Postulates and main theorems.
−−→
1. (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.
2. (The angle addition postulate.) If D is in the interior of ∠BAC, then
m∠BAC = m∠BAD + ∠DAC.
3. (The supplement postulate.) Sum of supplementary angles is 180.
4. (The vertical angle theorem.) If two angles form a vertical pair, then they
are congruent.
5. (SAS postulate.) If AB ∼
= A1 B1 , AC ∼
= A1 C1 , and ∠BAC ∼
= ∠B1 A1 C1 ,
∼
then 4ABC = 4A1 B1 C1 .
6. If two sides of a triangle are congruent, the angles opposite to them are
congruent.
7. (ASA theorem.) If BC ∼
= ∠C1 , then
= B1 C1 , ∠B ∼
= ∠B1 , and ∠C ∼
∼
4ABC = 4A1 B1 C1 .
8. (SSS theorem.) If AB ∼
= A1 B1 , AC ∼
= A1 C1 , and AC ∼
= A1 C2 1, then
∼
4ABC = 4A1 B1 C1 .
9. Every angle has exactly one bisector.
10. Given a line and a pint not on the line, then there is a line which passes
through the given point and is perpendicular to the given line.
11. Any exterior angle of a triangle is greater than each of its remote interior
angles.
12. If two sides of a triangle are not congruent, then the angles opposite them
are not congruent, and the greater angle is opposite the longer side.
13. The shortest segment joining a point to a line is the perpendicular segment.
14. In any triangle, the sum of the lengths of any two sides is greater than the
length of the third side.
15. (SAA theorem.) If AB ∼
= A1 B1 , ∠A ∼
= ∠A1 , and ∠C ∼
= ∠C1 , then
∼
4ABC = 4A1 B1 C1 .
16. Given a correspondence between two right triangles. If the hypotenuse
and one leg of one of the triangles are congruent to the corresponding
parts of the other triangle, then the correspondence is a congruence.
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