Download MAT 122 Postulates, Theorems, and Corollaries

MAT 122
Postulates, Theorems, and Corollaries
Postulate 1: Through two distinct points, there is exactly one line.
Postulate 2: (Ruler Postulate) The measure of any line segment is a unique positive
number.
Postulate 3: (Segment-Addition Posutlate) If X is a point of AB and A − X − B,
then AX + XB = AB.
Postulate 4: If two lines intersect, they intersect at a point.
Postulate 5: Through three noncollinear points, there is exactly one plane.
Postulate 6: If two distinct planes intersect, then their intersection is a line.
Postulate 7: Given two distinct points in a plane, the line containing these points also lies
in the plane.
Theorem 1.3.1: The midpoint of a line segment is unique.
Postulate 8: (Protractor Postulate) The measure of an angle is a unique positive number.
Postulate 9: (Angle-Addition Postulate) If a point D lies in the interior of ∠ABC,
then m∠ABD + m∠DBC = m∠ABC.
Theorem 1.4.1: There is one and only one angle bisector for a given angle.
Theorem 1.6.1: If two lines are perpendicular, then they meet to form right angles.
Theorem 1.6.2: If two lines intersect, then the vertical angles formed are congruent.
Theorem 1.6.3: In a plane, there is exactly one line perpendicular to a given line at any
point on the line.
Theorem 1.6.4: The perpendicular bisector of a line segment is unique.
Theorem 1.7.1: If two lines meet to form a right angle, then these lines are perpendicular.
Theorem 1.7.2: If two angle are complementary to the same angle (or to congruent angles),
then these angles are congruent.
Theorem 1.7.3: If two angle are supplementary to the same angle (or to congruent angles),
then these angles are congruent.
Theorem 1.7.4: Any two right angles are congruent.
Theorem 1.7.5: If the exterior sides of two adjacent acute angle form perpendicular rays,
then these angles are complementary.
Theorem 1.7.6: If the exterior sides of two adjacent angles form a straight line, then these
angles are supplementary.
Theorem 1.7.7: If two line segments are congruent, then their midpoints separate these
segments into four congruent segments.
Theorem 1.7.8: If two angles are congruent, then their bisectors separate these angles into
four congruent angles.
Theorem 2.1.1: From a point not on a given line, there is exactly one line perpendicular
to the given line.
Postulate 10: (Parallel Postulate) Through a point not on a line, exacly one line is
parallel to the given line.
Postulate 11: (Corresponding Angles) If two parallel lines are cut by a transversal,
then the corresponding angles are congruent.
Theorem 2.1.2: (Alternate Interior Angles) If two parallel lines are cut by a transversal,
then the alternate interior angles are congruent.
Theorem 2.1.3: (Alternate Exterior Angles) If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.
Theorem 2.1.4: (Same-Side Interior Angles) If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary.
Theorem 2.1.5: (Same-Side Exterior Angles) If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary.
Theorem 2.3.1: (Corresponding Angles ∼
=) If two lines are cut by a transversal so that
the corresponding angles are congruent, then these lines are parallel.
Theorem 2.3.2: (Alternate Interior Angles ∼
=) If two lines are cut by a transversal so
that the alternate interior angles are congruent, then these lines are parallel.
Theorem 2.3.3: (Alternate Exterior Angles ∼
=) If two lines are cut by a transversal so
that the alternate exterior angles are congruent, then these lines are parallel.
Theorem 2.3.4: (Same-Side Interior Angles Supplementary) If two lines are cut by a
transversal so that the interior angles on the same side of the transversal are supplementary,
then these lines are parallel.
Theorem 2.3.5: (Same-Side Exterior Angles Supplemantary) If two lines are cut by a
transversal so that the exterior angles on the same side of the transversal are supplementary,
then these lines are parallel.
Theorem 2.3.6: If two lines are each parallel to a third line, then these lines are parallel
to each other.
Theorem 2.3.7: If two coplanar lines are each perpendicular to a third line, then these
lines are parallel to each other.
Theorem 2.4.1: In a triangle, the sum of the measure of the interior angles is 180◦ .
Corollary 2.4.2: Each angle of an equiangular triangle measures 60◦ .
Corollary 2.4.3 The acute angles of a right triangle are complementary.
Corollary 2.4.4: If two angles of one triangle are congruent to two angles of another
triangle, then the third angles are also congruent.
Corollary 2.4.5: The measure of an exterior angle of a triangle equals the sum of the
measures of the two nonadjacent interior angles.
Theorem 2.5.1: The total number of diagonals D in a polygon of n sides is given by the
n(n − 3)
formula D =
.
2
Theorem 2.5.2: The sum S of the measure of the interior angles of a polygon with n sides
is given by S = (n − 2) · 180◦ . Note that n > 2 for any polygon.
Corollary 2.5.3: The measure I of each interior angle of a regular polygon or equiangular
(n − 2) · 180◦
polygon of n sides is I =
.
n
Corollary 2.5.4: The sum of the four interior angles of a quadrilateral is 360◦ .
Corollary 2.4.5: The sum of the measure of the exterior angles of a polygon, one at each
vertex, is 360◦ .
Corollary 2.5.6: The measure E of each exterior angle of a regular polygon or equiangular
360◦
.
polygon of n sides is E =
n