Geometry:
From the syllabus:
13.2 Theorems and Proofs
Students will be expected to understand the meaning of the following terms related to logic
and deductive reasoning: Theorem, proof, axiom, corollary, converse, implies, is
equivalent to, if and only if, proof by contradiction.
A knowledge of the Axioms, concepts, Theorems and Corollaries prescribed for JC-HL will
be assumed.
Students will study all the theorems and corollaries prescribed for LC-OL, but will not, in
general, be asked to reproduce their proofs in examination.
However, they may be asked to give proofs of the Theorems 11, 12, 13, concerning ratios,
which lay the proper foundation for the proof of Pythagoras studied at JC.
They will be asked to solve geometrical problems (so-called \cuts") and write reasoned
accounts of the solutions. These problems will be such that they can be attacked using the
given theory. The study of the propositions may be a useful way to prepare for such
examination questions.
LC-OL: Students will study proofs of Theorems 7, 8, 11, 12, 13, 16, 17, 18, 20, 21, and
Corollary 6.
(Essentially you should only be asked the proof of Theorem: 11, 12 and 13).
13.1 Constructions
(Basically all the constructions but take note of the exclusively HL)
Knowledge of the constructions prescribed for JC-HL will be assumed, and may be
examined. In addition, students will study the constructions prescribed for LC-OL, and
construction 22.
LC-OL: In addition, students will study the constructions prescribed for LC-FL, and
constructions 16, 17, 21.
LC-FL: students will study constructions 18, 19, 20.
JC-HL: Students will study constructions 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15
(Basically at higher level you study all the constructions but take note of the exclusively HL.
Note also Construction 16 – 22 are only Leaving Certificate)
List of Axioms, Theorems and Corollaries:
Axiom 1.
Axiom 2.
Axiom 3.
Axiom 4.
Axiom 5.
[Two points axiom] There is exactly one line through any two given points.
[Ruler axiom] The properties of the distance between points
[Protractor Axiom] The properties of the degree measure of an angle
Congruent triangles (SAS, ASA and SSS)
[Axiom of Parallels] Given any line l and a point P, there is exactly one line
through P that is parallel to l.
Theorem 1. Vertically opposite angles are equal in measure.
Theorem 2. In an isosceles triangle the angles opposite the equal sides are equal.
Conversely, if two angles are equal, then the triangle is isosceles.
Theorem 3. If a transversal makes equal alternate angles on two lines then the lines are
parallel, (and converse).
Theorem 4. The angles in any triangle add to 180˚.
Theorem 5. Two lines are parallel if and only if, for any transversal, the corresponding
angles are equal
Theorem 6. Each exterior angle of a triangle is equal to the sum of the interior opposite
angles.
Theorem 7. The angle opposite the greater of two sides is greater than the angles
opposite the lesser. Conversely, the side opposite the greater of two angles
is greater than the side opposite the lesser angle.
Theorem 8. Two sides of a triangle are together greater than the third.
Theorem 9. In a parallelogram, opposite sides are equal and opposite angles are equal (and
converses).
Theorem 10. The diagonals of a parallelogram bisect each other.
Theorem 11. If three parallel lines cut off equal segments on some transversal line, then
they will cut off equal segments on any other transversal. **
Theorem 12. Let 𝐴𝐵𝐶 be a triangle. If a line l is parallel to BC and cuts [𝐴𝐵] in the ratio
𝑠: 𝑡, then it also cuts [𝐴𝐶] in the same ratio (and converse). **
Theorem 13. If two triangles are similar, then their sides are proportional, in order (and
converse). **
Theorem 14. [Theorem of Pythagoras] In a right-angled triangle the square of the
hypotenuse is the sum of the squares of the other two sides.
Theorem 15. If the square of one side of a triangle is the sum of the squares of the other two
sides, then the angle opposite the first side is a right angle.
Theorem 16. For a triangle, base 𝒙 height does not depend on the choice of base.
Theorem 17. A diagonal of a parallelogram bisects the area.
Theorem 18. The area of a parallelogram is the base 𝒙 height.
