Postulates are mathematical statements that are accepted as true without proof.
Unit 1 postulates
1. The intersection of two lines is a point.
2. The intersection of two planes is a line.
3. Through any two points there is exactly one line.
4. Through any three noncollinear points there is exactly one plane.
5. If two points are in a plane, the the line containing them is in that plane.
6. Segment Congruence Postulate:
a. If two segments have the same length as measured by a fair ruler (a ruler that has evenly spaced segments marked), then the segments are congruent.
b. If two segments are congruent, then they have the same length as measured by a fair ruler.
7. Segment Addition Postulate:
If a line contains three points (let us call the points A, B, and C) and point B is between points A and C, then AB + BC = AC. AB is the length of segment AB, and BC is the length of segment BC and AC is the length of segment AC.
8. Angle Congruence Postulate:
If two angles have the same measure (the rays do not have to have the same length), then they are congruent. If two angles are congruent, then they have the same measure.
9. Angle Addition Postulate:
Suppose we have three noncollinear points, A, B, and C, and through each of these points there is a ray. Then suppose these three rays meet at a fourth point, D. If point B is in the interior of angle ADC, then the measure of angle ADB + the measure of angle BDC = the measure of angle ADC.
Unit 2 Postulates
Euclid's common notion: If equals are added to equals, then the wholes are equal.
Corresponding Angles Postulate: If two lines cut by a transversal are parallel, then corresponding angles are congruent
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Polygon Congruence Postulate: Two polygons are congruent if and only if there is a correspondence between their sides and angles such that:
Each pair of corresponding angles is congruent.
Each pair of corresponding sides is congruent.
SSS (Side-Side-Side) Postulate
If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent.
SAS (Side-Angle-Side) Postulate
If two sides and their included angle in one triangle are congruent to two sides and their included angle of another triangle, then the two triangles are congruent.
ASA (Angle-Side-Angle) Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Midpoint formula: The midpoint of a segment with endpoints
has the following coordinates: