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
