Postulates And Theoroms Essay, Research Paper
P1-Ruler Postulate.
P2-seg. add. postulate.
P3-Protractor postulate.
P4-angle add. postulate.
P5- A line contains at least two points; a plane contains at least 3 points not all in one line; space contains at least 4 pints not all in one plane.
P6- Through any 2 points their is excatly 1 line.
P7-Through any 3 points there is at least one plane , and through any three noncollinear points there is exactly one plane.
P8- If two points are in a plane the the line that contains the points is in that plane.
P9-If two intersect, then their intersection is a line.
T1-1-If tow lines intersect then they intersect in exactly one point.
T1-2-Through a line and a point not in the line there is exactly one plane.
T1-3- If 2 lines intersect then exactly one plane contains the lines.
Properties of equality
Add. Prop-if a=b and c=d then a+c=b+d
Subtraction Prop-if a=b and c=d then a-c=b-d
Mult. Prop- if a=b then ca=cb
Div Prop.-if a=b and c doesnt = 0 then a/c=b/c
Substitution prop- if a=b then either a or b may be substituded for the other in any equation.
Reflexive Property-a=a
Symmetric Property- if a=b then b=a
Transitive Prop.-if a=b and b=c then a=c.
Properties of Congruence
Reflexive Prop-Line DE is congruent to line DE. angle D=angle D
Symmetric Prop.- Line DE=FG then FG=DE. angle D=F then angle F=D.
Transitive Prop.- Line DE is congruent to line FG and line FG is congruent to JK then line DE is congruent to JK.
Distributive Prop.-a(b+c)=ab+ac
T2-1- IF M is the midpoint of line ab then am = half ab and mb = half ab line amb.
T2-2- If ray bx is the bisector of angle abc then m of angle abx=half the measure of angle abc and measure of angle xbc =half m angle abc.
T2-3- Vert. angle are congruent.
T2-4- If 2 lines are perpendicular then they form congruent adjacent angles.
T2-5- IF 2 lines form congruent adjacent angles then the lines are perpendicular.
T2-6- If the exterior sides of two adjacent acute angles are perpendicular then the angles are complementary.
T2-7- IF 2 angles are supplements of congruent angles then the 2 angles are congruent.
T2-8- IF 2 angles are complements of congruent angles then the 2 angles are congruent.
T3-1- IF 2 parallel planes are cut bty a 3rd plane then the lines of intersection are parallel.
P10- If 2 parallel lines are cut by a transversal, then corresponding angles are congruent.
T3-2- IF 2 parallel lines are cut by a transversal then alternate interior angles are congruent.
T3-3- IF 2 parallel lines are cut by a transveral then same side interior angles are supplementary.
T3-4-If a transversal is perpendicular to one of 2 parallel lines then it is perpendicular to the other one also.
P11- IF 2 lines are cut by a transversal and corresponding angles are congruent then the lines are parallel.
T3-5- If 2 lines are cut by a transversal and alternate interior angles are congrunt then the lines are parallel.
T3-6- If 2 lines are cut by a transversal and same side interior angles are supplementary then the lines are parallel.
T3-7- In a plane 2 lines perpendicular to the same line are parallel.
T3-8- Through a piont outside a line there is exactly one line parallel to the given line.
T3-9- Through a point outside a line there is exactly one line perpendicular to the given line.
T3-10- Two lines parallel to a 3rd line are parallel to each other.
T3-11- The sum of the measures of the angles of a triangle is 180.
C1- If 2 angles of one triangle are congruent to 2 angles of another triangle then the 3rd angless are congruent.
C2- Each angle of an equianglular triangle has measure 60.
C3- In a triangle there can be at most one right angle or obtuse angle.
C4- the acute of a right triangle ar complementary.
T3-12- The measure of an exterior angle of a triangle equals the sum of the measure of the 2 remote interior angles.
Scalene- no sides congruent.
Isosceles- at least 2 sides congruent.
Equilateral- all sides congruent.
Acute- 3 acute angles.
Obtuse- 1 obtuse angle.
Right- 1 right angle.
Equilangular- all angles congruent.
Ways To prove to lines are parallel.
1. Show that a pair of corresponding angles are congruent.
2. Show that a pair of alternate interior angles are congruent.
3. Show that a pair of same side interior angles are supplementary.
4. In a plane show that both lines are perpendicular to a third line.
5. Show that both lines are parallel to a third line.
Alternate Interior angles- are 2 nonadjacent interior angles on opposite sides of the transversal. Z
Same Side Interior angles- are 2 interior angles on the same side of the transversal. U
Corresponding angles- are 2 angles in corresponding postitions relative to the 2 lines. F
thier will be more to come