Strategy: Draw a line through the apex to the centre and extend it. Use the exterior angle of a triangle theorem on the two resulting isosceles triangles (radii are equal).
The opposite angles of a cyclic quadrilateral are supplementary (add up to 180°).
Proof Strategy: Connect the centre to two vertices. Use the "angle at centre = 2x angle at circumference" theorem on both the reflex and obtuse angles at the centre, which sum to 360°.
The angle between the tangent to a circle and the chord drawn from the point of contact is equal to the angle in the alternate segment.
Proof Strategy: Draw a diameter from the point of tangency. Use the theorems "tangent ⊥ radius" and "angle in semi-circle = 90°".
A line drawn parallel to one side of a triangle divides the other two sides proportionally.
Proof Strategy: Construct heights in the smaller triangle. Use the area of a triangle formula (½ b h) and compare areas of triangles sharing the same heights and bases.
If two triangles are equiangular, then their corresponding sides are in proportion (and consequently the triangles are similar).
Proof Strategy: Superimpose the smaller triangle onto the larger one by cutting off equal lengths. Prove the resulting line is parallel using corresponding angles, then use the Proportion Theorem.
| Theorem / Corollary | DBE Acceptable Reason(s) |
|---|---|
| Line from centre to midpt of chord is ⊥ to chord. | line from centre to midpt of chord |
| Perpendicular bisector of a chord passes through centre. | perp bisector of chord |
| Angle subtended by diameter at circumference is 90°. | ∠s in semi circle OR diameter subtends right angle OR ∠ in ½ ⊙ |
| Angles subtended by a chord on the same side are equal. | ∠s in the same seg |
| Equal chords subtend equal angles at circumference. | equal chords; equal ∠s |
| Equal chords subtend equal angles at centre. | equal chords; equal ∠s |
| Exterior angle of cyclic quad is equal to interior opposite angle. | ext ∠ of cyclic quad |
| Tangent is perpendicular to radius at point of contact. | tan ⊥ radius OR tan ⊥ diameter |
| Two tangents drawn from the same point outside circle are equal. | Tans from common pt OR Tans from same pt |
| Theorem / Axiom | DBE Acceptable Reason(s) |
|---|---|
| Interior angles of a triangle are supplementary. | ∠ sum in ∆ OR sum of ∠s in ∆ OR Int ∠s ∆ |
| Exterior angle of triangle = sum of interior opposite angles. | ext ∠ of ∆ |
| Angles opposite equal sides in isosceles triangle are equal. | ∠s opp equal sides |
| Sides opposite equal angles in isosceles triangle are equal. | sides opp equal ∠s |
| Pythagoras Theorem. | Pythagoras OR Theorem of Pythagoras |
| Midpoint Theorem: Line joining midpoints is || to 3rd side and half its length. | Midpt Theorem |
| If sides of two triangles are proportional, they are equiangular. | Sides of ∆ in prop |
| Triangles on same base and between same parallel lines have equal area. | same base; same height OR equal bases; equal height |
| Concept | DBE Acceptable Reason(s) |
|---|---|
| Adjacent angles on a straight line are supplementary. | ∠s on a str line |
| Vertically opposite angles are equal. | vert opp ∠s = |
| Alternate angles (requires parallel lines). | alt ∠s; AB || CD |
| Corresponding angles (requires parallel lines). | corresp ∠s; AB || CD |
| Co-interior angles (requires parallel lines). | co-int ∠s; AB || CD |
| Interior angles of a quadrilateral add up to 360°. | sum of ∠s in quad |
| Opposite sides of a parallelogram are equal / parallel. | opp sides of ||m |
| Opposite angles of a parallelogram are equal. | opp ∠s of ||m |
| Diagonals of a parallelogram bisect each other. | diag of ||m |
| Diagonals of a rhombus bisect at right angles. | diags of rhombus |
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