POSTULATES, THEOREMS, AND COROLLARIES FROM
GEOMETRY
Postulate 1 For any two points, there is exactly one line containing them.
Theorem 1 Two lines intersect in at most one point.
Postulate 2 Three noncollinear points are contained in exactly one plane.
Postulate 3 If two points of a line are in a given plane, then the line is in the plane.
Postulate 4 If two planes intersect, then they intersect in exactly one line.
Postulate 5 Space is determined by at least four points not all in the same plane.
Theorem 2 A line and a point not on the line are contained in exactly one plane.
Theorem 3 If
a line intersects a plane, but is not contained in the plane, then the
intersection
is exactly one point.
Theorem 4 Two intersecting lines are contained in exactly one plane.
Postulate 6 On every
line, there is a segment with a given point as an endpoint
congruent
to any given segment.
Postulate 7 For every
ray, there is an angle with the given ray as a side congruent to
any
given angle.
Postulate 8 Every segment has exactly one midpoint.
Postulate 9 Every angle, except a straight angle, has exactly one bisector.
Postulate 10 Angle Addition Postulate: If D is in the interior of / ABC, then
m / ABC = m / ABD + m / DBC.
Postulate 11 The sum of
the measures of the angles with the same vertex on one side
of
a line and with no interior points in common is 180 degrees.
The
sum of the measures of all angles around a common vertex and with
no
interior points in common is 360 degrees.
Theorem 5 All right angles are congruent.
Theorem 6 Two perpendicular lines form four congruent right angles.
Postulate 12 For a
given point and a line in a plane, there is exactly one line through
the
point that is perpendicular to the given line.
Theorem 7 If
two lines form congruent adjacent angles, then the lines are
perpendicular.
Theorem 8 Supplements of congruent angles are congruent.
Corollary Supplements of the same angle are congruent.
Theorem 9 Complements of congruent angles are congruent.
Corollary Complements of the same angle are congruent.
Theorem 10 Vertical angles are congruent.
Postulate 13 Alternate
Interior Angle Postulate: If two parallel lines are intersected
by
a transversal, then the alternate interior angles are congruent.
Theorem 11 If
two parallel lines are intersected by a transversal, then the
corresponding angles
are congruent.
Theorem 12 If two parallel lines are intersected by a transversal, then the consecutive
interior angles are supplementary.
Postulate 14 If two
lines are intersected by a transversal so that the alternate interior
angles are
congruent, then the lines are parallel.
Theorem 13 If
two lines are intersected by a transversal so that the corresponding
angles
are congruent, then the lines are parallel.
Corollary In a plane, two lines perpendicular to the same line are parallel.
Theorem 14 If
two lines are intersected by a transversal so that the consecutive
interior
angles are supplementary, then the lines are parallel.
Postulate 15 The Parallel
Postulate: Through a point not on a line, there is exactly
one
line parallel to the given line.
Theorem 15 The sum of the measures of the angles of a triangle is 180 degrees.
Theorem 16 If
two angles of one triangle are congruent to two angles of a second
triangle then
the third angles are congruent.
Theorem 17 In a plane, two lines parallel to the same line are parallel to each other.
Theorem 18 The
measure of an exterior angle of a triangle is equal to the sum of the
measures
of its two remote interior angles.
Corollary The
measure of an exterior angle of a triangle is greater than either
remote
interior angle.
Postulate 16 SSS:
If the three sides of one triangle are congruent to the
corresponding three
sides of another triangle, then the triangles are
congruent.
Postulate 17 SAS:
If two sides and the included angle of one triangle are congruent
to the
corresponding sides and angle in another triangle, then the
triangles
are congruent.
Postulate 18 ASA: If
two angles and the included side of one triangle are congruent
to the
corresponding angles and sides in another triangle, then the
triangles
are congruent.
Theorem 19 AAS: In
a triangle, if two angles and a side opposite one of them are
congruent to
the corresponding angles and side in another triangle, then
the
triangles are congruent.
Theorem 20 If two sides of a triangle are congruent, then the angles opposite them are
Corollary 1 An equilateral triangle is also equiangular.
Corollary 2 The measure of each angle in an equilateral triangle is 60 degrees.
Theorem 21 If
two angles of a triangle are congruent, then the sides opposite those
angles are
congruent.
Corollary An equiangular triangle is also equilateral.
Theorem 22 LL: If
the two legs of a right triangle are congruent to two legs of
another right
triangle, then the right triangles are congruent.
Theorem 23 LA: If a leg and an acute angle of one right triangle are congruent to the
corresponding
leg and angle of another right triangle, then the triangles
are congruent.