Theorem 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.
Theorem 20.
(i)
Each tangent is perpendicular to the radius that goes to the point of
contact. (proved by contradiction)
(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.
Theorem 21.
(i)
(ii)
Corollaries:
1.
2.
3.
4.
5.
6.
The perpendicular from the centre to a chord bisects the chord.
The perpendicular bisector of a chord passes through the centre.
A diagonal divides a parallelogram into 2 congruent triangles.
All angles at points of the circle, standing on the same arc, are equal.
Each angle in a semi-circle is a right angle.
If the angle standing on a chord [BC] at some point of the circle is a rightangle, then [BC] is a diameter.
If ABCD is a cyclic quadrilateral, then opposite angles sum to 180˚, (and
converse).
If two circles intersect at one point only, then the two centres and the point
of contact are collinear.
Bold represents specific leaving certificate theorems at OL and HL
** Represents proofs required (only 3).
Constructions:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Bisector of a given angle, using only compass and straight edge.
Perpendicular bisector of a segment, using only compass and straight edge.
Line perpendicular to a given line l , passing through a given point not on l .
Line perpendicular to a given line l , passing through a given point on l .
Line parallel to given line, through given point.
Division of a segment into 2, 3 equal segments, without measuring it.
Division of a segment into any number of equal segments, without measuring it.
Line segment of given length on a given ray.
Angle of given number of degrees with a given ray as one arm.
Triangle, given lengths of three sides.
Triangle, given SAS data.
Triangle, given ASA data.
Right-angled triangle, given the length of the hypotenuse and one other side.
Right-angled triangle, given one side and one of the acute angles (several cases).
Rectangle, given side lengths.
Exclusively LC constructions:
16. Circumcentre and circumcircle of a given triangle, using only straight edge and
compass.
17. Incentre and incircle of a given triangle, using only straight-edge and compass.
18. Angle of 60 o , without using a protractor or set square.
19. Tangent to a given circle at a given point on it.
20. Parallelogram, given the length of the sides and the measure of the angles.
21. Centroid of a triangle.
22. Orthocentre of a triangle.
Definitions / Terminology:
1. An axiom is a statement accepted without proof, as a basis for argument
2. A theorem is a statement deduced from the axioms by logical argument. Theorems
can also be deduced from previously established theorems.
3. Corollary: a proposition that follows directly from the proof of another proposition.
4. A proposition is a useful or interesting statement that could be proved at this point,
but whose proof is not stipulated as an essential part of the programme.
5. Two objects are equivalent to if they are able to be placed in one-to-one
correspondence. For example: 𝑠𝑖𝑛180° is equivalent to 𝑠𝑖𝑛𝜋 or 3:4 is equivalent to
6: 8
6. if and only if: symbols:  or iff
a. if and only if means that two statements are "bi-conditional". "A  B" or “A
iff B” means that:
b. "A" is true if "B" is true
and
"B" is true if "A" is true.
c. For example: For any real numbers 𝑥 and 𝑦, 𝑥 = 𝑦 + 1 𝑖ff 𝑦 = 𝑥 − 1.
d. A person is a bachelor iff that person is a marriageable man who has never
married
7. Proof by contradiction:
a. Example: Proposition: There is no more than 1 right angle in a triangle:
 Proof: Given ABC
 Assume: A and B are both 90 o (i.e. there is more than one right
angle
  A  B  C  180 (as all angles in triangle add up to 180 o ).
  90  90  C  180
  C  0o
 However this means that ABC is not a triangle which contradicts the
fact that ABC is a triangle.
  Can only have 1 right angle in a triangle
b. Explanation of Proof by contradiction: we assume that the statement we
want to prove is false, and then show that this assumption leads to nonsense.
We are then led to conclude that we were wrong to assume the statement was
false, so the statement must be true.
c. In fancier English: A proof by contradiction establishes the truth of a given
proposition by the supposition that it is false and the subsequent drawing of a
conclusion that is contradictory to something that is proven to be true.
Example 2 proof by contradiction:
Proposition: The number √𝟐 irrational