Leaving Cert Geometry
Axioms and Theorems (proofs at Higher Level) (supported by 46 definitions; 20 propositions; 5 axioms; 21 theorems) (Note: Listed here are 26 theorem (converses included) ideas + 6 corollary ideas + 5 axioms + 1 Definition) Propositions are useful ideas which deserve mention and are listed separately. Definitions are listed separately.		F (B)	O (G)	H (A)
	Axioms
1	There is exactly one line through any two given points [Two Points Axiom].
2	The 4 properties of the distance between points [Ruler Axiom]. (i) The distance |AB| is never negative; (ii) |AB| = |BA|; (iii) if C lies on AB, between A and B, then |AB| = |AC| + |CB|; (iv) Given any ray from A, and given any real number k ≥0, there is a unique point B on the ray whose distance from A is k [Marking off distance].
3	The number of degrees in an angle is always between 0^o and 360^o [Protractor Axiom]. The number of degrees of an ordinary angle is less than 180^o. A straight angle has 180^o.
4	Congruent triangles conditions [SSS, SAS, ASA]. [If these conditions apply then the triangles are congruent].
5	Given any line l and a point P, there is exactly one line through P that is parallel to l [Axiom of Parallels].
	Theorems (Linked to dynamic geometry illustrations)
1	Vertically opposite angles are equal in measure [or See In Geogebra file here ]
2	In an isosceles triangle the angles opposite the equal sides are equal And Conversely, if two angles are equal, then the triangle is isosceles.
3	If a transversal makes equal alternate angles on two lines then the lines are parallel And Converse
4	The angles in any triangle add to 180^o [Proof Required]
5	Two lines are parallel if, and only if, for any transversal, the corresponding angles are equal
6	Each exterior angle of a triangle is equal to the sum of the interior opposite angles. [PR]
7	In a triangle, the angle opposite the greater of two sides is greater than the angle opposite the lesser. Conversely, the side opposite the greater of two angles is greater than the side opposite the lesser angle.
8	Two sides of a triangle are together greater than the third
9	In a parallelogram, opposite sides are equal, And opposite angles are equal. [PR]
Cor 1	A diagonal divides a parallelogram into two congruent triangles
10	The diagonals of a parallelogram bisect one another
11	If three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal [PR]
12	Let ABC be a triangle, if a line l is parallel to BC and cuts [AB] in the ratio m:n, then it also cuts [AC] in the same ratio. [PR]
13	If two triangles are similar, then their sides are proportional, in order. [PR]
14	Theorem of Pythagoras: Ina a right-angled triangle the square of the hypotenuse is the sum of the squares of the other two sides [PR]
15	Converse of Pythagoras: If the square of one side of triangle is the sum of the squares of the other two, then the angle opposite the first side is a right angle
16	For a triangle, base times height does not depend on the choice of base.
DEF38	The area of a triangle is half the base by the height
17	A diagonal of a parallelogram bisects the area
18	The area of a parallelogram is the base by the height
19	The angle at the centre of a circle standing on a given arc is twice the angle at any point of the circle standing on the same arc [PR]
Cor 2	All angles at points of the circle, standing on the same arc, are equal
Cor 3	Each angle in a semi-circle is a right angle
Cor 4	If the angle standing on a chord [BC] at some point of the circle is a right-angle, then [BC] is a diameter
Cor 5	If ABCD is a cyclic quadrilateral, then opposite angles sum to 180^o
20	(i) Each tangent is perpendicular to the radius that goes to the point of contact
20	(ii) If P lies on the circle S, and a line l is perpendicular to the radius to P, then l is a tangent to S
Cor 6	If two circles intersect at one point only, then the two centres and the point of contact are collinear
21	(i) The perpendicular from the centre to a chord bisects the chord
21	(ii) The perpendicular bisector of a chord passes through the centre

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