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Main Postulates
SS-1 Given a plane is space. The set of all points that do not lie in the plane is the union od two sets H1 and
H2 such that: (a) each of the sets is convex (b) if P ∈ H1 and Q ∈ H2 , then P Q intersects the plane.
−−→
M-3 (The angle construction postulate.) Let AB be a ray on the edge of the half plane H. For every number
−→
r between 0 and 180, there is exactly one ray AP , with P in H, such that m∠P AB = r.
M-4 (The angle-addition postulate.) If D is in the interior of ∠BAC, then m∠BAC = m∠BAD + m∠DAC.
M-5 (The supplement postulate.) Sum of supplementary angles is 180.
SAS If AB ∼
= A1 B1 , AC ∼
= A1 C1 , and ∠BAC ∼
= ∠B1 A1 C1 , then △ABC ∼
= △A1 B1 C1 .
Main Theorems
1. If P and Q are on opposite sides of the line L, and Q and T are on opposite sides of L, then P and T are
on the same side of L.
2. If P and Q are on opposite sides of the line L, and Q and T are on the same side of L, then P and T are
on opposite sides of L.
3. Given a line, and a ray which has its end point on the line but does not line on the line. Then all points
of the ray, except for the end point are on the same side of the line.
4. Every side of a triangle lies, except for its end points, in the interior of the opposite angle.
−→
5. If F is the interior of ∠BAC, then AF − A lies in the interior of ∠BAC.
6. Let L be a line, let A and F be two points of L, and let B and G be points on opposite sides of L. Then
−−→
F B does not intersect AG.
7. In △F BC, let A be a point between F and C, and let D be a point such that D and B are on the same
←→
−−→
side of F C. Then AD intersects either F B or BC.
−−→
8. (The Crossbar Theorem.) If D is in the interior of △ABC then AD intersects BC in a point between B
and C.
9. Let H be a half plane with edge L, and let E be a plane that contains L but not H. Then all points of
H are on the same side of E.
10. The interior of a dihedral angle is always a convex set.
11. If P and Q are in different sides of a dihedral angle, then every point between P and Q is in the interior
of the dihedral angle.
12. The diagonals of a convex quadrilateral always intersect each other.
13. (The vertical angle theorem.) If two angles for a vertical pair, then they are congruent.
14. (The isosceles triangles theorem.) If two sides of a triangle are congruent, then the angles opposite to
them are congruent.
15. (ASA theorem.) If BC ∼
= B1 C1 , ∠B ∼
= ∠B1 , and ∠C ∼
= ∠C1 , then △ABC ∼
= △A1 B1 C1 .
16. (SSS theorem.) If AB ∼
= A1 B1 , AC ∼
= A1 C1 , and BC ∼
= B1 C1 , then △ABC ∼
= △A1 B1 C1 .
17. Every angle has exactly one bisector.
18. Given a line and a point not on the line, then there is a line which passes through the given point and is
perpendicular to the given line.
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