# Chapter 4: Congruence of Line Segments, Angles, and Triangles

• CHAPTER 4 POSTULATES, THEOREMS, and DEFINITIONS
A line segment can be extended to any length in either direction.

Through two given points, one and only one line can be drawn.   *Two points determine a line.

Two lines cannot intersect in more than one point.

One and only one circle can be drawn with any given point as center and the length of any given line segment as a radius.

At a given point on a given line, one and only one perpendicular can be drawn to the line.

From a given point not on a given line, one and only one perpendicular can be drawn to the line.

From any two distinct points, there is only one positive real number that is the length of the line segment joining the two points.

The shortest distance between two points is the length of the line segment joining these two points.

A line segment has one and only one midpoint.

An angle has one and only one bisector.

Thm:  If two angles are right angles, then they are congruent.

Def’n:  Adjacent angles are two angles in the same plane that have a common vertex and a common side but do not have any interior points in common.

Def’n: Complementary angles are two angles, the sum of whose degree measures is 90.

Def’n: Supplementary angles are two angles, the sum of whose degree measures is 180.

Thm: If two angles are complements of the same angle, then they are congruent.

Thm: If two angles are congruent, then their complements are congruent.

Thm: If two angles are supplements of the same angle, then they are congruent.

Thm: If two angles are congruent, then their supplements are congruent.

Def’n:  A linear pair of angles are two adjacent angles whose sum is a straight angle.

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

Thm:  If two lines intersect to form congruent adjacent angles, then they are perpendicular.

Def’n: Vertical angles are two angles in which the sides of one angle are opposite rays to the sides of the second angle.

Thm: If two lines intersect, then the vertical angles are congruent.

Def’n: Two polygons are congruent if and only if there is a one-to-one correspondence between their vertices such that corresponding angles are congruent and corresponding sides are congruent.   “Corresponding parts of congruent polygons are congruent.”

Any geometric figure is congruent to itself.  (Reflexive Property)

A congruence may be expressed in either order.  (Symmetric Property)

Two geometric figures congruent to the same geometric figure are congruent to each other.  (Transitive Property)

Two triangles are congruent if two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of the other.  (SAS)

Two triangles are congruent if two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of the other. (ASA)

Two triangles are congruent if the three sides of one triangle are congruent, respectively, to the three sides of the other.  (SSS)