Download defintions and theorems 1 - The Bronx High School of Science

Bronx High School of Science
M$4
Mathematics Department
Ms. Abbott
Unit 1: Congruent Triangles
DEFINITIONS
 A line segment is a part of a line consisting of two points, called endpoints, and
all points between them.
 A ray is part of a line consisting of one endpoint and all points of the line on one
side of the endpoint.
 An angle is the union of 2 rays with a common endpoint.
 An acute/right/obtuse/straight angle is an angle whose measure is <90 and
>0/=90/>90 and <180/=180.
 Two line/line segments/rays are perpendicular is they intersect to form a right
angle.
 Two angles are congruent if they have the same measure.
 Two line segments are congruent if they have the same length.
 Two angles are complementary/supplementary if the sum of their measures is
90/180.
 A point B is between points A and C if A, B, and C are collinear, and
AB+BC=AC.
 If B is between A and C then ray BA and ray BC are opposite rays.
 Vertical angles are two angles in which the sides of one are opposite rays to the
sides of the second.
 A scalene/isosceles/equilateral/equiangular/acute/right/obtuse triangle is a
triangle with no congruent sides/2 congruent sides/3 congruent sides/3 congruent
angles/3 acute angles/one right angle/one obtuse angle.
 A point M is the midpoint of CD is M is between C and D and MC=MD.
 A line/ray/segment/plane that intersects a segment at its midpoint is a bisector of
that segment.
 An angle bisector divides an angle into two congruent angles.
 A median of a triangle is a segment drawn from any vertex of the triangle to the
midpoint of the opposite segment.
 An altitude of a triangle is a segment drawn from any vertex of the triangle
perpendicular to the opposite side.
 A linear/orthogonal pair is two adjacent angles whose exterior sides are
opposite/perpendicular rays.
 [Definition of congruent triangles] Corresponding parts of congruent triangles
are congruent.
POSTULATES
 Reflexive Postulate: Every line segment/angle is congruent to itself.
 Symmetric Postulate: If x=y, then y=x. (Also true for congruent)
 Transitive Postulate: If x=y and y=z, then x=z. (Also true for congruent, <, >,
,  )
 A line segment has one and only one midpoint.
Bronx High School of Science
M$4
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Mathematics Department
Ms. Abbott
An angle has one and only one bisector.
Substitution Postulate: A quantity may be substituted for its equal in any
expression.
Addition Postulate: If equal quantities are added to equal quantities, the sums are
equal.
Subtraction Postulate: If equal quantities are subtracted from equal quantities, the
differences are equal.
Multiplication Postulate: If equal quantities are multiplied by equal quantities, the
products are equal.
o Doubles Postulate: Doubles of equal quantities are equal.
Division Postulate: If equal quantities are divided by equal quantities, the
quotients are equal.
o Halves Postulate: Halves of equal quantities are equal.
SSS Postulate: If three sides of one triangle are congruent to three sides of another
triangle, then the triangles are congruent.
SAS Postulate: If two sides and the included angle of one triangle are congruent
to two sides and the included angle of another triangle, then the triangles are
congruent.
ASA Postulate: If two angles and the included side of one triangle are congruent
to two angles and the included side of another triangle, then the triangles are
congruent.
THEOREMS and COROLLARIES
 If two angles are congruent, their supplements/complements are congruent.
 If two angles are supplementary/complementary to the same angle, they are
congruent.
 If two angles are vertical angles, then they are congruent.
 If two angles form a linear pair, they are supplementary.
 If two angles form an orthogonal pair, they are complementary.
 All right angles are congruent.
 If two sides of a triangle are congruent, then the angles opposite them are
congruent. (The base angles of an isosceles triangle are congruent)
 The bisector of the vertex angle of an isosceles triangle bisects the base.
 The bisector of the vertex angle of an isosceles triangle is perpendicular to the
base.
 All equilateral triangles are equiangular.
 If two angles of a triangle are congruent, then the sides opposite them are
congruent.
 All equiangular triangles are equilateral.