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