Honors Geo Chap 5 Postulates, Theorems, & Definitions

Definition of a Bisector

A point, line or line segment that divides a segment or angle into two congruent pieces

angle bisector of a triangle

A segment that bisects an angle of the triangle and has one endpoint at a vertex of the triangle and the other endpoint at another point on the triangle.

congruence statement

A statement that indicates that two triangles or polygons are congruent by listing the vertices in the order of correspondence.

Definition of an equilateral triangle

A triangle with three congruent sides. An equilateral triangle is also an isosceles triangle

Isosceles Triangle Theorem

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

included angle (of a triangle)

the angle between two sides of a triangle. Angle Q is the included angle of sides PQ and SQ

converse

the statement formed by exchanging the hypothesis and conclusion of a conditional statement (if-then statement)

Definition of Linear Pair

two adjacent angles that form a line

Definition of Perpendicular

two lines, segments, or rays are perpendicular if they intersect to form right angles

Definition of an Isosceles Triangle

a triangle with at least two congruent sides. The two angles that include the base are the base angles. The angle opposite the base is the vertex angle.

scalene triangle

a triangle with no congruent sides

Supplements Theorem

if <A and <B are supplementary and <A and <C are supplementary, then <B≅<C

corollary

a theorem which is an immediate consequence of another theorem

Right Angle Theorem

All right angles are congruent

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

Angle Bisector Theorem

Every angle has one and only one bisector. (a triangle has 3 angles so a triangle contains 3 angle bisectors)

Equilateral Triangle Corollary

If a triangle is equilateral, then it is equiangular. If a triangle is equiangular, then it is equilateral

Theorem 4-2 (write the actual theorem out in a proof, don't use the number)

If the angles in a linear pair are congruent, then each of them is a right angle.

Theorem 4-3 (write the actual theorem out in a proof, don't use the number)

If two angles are both congruent and supplementary, then each is a right angle.

Supplements Postulate

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

Converse of the Isosceles Triangle Theorem

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

Congruence Postulates for Triangles

SSS, SAS, ASA, AAS can all be used to prove triangles congruent. (AAA & SSA do not work to prove triangles congruent)

Definition of Midpoint

The midpoint M of line PQ is the point between P and Q (P-M-Q) such that PM = MQ. It divides the segment into two equal pieces

included side (of a triangle)

The segment of a triangle whose endpoints are the vertices of two of the angles of the triangle. XY is the included side of angle X and angle Y.

Vertical Angles Theorem

Vertical angles are congruent

quadrilateral

a four-sided polygon

Equiangular

a polygon or triangle in which all of its angles are congruent

rectangle

a quadrilateral with four right angles

square

a quadrilateral with four right angles and four congruent sides

Definition of an angle bisector

a ray that goes through a point in the interior of an angle and divides the angle into two angles that are congruent

equivalence relation

a relation that is reflexive, symmetric, and transitive. Congruence for segments and Congruence for triangles are both equivalence relations

median of a triangle

a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side

conditional statement

a statement that can be written in if-then form