Honors Geometry Fall Final (A star * means you should know how to prove the theorem as well)
Converse of the Side Angle Relation Theorem
If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle
Converse of the Alternate Interior Angles Theorem*
If two lines and a transversal form Alternate interior angles, and those angles are congruent, then the lines are parallel.
Converse of the Same-Side Interior Angles Postulate*
If two lines and a transversal form Same-Side Interior angles, and those angles are supplements, then the lines are parallel.
Converse of the Alternate Exterior Angles Theorem*
If two lines and a transversal form alternate exterior angles, and those angles are congruent, then the lines are parallel.
Converse of the corresponding angles theorem
If two lines and a transversal form corresponding angles and they are congruent, then the lines are parallel.
SAS
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, than the two triangles are congruent
Base Angles Theorem*
If two sides of a triangle are congruent, then the angles opposite those sides are congruent
Side Angle relation Theorem
If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side
conditional statement
a statement that joins a statement called the hypothesis with a statement called the conclusion using the words "if"..."then"; it is only false when the hypothesis is true and the conclusion is false
disjunction
a statement that joins two other statements using the word "or"; it is only false if both statements are false.
Isosceles Triangle
a triangle with two congruent sides
Exterior angle of a polygon
angle formed by a side of the polygon and an extension of an adjacent side.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side
Triangle Sum Theorem*
The sum of the measures of the interior angles of a triangle is 180 degrees
Triangle Exterior Angle Theorem*
The sum of the measures of two remote interior angles of a triangle to an exterior angle is equal to the measure of the exterior angle
Parallel Postulate
Through a point not on a line, there is on and only one line parallel to the given line.
Perpendicular Postulate
Through a point not on a line, there is one and only one line perpendicular to the given line
coordinate
a Real number value given to a point
transversal
a line that intersect two coplanar lines at two distinct points
altitide of a triangle
a perpendicular segemt from the vertex of a triangle to the LINE containing the opposite side
segment bisector
a point, line, ray, or other segment that intersects a segment at its midpoint
angle bisector
a ray that divides an angle into two congruent angles
midsegment of a triangle
a segment that connects the midpoints of two sides of a triangle
median of a triangle
a segment that connects the vertex of a triangle with the midpoint of the opposite side
statement
a sentence that can be assigned a truth values; the truth value is either true of false but not both
conjunction
a statement formed by tow other statements using the word "and"; it is only true when both statements are true.
tautology
a statement that is always true
Parallel lines
coplanar lines that do not intersect
CPCTC
corresponding parts of congruent triangles are congruent
inverse
derived from a conditional statement where the hypothesis and conclusion are negated
Converse
derived from a conditional statement where the hypothesis and conclusion are switched
contrapositive
derived from a conditional statement where the hypothesis and conclusion are switched and negated
midpoint
divides a segment into two congruent segments
equidistant
equal distance
centroid theorem
if a point is a centroid, then it is two thirds the distance FROM the vertext TO the midpoint of the opposite side.
incenter theorem*
if a point is an incenter then it is equidistant from all three sides of the triangle
Converse of the perpendicular bisector theorem*
if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
perpendicular bisector*
if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Point
indicates a location and has no size
Remote Interior Angle
interior angles of a polygon that are nonadjacent to a given remote angle
linear pairs
is a pair of adjacent angles whose non common sides are opposite rays
angle bisector
is a ray that divides an angle into two congruent segments
biconditional statement
is a statement that joins two other statements using the words "if and only if"; It is true when the truth values of each statement match
Corresponding Angles
non adjacent angles that are on the same side of the tansversal and exactly one angle is between the two lines. Lines do not need to be parallel!
anlternate interior angles
nonadjacent angles that are on different sides of the transversal and are between the other two lines. Lines do not need to be parallel!
alternate exterior angles
nonadjacent angles that are on different sides of the transversal and are not between the other two lines. Lines do not need to be parallel!
same-side interior angles
nonadjacent angles that are on the same side of the transversal and are between the two other lines. Lines do not need to be parallel!
skew lines
noncoplaner lines that do not intersect
ray
part of a line that consists of one end point and all the points of the line on one side of the endpoint
line segment
part of a line that consists of two endpoints and all points between them
incenter
point where all three angle bisectors of a triangle intersect
circumcenter
point where all three perpendicular bisectors of a triangle intersect
Congruent polygons
polygons that congruent corresponding parts
plane
represents a flat surface that extends without end and has no thickness
line
represents a straight path that extends in two opposite directions without end and has no thickness
distance from a point to a line
the length of the perpendicular segment from the point to the line.
orthocenter of a triangle
the point where all the altitudes intersect
centroid
the point where all the medians of a triangle intersect
point of concurrency
the point where three or more lines intersect
vertical angles
two angles whos sides are opposite rays
supplementary angles
two angles whose measures have a sum of 180 degrees
complementary angles
two angles whose measures have a sum of 90 degrees
adjacent angle
two coplanar angles, with a common side, a common vertex, and no common interior points
opposite rays
two rays that share the same endpoint and form a line
angle
two rays with the same endpoint
congruent segments
two segments that have the same length
negation
uses the word not to change the truth value of a statement
Angle Addition Postulate
If B is on the interior of <AOC, then m<AOB + m<BOC = m<AOC
Converse of the angle bisector theorem*
If a point in the interior of an angle is equidistant from the sides of an angle, then the point is on the angle bisector
angle bisector theorem*
If a point is on the angle bisector of an angle, then the point is equidistant from the sides of the angle
circumcenter theorem*
If a point is the circumcenter of a triangle, then it is equadistant from all three verticies of the triangle
Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long.
Alternate Exterior Angles Theorem*
If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.
Alternate Interior Angles Theorem*
If a transversal intersects two parallel lines, then the alternate interior angles are congruent.
Corresponding Angles Theorem*
If a transversal intersects two parallel lines, then the corresponding angles are congruent.
Same-Side Interior Angles Postulate
If a transversal intersects two parallel lines, then the same-side interior angles are supplements
Law of Syllogism*
If a=>b and b=>c then a=>c.
Transitive Property
If a=b and b=c Then a=c and b is the train
comparison property of inequality*
If a=b+c and c>0, then a>b
HL
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent
SSS
If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.
Segment Addition Postulate
If three points A, B, and C are colinear and B is between A and C, then AB + BC = AC
Vertical Angles Theorem*
If two angels are verticle, then they are congruent.
AAS*
If two angles and a nonincluded angle of one triangle are congruent to two angles and a nonincluded angle of another triangle, then the two triangles are congruent.
ASA
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
Congruent Supplements Theorem*
If two angles are congruent to the same angle, then the two angles are congruent.
All Right Angles are Congruent
If two angles are right, then they are congruent.
Linear Pairs Postulate
If two angles form linear pairs, then they are supplementary
Third Angles Theorem*
If two angles of a triangle are congruent to two angles of another triangle, then the third angles are congruent
Converse of the Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent
Contradiction
A statement that is always false
Hypothesis
The first statement of a conditional statement
Conclusion
The last statement in a conditional statement