Synthetic Geometry

Geometry Axioms

Back to top ↑

What is an axiom?

An axiom is a basic fact we accept as true without proof. It is a starting point for geometry.

  • We use axioms to build theorems and constructions.
  • Axioms describe how points, lines, distances and angles behave.
  • All geometry in this course is built on these foundation rules.

Axiom 1 — Two Points Axiom

  • There is exactly one straight line through any two given points.
  • We name this line by the two points, e.g. line \(AB\).
  • This ensures that a line is uniquely determined by two points.

Axiom 2 — Ruler Axiom

  • Distances are always \(\ge 0\) (never negative).
  • \(|AB| = |BA|\): distance is the same in both directions.
  • If \(C\) lies between \(A\) and \(B\), then \(|AB| = |AC| + |CB|\).
  • This axiom underpins all length measurement in geometry.

Axiom 3 — Protractor Axiom

  • A straight angle has \(180^\circ\).
  • Every angle between \(0^\circ\) and \(180^\circ\) has a unique degree measure.
  • If a point \(D\) lies inside \(\angle BAC\), then \(\;|\angle BAC| = |\angle BAD| + |\angle DAC|\).
  • This axiom justifies how we measure and add angles correctly.

Language of Proof

Students should be able to use and explain the following terms:

Theorem
Tap to flip
A theorem is a mathematical statement that has been proved from axioms and known results.
Proof
Tap to flip
A proof is a logical argument demonstrating that a statement must be true.
Axiom
Tap to flip
An axiom is a basic assumption accepted without proof.
Corollary
Tap to flip
A corollary is a result that follows almost immediately from a theorem.
Converse
Tap to flip
The converse reverses the “if” and “then” parts of a theorem and usually requires its own proof.
Implies
Tap to flip
Implies indicates that one statement leads logically to another, written using “⇒”.

16 Theorems and 5 Corollaries

Back to top ↑

Parallel Lines and a Transversal

Parallel lines cut by a transversal with labelled angles A, B, C, D, E, F, G, H.

CORRESPONDING ANGLES

∠A = ∠E,   ∠B = ∠F,   ∠C = ∠G,   ∠D = ∠H.

ALTERNATE INTERIOR ANGLES

∠C = ∠F,   ∠D = ∠E.

ALTERNATE EXTERIOR ANGLES

∠A = ∠G,   ∠B = ∠H.

CO-INTERIOR (CONSECUTIVE) ANGLES

∠C + ∠E = 180°,   ∠D + ∠F = 180°.

Explore with GeoGebra

Drag the points to move the transversal and the parallel lines. Watch how the corresponding, alternate and co-interior angle relationships stay true.

Try to match each angle pair in the bubbles above with the angles in the interactive.

Theorem 4 — Angles in a Triangle Sum to 180°

Statement:
In any triangle, the sum of the measures of the three interior angles is 180°

Theorem 6 — Exterior Angle Theorem (Interactive)

Statement:
The exterior angle of a triangle is equal to the sum of the two interior opposite angles.

Theorem 9 — Parallelogram Theorem

Statement:
In a parallelogram:
  • Opposite sides are equal in length.
  • Opposite angles are equal in measure.

Theorem 14 — The Theorem of Pythagoras (Activity)

Statement:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

\[ c^2 = a^2 + b^2 \]

Pythagoras Quiz — Right-angled Triangles

Score: 0 / 0

Question diagram

Calculator

0

Choose the correct answer

    Theorem 19 — Angle at the Centre

    Statement:
    The angle at the centre of a circle is twice the angle standing on the same arc at the circumference.

    GT.3 — Geometry: Axioms & Proof

    Back to top ↑

    B. Axioms, Theorems, Corollaries & Converses

    Students recall and use the concepts, axioms, theorems, corollaries and converses specified in the geometry handbook.

    I. Axioms

    • Axioms 1, 2, 3, 4, 5

    II. Theorems

    Ordinary Level uses section 9; Higher Level uses section 10.

