# Chapters 1 & 2 Postulates, Theorems, and Formulas

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### Postulate 2.8 "Ruler Postulate" (HW Postulate 1)

The points on any line or line segment can be paired with real numbers so that given any 2 points, A and B on a line, A corresponds to 0 and B corresponds to a positive real number.

### Postulate 2.9 "Segment Addition Postulate" (HW Postulate 2)

If A, B, and C are collinear, and B is between A and C, then AB+BC=AC. or If AB+BC=AC, then B is between A and C.

### Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

### Midpoint Formula

(x₁+x₂)/2, (y₁+y₂)/2

### Postulate 2.10 "Protractor Postulate" (HW Postulate 3)

Given line AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of line AB, such that the measure of the angle formed is r.

### Postulate 2.11 "Angle Addition Postulate"

If B lies in the interior of <AOC, then m<AOB + m<BOC = m<AOC. If <AOC is a straight angle, and B is any point not on ray AC, then m<AOC + m<BOC = 180.

### HW Postulate 5

A line contains at least 2 points; a plane contains at least 3 points not all in one line; space contains at least 4 points not all in one plane.

### HW Postulate 6

Through any 2 points there is exactly one line.

### HW Postulate 7

Through any 3 points, there is at least one plane, and though any 3 noncollinear points there is exactly one plane.

### HW Postulate 8

If 2 points are in a plane, then the line that contains the points is in that plane.

### HW Postulate 9

If 2 planes intersect, then their intersection is a line.

### HW Theorem 1-1

If two lines intersect, then they intersect in exactly one point.

### HW Theorem 1-2

Through a line and a point not in the line, there is exactly one plane.

### HW Theorem 1-3

If 2 lines intersect, then exactly one plane contains the lines.

### HW Theorem 2-1 "Midpoint Theorem"

If M is the midpoint of segment AB, then AM=1/2AB and MB=1/2AB.

### HW Theorem 2-2 "Angle Bisector Theorem"

If ray BX is the bisector of <ABC, then m<ABX=1/2M<ABC and m<XBC=1/2m<ABC.

### HW Theorem 2-3 "Vertical Angles Theorem"

Vertical angles are congruent.

### HW Theorem 2-4

If 2 lines are perpendicular, then they form congruent adjacent angles.

### HW Theorem 2-5

If 2 lines form congruent adjacent angles, then the lines are perpendicular.

### HW Theorem 2-6

If the exterior sides of 2 adjacent acute angles are perpendicular, then the angles are complementary.

### Theorem 2.6 (HW Theorem 2-7)

Angles supplementary to the same angle or to congruent angles are congruent.
Abbr. <'s suppl. to same < or congruent <'s are congruent

### Theorem 2.7 (HW Theorem 2-8)

Angles complementary to the same angle or to congruent angles are congruent.
Abbr. <'s compl. to same < or congruent <'s are congruent

Example:

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