Theorem: A general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths.

 

1. Vertically opposite angles are equal in measure.

 

2. In an isosceles triangle the angles opposite the equal sides are equal. 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. Conversely, if two lines are parallel, then any transversal will make equal alternate angles with them.

 

4.* The angles in any triangle add to 180.

 

 

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.

 

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.

 

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. Conversely, (1) if the opposite angles of a convex quadrilateral are equal, then it is a parallelogram; (2) if the opposite sides of a convex quadrilateral are equal, then it is a parallelogram. (Corollary 1)

10. The diagonals of a parallelogram bisect each other. Conversely, if the diagonals of a quadrilateral bisect one another, then the quadrilateral is a parallelogram.

11.* If three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal.

 

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. Conversely, if the sides of two triangles are in proportion, then the two triangles are similar.

13.* If two triangles are similar, then their sides are proportional, in order (and converse).

 

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.

 

 

15. [Converse to Pythagoras]. If the square of one side of a 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 x height does not depend on the choice of base.

 

17. A diagonal of a parallelogram bisects the area.

 

18. The area of a parallelogram is the base x 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. (Corollary 2-5)

 

20. (i) Each tangent is perpendicular to the radius that goes to the point of contact.

(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. (Corollary 6)

 

21. (i) The perpendicular from the centre to a chord bisects the chord.

(ii) The perpendicular bisector of a chord passes through the centre.