JCHL Geometry 1: Theorems, axioms and Corollaries

Learning work

• An axiom is a statement we accept without proof (the starting point) — learn one: a straight angle is 180°
• A theorem is a statement that is PROVED true by a chain of reasoning
• A corollary is a statement that follows directly from a theorem (an extra extension)
• A converse swaps the "given" and the "to prove"; the converse is NOT always true (all rectangles are parallelograms, but not all parallelograms are rectangles)
• The course has 14 theorems (1–6, 9–15, 19), 5 corollaries (1–5) and 5 axioms (1–5): you need to recognise and use them ALL
• The five proofs (Theorems 4, 6, 9, 14 and 19) are NOT required on the current Junior Cycle course — they are included as good practice, and they set you up for Leaving Cert
• The 5 axioms: (1) exactly one line through any two points; (2) Ruler axiom — distances add along a line and |AB| = |BA|; (3) Protractor axiom — an angle measures between 0° and 360°, a straight angle is 180°; (4) Congruent-triangles axiom (SAS, ASA, SSS); (5) Axiom of parallels — exactly one line through a point parallel to a given line
• Theorem 1: vertically opposite angles are equal in measure
• Theorem 2: in an isosceles triangle, the angles opposite the equal sides are equal
• Theorem 3: if a transversal makes equal alternate angles on two lines, then the lines are parallel
• Theorem 4: the three angles of any triangle add to 180° (examinable proof)
• Theorem 5: two lines are parallel if and only if, for any transversal, corresponding angles are equal
• Theorem 6: each exterior angle of a triangle equals the sum of the two interior opposite angles (examinable proof)
• Theorem 9: in a parallelogram, opposite sides are equal and opposite angles are equal (examinable proof)
• Theorem 10: the diagonals of a parallelogram bisect one another
• Theorem 11: if three parallel lines cut off equal segments on one transversal, they cut off equal segments on any transversal
• Theorem 12: in triangle ABC, a line parallel to BC that cuts [AB] in the ratio s:t cuts [AC] in the same ratio
• Theorem 13: if two triangles are similar, then their sides are proportional, in order
• Theorem 14: Pythagoras — in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides (examinable proof)
• Theorem 15: converse of Pythagoras — if a² + b² = c² then the angle opposite the longest side is a right angle
• Theorem 19: the angle at the centre of a circle is twice the angle at any point of the circle standing on the same arc (examinable proof)
• Corollary 1: a diagonal divides a parallelogram into two congruent triangles
• Corollary 2: all angles at points of the circle standing on the same arc are equal
• Corollary 3: each angle in a semicircle is a right angle
• Corollary 4: if the angle standing on a chord is a right angle, then the chord is a diameter
• Corollary 5: in a cyclic quadrilateral, opposite angles sum to 180°