Notation, Axioms & Theorems

1. A straight angle has 180◦

2. Given a ray [AB, and a number d between 0 and 180, there is exactly one ray from A on each side of the line AB that makes an (ordinary) angle having d degrees with the ray [AB.

3. If D is a point inside an angle ∠BAC, then |∠BAC| = |∠BAD| + |∠DAC|.

Null angles are assigned 0◦, full angles 360◦, and reflex angles have more than 180◦. To be more exact, if A, B, and C are noncollinear points, then the reflex angle “outside” the angle ∠BAC measures 360◦ −|∠BAC|, in degrees.

Axiom 4 (SAS+ASA+SSS).

If (1) |AB| = |A'B'|, |AC| = |A'C'| and |∠A| = |∠A'|,

or

(2) |BC| = |B'C'|, |∠B| = |∠B'|, and |∠C| = |∠C'|,

or

(3) |AB| = |A'B'|, |BC| = |B'C'|, and |CA| = |C'A'|

then the triangles ∆ABC and ∆A'B'C' are congruent.

Notation 4

We write l || m for “ l is parallel to m”.

Axiom 5 (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).

Vertically opposite angles are equal in measure.

Theorem 2 (Isosceles Triangles).

(1) In an isosceles triangle the angles opposite the equal sides are equal.

(2) Conversely, If two angles are equal, then the triangle is isosceles.

Theorem 3 (Alternate Angles). Suppose that A and D are on opposite sides of the line BC.

(1) If |∠ABC| = |∠BCD|, then AB||CD. In other words, if a transversal makes equal alternate angles on two lines, then the lines are parallel.

(2) Conversely, if AB||CD, then |∠ABC| = |∠BCD|. In other words, if two lines are parallel, then any transversal will make equal alternate angles with them.

Theorem 4 (Angle Sum 180). The angles in any triangle add to 180◦

Theorem 5 (Corresponding Angles). Two lines are parallel if and only if for any transversal, corresponding angles are equal.

Theorem 6 (Exterior Angle). Each exterior angle of a triangle is equal to the sum of the interior opposite angles.

Theorem 7.

(1) In ∆ABC, suppose that |AC| > |AB|. Then |∠ABC| > |∠ACB|. In other words, the angle opposite the greater of two sides is greater than the angle opposite the lesser side.

(2) Conversely, if |∠ABC| > |∠ACB|, then |AC| > |AB|. In other words, the side opposite the greater of two angles is greater than the side opposite the lesser angle.

Theorem 8 (Triangle Inequality).

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.

Converse 1 to Theorem 9: If the opposite angles of a convex quadrilateral are equal, then it is a parallelogram.

Converse 2 to Theorem 9: If the opposite sides of a convex quadrilateral are equal, then it is a parallelogram.

Corollary 1. A diagonal divides a parallelogram into two congruent triangles

Theorem 10. The diagonals of a parallelogram bisect one another.

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.

Theorem 13. If two triangles ∆ABC and ∆A'B'C'0 are similar, then their sides are proportional, in order:

|AB| / |A'B'| = |BC| / |B'C'| = |CA| / |C'A'|.

Theorem 14 (Pythagoras). In a right-angle triangle the square of the hypotenuse is the sum of the squares of the other two sides.

Theorem 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.

Theorem 16. For a triangle, base times 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 by the 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

Corollary 3. Each angle in a semicircle is a right angle. In symbols, if BC is a diameter of a circle, and A is any other point of the circle, then ∠BAC = 90◦.

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◦

Theorem 20.

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

(2) If P lies on the circle s, and a line l is perpendicular to the radius to P, then l is tangent to s.

Corollary 6. If two circles share a common tangent line at one point, then the two centres and that point are collinear

Theorem 21.

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

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