Theorem 24 HA: If
the hypotenuse and an acute angle of one right triangle are
congruent to
the hypotenuse and corresponding acute angle of another
right
triangle then the triangles are congruent.
Theorem 25 HL: If
the hypotenuse and a leg of one right triangle are congruent to
the hypotenuse
and corresponding leg of another right triangle, then the
triangle are
congruent.
Theorem 26 Perpendicular
Bisector Theorem: Any point on a perpendicular bisector
of
a segment is equidistant from the endpoints of the segment.
Theorem 27 A line containing
two points that are each equidistant from the endpoints
of
a segment is the perpendicular bisector of the segment.
Theorem 28 Corresponding medians of congruent triangles are congruent.
Theorem 29 Corresponding altitudes of congruent triangles are congruent.
Theorem 30 The
sum of the measures of the angles of a convex quadrilateral is 360
degrees.
Theorem 31 The sum of the measures of the angles of a convex polygon with n sides is
(n - 2)180 degrees.
Corollary The
measure of an angle of a regular polygon with n sides is (n - 2)180 .
n
Theorem 32 A diagonal of a parallelogram forms two congruent triangles.
Corollary Opposite sides of a parallelogram are congruent.
Corollary 2 Opposite angles of a parallelogram are congruent.
Theorem 33 The diagonals of a parallelogram bisect each other.
Theorem 34 If both pairs of opposite sides of a quadrilateral are congruent, then
the quadrilateral is a parallelogram.
Theorem 35 If both pairs of opposite angles of a quadrilateral are congruent, then
the quadrilateral is a parallelogram.
Theorem 36 If two
sides of a quadrilateral are parallel and congruent, then the
quadrilateral is
a parallelogram.
Theorem 37 If the
diagonals of a quadrilateral bisect each other, then the
quadrilateral
is a parallelogram.
Theorem 38 The segment
joining the midpoints of two sides of a triangle is parallel to
the third
side, and its length is half the third side.
Theorem 39 If three
parallel lines cut off congruent segments on one transversal, then
they cut
off congruent segments on every transversal.
Postulate 19 The shortest path between any two points is a segment.
Theorem 40 Parallel lines are equidistant at all points.
Postulate 21 A line
of symmetry of a symmetric polygon is the perpendicular bisector
of
any segment joining a pair of corresponding points of the polygon.
Postulate 22 AA: If
two angles in a triangle are congruent to the two corresponding
angles in
another triangle, then the triangles are similar.
Theorem 41 If a line
is parallel to a side of a triangle, and it intersects the other two
sides, then
it divides the two sides proportionally.
Theorem 42 If a line
divides two sides of a triangle proportionally, then it is parallel to
the third
side.
Theorem 43 SAS Similarity: If
an angle of one triangle is congruent to an angle of
another triangle,
and the lengths of the corresponding sides including
these
angles are proportional, then the triangles are similar.
Theorem 44 SSS Similarity: If
the sides of a triangle are proportional to the
corresponding sides
of another triangle, then the triangles are similar.
Theorem 45 Corresponding
medians of similar triangles are proportional to
corresponding sides.
Theorem 46 Corresponding
altitudes of similar triangles are proportional to
corresponding sides.
Theorem 47 The bisector
of an angle of a triangle divides the opposite side into
segments proportional
to the other sides of the triangle.
Theorem 48 In a right
triangle, the altitude to the hypotenuse forms two triangles that
are
similar to the original triangle, and that are similar to each other.
Corollary 1 In
a right triangle, the length of the altitude to the hypotenuse is the
geometric mean
of the lengths of the segments of the hypotenuse that are
formed.
Corollary 2 If
the altitude is drawn to the hypotenuse of a right triangle, then the
length of
either leg is the geometric mean of the length of the
hypotenuse,
and the length of the segment of the hypotenuse adjacent to
that
leg.
Theorem 49 Pythagorean
Theorem: In a right triangle, the sum of the squares of the
lengths of
the two legs equals the sum of the square of the hypotenuse.
Theorem 50 If the
sum of the squares of the lengths of two sides of a triangle equals
the square
of the third side, then the triangle is a right triangle.
Theorem 51 In a 45
- 45 (isosceles) right triangle, the length of the hypotenuse equals
__
the
length of a leg times 
2.
Theorem 52 In a 30
- 60 - 90 right triangle, the hypotenuse is twice as long as the leg
opposite the
30 degree angle, and the leg opposite the 60 degree angle is
__
3
times as long as the leg opposite the 30 degree angle.