Theorems & Corollaries

Theorems & Corollaries

1. 1. Vertically opposite angles are equal in measure.

2. 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. 3. If a transversal makes equal alternate angles on two lines then the lines are parallel (and converse).

4. 4. The angles in any triangle add to 180⁰

5. 5. Two lines are parallel if, and only if, for any transversal, the corresponding angles are equal.

6. 6. Each exterior angle of a triangle is equal to the sum of the interior opposite angles.

7. 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. 8. Two sides of a triangle are together greater than the third.

9. 9. In a parallelogram, opposite sides are equal, and opposite angles are equal.

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

11. 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.

12. 13. If two triangles are similar, then their sides are proportional, in order.

13. 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.

14. 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.

15. 16. For a triangle, base x height does not depend on the choice of base.

16. 17. A diagonal of a parallelogram bisects the area.

17. 18. The area of a parallelogram is the base x height.

18. 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†: 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 to 180 ⁰.

19. 20. 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: If two circles intersect at one point only, then the two centres and the point of contact are collinear.

20. 21.The perpendicular from the centre to a chord bisects the chord. (ii) The perpendicular bisector of a chord passes through the centre.

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