line
Use arrow tips on both ends. Name using a lower case letter, or with 2 captial letters that are points on the line.
Point
represented by a dot. Use capital letters
to name.
Plane
represented by a thin flat surface that extends in all directions without ending. Use a capital letter to name.
Horizontal plane
One pair of horizontal lines
Vertical plane
One pair of vertical lines
Space
Set of all points.
Collinear points
Points all in one line.
Noncollinear points
Points not all in one line
Coplanar points
Points all in one plane.
Noncoplanar points
Points not all in one plane
Intersection
Set of points that are in all figures
Between
Points A and B are between points C and D
segment
Part of a line with endpoints
Ray
Starts at a endpoint or start point and goes on forever in one direction
Opposite Rays
Rays that go in opposite directions
AB
number; Is read "the length of line segment AB".
Postulates or Axioms
Accepted facts without proof.
Ruler Postulate
You can make a ruler and then find any point and its coordinate on it.
Segment Addition Postulate
If B is between A and C, then
AB + BC = AC
Congruent
Same size and shape.
Congruent Segments
Segments that have the same length.
Midpoint of a segment
The point that divides
the segment into two congruent segments.
F B is the midpoint of AC, THEN AB BC
IF AB BC, THEN B is the midpoint of AC
Angle
Figure formed by 2 rays with a common
endpoint.
Vertex
Where the endpoints meet on an angle.
Angles classified by their measures
Acute angle: <0 but >90
Right angle: 90
Obtuse angle: >90 but <180
Straight angle: 180
Angle Addition Postulate
If B lies in the interior
of AOC, then
mAOB + mBOC = mAOC
Congruent angles
Angles that have the same
measure.
Adjacent angles
Two angles in a plane that have a common vertex and side, but no common interior points.
Bisector of an angle
A ray that divides the angle
into 2 congruent adjacent angles.
IF AD bisects BAC, THEN BAD DAC
IF BAD DAC, THEN AD bisects BAC.
1 POSTULATE
A line contains at least 2 points; a plane contains at least 3 noncollinear points: space contains at least 4 noncoplanar points.
2 POSTULATE
Through any 2 points there is exactly one line.
3 POSTULATE
Through any 3 points there is at least one plane, and through any 3 noncollinear points there is exactly one plane.
4 POSTULATE
If 2 points are in a plane, then the line that contains the points is in that plane.
5 POSTULATE
If 2 planes intersect, then there intersection is a line.
6 POSTULATE
If 2 lines intersect, then they intersect in exactly one point.
7 POSTULATE
Through a line and a point not in the line there is exactly one plane.
8 POSTULATE
If 2 lines intersect, then there is exactly one plane contains them.
Conditionals
If-then statements.
IF B is between A and C, THEN AB+BC=AC. hypothesis, Conclusion
Converse
Interchange the hypothesis and
conclusion.
Statement: IF P, THEN Q. Converse: IF Q, THEN P.
Counterexample
An example that makes a conditional FALSE.
Other ways to write conditionals
If P, Then Q.
P implies Q.
P only if Q.
Q if P
Biconditional
When a conditional and its
converse are both TRUE. P if and only if Q.
ALL DEFINTIONS ARE BICONDITIONALS
Congruent angles
Angles with equal measure.
Angles are congruent if and only if their measures are equal.
Addition property
If a= band c=d, then a+C=b+d
Subtraction property:
If a=b and c=d then a-c=b-d
multiplication property
if a=b then ca=cb
division property
if a=b and c is not equal to 0 then a/c = b/c
substitution property
if a=b then either can replace the other
distributive property
a(b+c)=ab+ac
reflexive property
a=a
symmetric property
if a=b then b=a
transitive property
if a=b and b=c then a=c
reflexive property of congruence
segment AB is congruent to segment AB
Symmetric property of congruence
if line segment AB is congruent to CD then CD is congruent to AB
Transitive property of congruence
if line segment AB is congruent to CD and CD is congruent to EF then AB is congruent to EF
Theorems
Theorems are statements that have been proven using postulates, definitions, and properties that have been accepted without proof.
Midpoint Theorem
If M is the midpoint of AB then AM=1/2AB and MB=1/2AB
Angle Bisector Theorem
If BX is the bisector of <ABC then <ABX = 1/2m<ABD and m<XBC=1/2m<ABC
Reasons to use proofs
given info, definitions, postulates, theorems that have already been proved
Complementary angles
Two angles whose sum
is 90. Each angle is a complement of the other. (90-x=complement of an angle)
Supplementary angles
Two angles whose sum
is 180. Each angle is a supplement of the other. (180-x=supplement of an angle)
Vertical angles
These are formed when two lines intersect.
Theorem: Vertical angles are congruent
Perpendicular lines
Two lines that intersect to
form right angles or 90 degree angles.
If perpendicular, then right angles.
If right angles, then perpendicular. If perpendicular, then 90 degrees.
If 90 degrees, then perpendicular.
Perpendicular Theorems
If two lines are perpendicular, then they form congruent adjacent angles. If two lines form congruent adjacent angles, then they are perpendicular.
Complementary Theorem
If the exterior sides of two acute adjacent angles are perpendicular, then the angles are complementary
Supplement Congruence Theorem
If 2 angles are supplements of congruent angles (or the same angle), then the 2 angles are congruent.
Complement Congruence Theorem
If 2 angles are complements of congruent angles (or the same angle), then the 2 angles are congruent.