    • Theorems 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 15
    • Theorems 11, 12, 19 (HL)
    • Relevant converses (including use of square roots)
    • 1. Vertically opposite angles are equal.
    • 2. Base angles in an isosceles triangle are equal (and converse).
    • 3. Alternate interior angles equal ⇒ parallel lines (and converse).
    • 4. The angles in a triangle sum to \(180^\circ\).
    • 5. Corresponding angles equal ⇒ parallel lines (and converse).
    • 6. The exterior angle equals the sum of the opposite interior angles.
    • 9. Opposite sides and angles of a parallelogram are equal.
    • 10. Diagonals of a parallelogram bisect each other.
    • 11. Three parallel lines cut equal segments on all transversals.
    • 12. If a line is parallel to BC, then it cuts the other sides in the same ratio.
    • 13. If two triangles are similar, their sides are proportional (and converse).
    • 14. Pythagoras’ theorem.
    • 15. Converse of Pythagoras: if \(a^2 + b^2 = c^2\), angle opposite side \(c\) is right.
    • 19. The angle at the centre is twice the angle at the circumference on the same arc.
    • Corollary 1. A diagonal divides a parallelogram into two congruent triangles.
    • Corollary 2. Angles in the same arc are equal.
    • Corollary 3. Each angle in a semicircle is a right angle.
    • Corollary 4. If an angle standing on chord BC is right, then BC is a diameter.
    • Corollary 5. Opposite angles in a cyclic quadrilateral sum to \(180^\circ\) (and converse).

    C. Language of Proof

    Students should be able to use and explain the following terms:

    • Theorem, Proof, Axiom, Corollary, Converse, Implies

    D. Creating & Evaluating Proofs

    Students should be able to create, justify, and evaluate proofs of geometrical propositions.

    E. Understanding Key Proofs

    Students should display understanding (not formal proofs) of:

    • Theorems 1, 2, 3, 4, 5, 6, 9, 10, 14, 15
    • Theorems 13 and 19
    • Corollaries 3, 4, 1, 2, 5

    GT.3 — Geometry: Axioms & Proof

    Back to top ↑

    B. Axioms, Theorems, Corollaries & Converses

    Students recall and use the concepts, axioms, theorems, corollaries and converses specified in the geometry handbook.

    I. Axioms

    • Axioms 1, 2, 3, 4, 5

    II. Theorems

    Ordinary Level uses the section 9 list; Higher Level uses the section 10 list.

    • Theorems 1, 2, 3, 4, 5, 6, 9, 10, 13, 14, 15
    • Theorems 11, 12, 19 (HL content)
    • Relevant converses, including operations involving square roots

    III. Corollaries

    • Corollaries 3, 4
    • Corollaries 1, 2, 5

    THEOREMS

    Formal proofs are not examinable.
    • Theorem 1: Vertically opposite angles are equal in measure.
    • Theorem 2: In an isosceles triangle the angles opposite the equal sides are equal; the converse also holds.
    • Theorem 3: Equal alternate interior angles imply lines are parallel (and converse).
    • Theorem 4: The angles in any triangle add to \(180^\circ\).
    • Theorem 5: Corresponding angles equal imply parallel lines (and converse).
    • Theorem 6: The exterior angle equals the sum of the two interior opposite angles.
    • Theorem 9: Opposite sides and opposite angles of a parallelogram are equal (and converses).
    • Theorem 10: The diagonals of a parallelogram bisect each other.
    • Theorem 11: Three parallel lines cut equal segments on all transversals.
    • Theorem 12: A line parallel to one side of a triangle divides the other two sides proportionally (and converse).
    • Theorem 13: If two triangles are similar, their sides are proportional (and converse).
    • Theorem 14: Pythagoras’ theorem.
    • Theorem 15: If \(a^2 + b^2 = c^2\), then the angle opposite side \(c\) is right-angled.
    • Theorem 19: The angle at the centre is twice the angle at the circumference on the same arc.

    COROLLARIES

    • Corollary 1: A diagonal divides a parallelogram into two congruent triangles.
    • Corollary 2: Angles at points on the same arc are equal (and converse).
    • Corollary 3: Each angle in a semicircle is a right angle.
    • Corollary 4: If an angle standing on chord BC is right, then BC is a diameter.
    • Corollary 5: Opposite angles in a cyclic quadrilateral sum to \(180^\circ\) (and converse).

    C. Language of Proof

    Students should be able to use and explain the following terms:

    • Theorem, Proof, Axiom, Corollary, Converse, Implies

    D. Creating & Evaluating Proofs

    Students should be able to create, justify, and evaluate proofs of geometrical propositions appropriate to Junior Cycle.

    E. Understanding Key Proofs

    Students should display understanding (not full formal proofs) of:

    • Theorems 1, 2, 3, 4, 5, 6, 9, 10, 14, 15
    • Theorems 13 and 19
    • Corollaries 3, 4, 1, 2, 5