# Honors Geometry Revised Semester One Exam Theorems and Postulates

## 84 terms

If B is between A and C, then AB+BC=AC

I, If P is in the interior of <RST, then m<RSP + m<PST = m<RST

Two angles that share a common vertex and side, but have no common interior points.

### Convex Polygon

No line that contains a side of the polygon contains a point in the interior of the polygon. (Concave is a nonconvex polygon)

### Conjecture

An unproven statement based on observations.

### Inductive Reasoning

Finding a pattern in specific cases and then write a conjecture for the general case.

### Negation

The opposite of the original statement.

### Biconditional statement

A statement containing the phrase "If and only if" (IFF)

### Law of Detachment

If the hypothesis of a true conditional statement is true, then the conclusion is also true.

### Law of Syllogism

If hypothesis p, then conclusion q
If hypothesis q, then conclusion r,
(If these statements are true,)
If hypothesis p, then conclusion r (then this is true)

### Deductive Reasoning

Uses facts, definitions, accepted properties, and the laws of logic to form a logical argument.

### Line

Through any two points there exists exactly one ____

### Points

A line contains at least two _____

### Point

If two lines intersect, then their intersection is exactly one _____.

### Plane

Through any three noncollinear points there exists exactly one _____.

### Plane

A _____ contains at least three noncollinear points

### Line

If two points lie in a plane, then the ____ containing them lies in the plane

### Planes

If two ______ intersect then their intersection is a line

### Right Angles Congruence Theorem

All right angles are congruent.

### Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles) then they are congruent

### Congruent Complements Theorem

If two angles are complementary to the same angle, then they are congruent

### Linear Pair Postulate

If two angles form a linear pair, then they are supplementary

### Vertical Angles Congruence Theorem

Vertical Angles are Congruent

### Skew Lines

Lines that do not intersect and are not coplanar

### Parallel Planes

Planes that do not intersect

### Parallel Postulate

If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

### Perpendicular Postulate

If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line

### Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent

### Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent

### Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent

### Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary

### Corresponding Angles Converse

If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel

### Alternate Exterior Angles Converse

If two lines are cut by a transversal so that the alternate exterior angles are congruent, then the lines are parallel

### Alternate Interior Angles Converse

If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel

### Consecutive Interior Angles Converse

If two lines are cut by a transversal so that consecutive interior angles are supplementary, then the lines are parallel

### Transitive Property of Parallel Lines

If two lines are parallel to the same line, then they are parallel to each other.

### Slopes of Parallel Lines

In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel.

### Slopes of Perpendicular Lines

In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. vertical and horizontal lines are perpendicular.

### Perpendicular

If two lines intersect to form a linear pair of congruent angles, then the lines are _____________.

### Right

If two lines are perpendicular, then they intersect to forms four _____ angles

### Complementary

If two sides of two adjacent acute angles are perpendicular, then the angles are _________________.

### Perpendicular Transversal Theorem

If a transversal is perpendicular to one of the two parallel lines, then it is perpendicular to the other

### Lines Perpendicular to a Transversal Theorem

In a plane, if two lines are perpendicular to the same line, then they are parallel

### Triangle Sum Theorem

The sum of the measures of the angles of a triangle is 180.

### Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

### Corollary to the Triangle Sum Theorem

The acute angles of a right triangle are complementary

### Third Angles Theorem

If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are also congruent

### SSS Congruence Postulate

If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent

### SAS Congruence Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent

### HL Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent

### ASA Congruence Postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle then the two triangles are congruent

### AAS Congruence Theorem

If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent.

### Base Angles Theorem

If two sides of a triangle are congruent, then the angles opposite them are congruent

### Converse of Base Angles Theorem

If two angles of a triangle are congruent, then the sides opposite them are congruent.

### Corollary to the Base Angles Theorem

If a triangle is equilateral, then it is equiangular

### Corollary to the Converse of Base Angles Theorem

If a triangle is equiangular, then it is equilateral

### Midsegment Theorem

The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long

### Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

### Converse of the Perpendicular Bisector Theorem

In a plane, if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

### Concurrency of Perpendicular Bisectors Theorem

The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle

### Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle

### Converse of the Angle Bisector Theorem

If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle

### Concurrency of Angle Bisectors of a Triangle

The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle

### Concurrency of Medians of a Triangle

The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side

### Concurrency of Altitudes of a Triangle

The lines containing the altitudes of a triangle are concurrent

### Larger

If one side of a triangle is longer than another side, then the angle opposite the longer side is _________ than the angle opposite the shorter side.

### Longer

If one angle of a triangle is larger than another angle, then the side opposite the larger angle is ______ than the side opposite the smaller angle.

### Triangle Inequality Theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

### Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second.

### Converse of the Hinge Theorem

If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.

### Cross Products Property

In a proportion, a/b=c/d, where b and d don't equal 0, then ad=bc

x squared=ab

### Perimeters of Similar Polygons

If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding lengths

### Corresponding Lengths in Similar Polygons

If two polygons are similar, then the ration of any two corresponding lengths in the polygon is equal to the scale factor of the similar polygon

### AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar

### SSS Similarity Theorem

If the lengths of the corresponding side lengths of two triangles are proportional then the triangles are similar

### SAS Similarity Theorem

If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar.

### Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally

### Converse of Triangle Proportionality Theorem

If a line divides two sides of a triangle proportionally, then it is parallel to the third side

### Proportionally

If three parallel lines intersect two transversals, then they divide the transversals _____________.

### Proportional

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are ____________ to the lengths of the other two sides.

### Inverse

The statement formed when you negate the hypothesis and conclusion of a conditional statement

### Converse

The statement formed by exchanging the hypothesis and conclusion of a conditional statement

### Contrapositive

The statement formed when you negate the hypothesis and conclusion of the converse of a conditional statement