Junior Certificate Theorems, Axioms and Corollaries
Axiom 1: There is exactly one line through any two given points
Axiom 2: The properties of the distance between points (Ruler Axiom)
Axiom 3: Protractor Axiom (properties of the degree measure of an angle)
Axiom 4: Congruent Triangles (SSS, SAS, ASA, RHS)
Axiom 5: 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.
Theorem 3: If a transversal makes equal alternate angles on two lines then the lines are parallel.
*Theorem 4: The angles in a triangle add up to 180 degrees.
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 angle opposite the lesser side, (and converse)
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.
Corollary 1: A diagonal bisects the area of a parallelogram
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 m : n 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.
*Theorem 14: Theorem of Pythagoras - In a right angled triangle the square of the hypotenuse is equal to sum
of the squares of the other two sides.
Theorem 15: Converse to Pythagoras - If the square of one side of a triangle is equal to the sum of the squares of the other two, then the angle opposite the first side is a right angle.
*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 standingon the same arc.
Corollary 2: All angles at points of a circle, standing on the same arc are equal
Corollary 3: Each angle in a semi-circle is a right angle
Corollary 4: If the angle standing on a chord [BC] at some point of the circle is a right angle then [BC] is a diameter
Corollary 5: If ABCD is a cyclic quadrilateral, then opposite angles sum up to 180 degrees.
Axiom 2: The properties of the distance between points (Ruler Axiom)
Axiom 3: Protractor Axiom (properties of the degree measure of an angle)
Axiom 4: Congruent Triangles (SSS, SAS, ASA, RHS)
Axiom 5: 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.
Theorem 3: If a transversal makes equal alternate angles on two lines then the lines are parallel.
*Theorem 4: The angles in a triangle add up to 180 degrees.
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 angle opposite the lesser side, (and converse)
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.
Corollary 1: A diagonal bisects the area of a parallelogram
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 m : n 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.
*Theorem 14: Theorem of Pythagoras - In a right angled triangle the square of the hypotenuse is equal to sum
of the squares of the other two sides.
Theorem 15: Converse to Pythagoras - If the square of one side of a triangle is equal to the sum of the squares of the other two, then the angle opposite the first side is a right angle.
*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 standingon the same arc.
Corollary 2: All angles at points of a circle, standing on the same arc are equal
Corollary 3: Each angle in a semi-circle is a right angle
Corollary 4: If the angle standing on a chord [BC] at some point of the circle is a right angle then [BC] is a diameter
Corollary 5: If ABCD is a cyclic quadrilateral, then opposite angles sum up to 180 degrees.
Leaving Certificate Ordinary Level Theorems
***Important to note that all theorems learned at Junior Cert are also examinable***