Chapter 1 – Discovering Points, Lines, Planes, and Angles
Postulate 1.1 (Ruler postulate): The points on any line can be paired with the real
numbers so that, given ant two points P and Q in the line, P corresponds to zero, and Q
corresponds to a positive numbers. (29)
Postulate 1.2 (Segment addition Postulate): If Q is between P and R, then PQ+QR=PR,
then Q is between P and R. (29)
Postulate 1.3 (Protractor Postulate): Given AB and a number r between 0 and 180, there is
exactly one ray with endpoint A, extending on each side of AB, such that the measure of
the angle formed is r. (46)
Postulate 1.4 (Angle Addition Postulate): If R is in the interior of angle PQS, then m
angle PQR+ m angle PQS = m angle PQS. If -----------Theorem 1.1 (Midpoint Theorem): If M is the midpoint of AB, then AM=MB. (39)
Chapter 2 – Connecting Reasoning and Proof
Postulate 2.1: Through any two points there is exactly one line. (79)
Postulate 2.2: Through any three points not on the same line there is exactly one plane.
(79)
Postulate 2.3: A line contains at least two points. (79)
Postulate 2.4: A plane contains at least three points not on the same line. (79)
Postulate 2.5: If two points lie in a plane, then the entire line containing those two points
lies in that plane. (79)
Postulate 2.6: If two planes intersect, then their intersection is a line. (79)
Theorem 2.1: Congruence of segments is reflexive, symmetric, and transitive.
Theorem 2.2: If two angles form a linear pair, then they are supplementary angles.
Theorem 2.3: Congruence of angles is reflexive, symmetric, and transitive.
Theorem 2.4: Angles supplementary to the same angle or to congruent angles are
congruent.
Theorem 2.5: Angles complementary to the same angle or to congruent angles are
congruent. (108)
Theorem 2.6: All right angles are congruent. (109)
Theorem 2.7: Vertical angles are congruent. (109)
Theorem 2.8: Perpendicular lines intersect to form four right angles. (110)
Chapter 3 – Using Perpendicular and Parallel Lines.
Postulate 3.1(Corresponding Angles Postulate): It two parallel lines are cut by a
transversal, then each pair of corresponding angles is congruent. (132)
Theorem 3.1 (Alternate Interior Angles Theorem): If two parallel lines are cut by a
transversal, then each pair of alternate interior angles is congruent. (132)
Theorem 3.2 (Consecutive Interior Angles Theorem): It two parallel lines are cut by a
transversal, then each pair of consecutive interior angles is supplementary. (132)
Theorem 3.3(Alternate Exterior Angles Theorem): If two parallel lines are cut by a
transversal, then each pair of alternate exterior angles is congruent. (132)
Theorem3.4 (Perpendicular Transversal Theorem): In a plane, if a line is perpendicular to
one of two parallel lines, then it is perpendicular to the other.
Postulate 3.2: Two nonvertical lines have the same slope if and only of they are parallel.
Postulate 3.3: Two nonvertical lines are perpendicular if and only if the product of their
slopes is -1. (139)
Postulate 3.4: If two lines in a plane are cut by a transversal so that the corresponding
angles are congruent, then the lines are parallel. (147)
Postulate 3.5(Parallel Postulate): If there is a line and a point not on the line, then there
exists exactly one line through the point that is parallel to the given line. (147)
Theorem 3.5: If two lines in a plane are cut by a transversal so that a pair of alternate
exterior angles is congruent, then the two lines are parallel. (147)
Theorem 3.6: If two lines in a plane are cut by a transversal so that a pair of consecutive
interior angles is supplementary, then the lines are parallel. (147)
Theorem 3.7: If two lines in a plane are cut by a transversal so that a pair of alternate
interior angles is congruent, then the lines are parallel. (147)
Theorem 3.8: In a plane, if two lines are perpendicular to the same line, then they are
parallel. (147)
Chapter 4- Identifying Congruent Triangles
Theorem 4.1(Angle Sum Theorem): The sum of the measures of the angles of a triangle
is 180. (189)
Theorem 4.2(Third Angle Theorem): If two angles of one triangle are congruent to two
angles of a second triangle, then the third angles of the triangles are congruent.
Theorem 4.3(Exterior Angle Theorem): The measure of an exterior angle of a triangle is
equal to the sum of the measures of the two remote interior angles.
Corollary 4.1: The acute angles of a right triangle are complementary.
Corollary 4.2: There can be at most on right or obtuse angle in a triangle.
Theorem 4.4: Congruence of triangles is reflexive, symmetric, and transitive.
Postulate 4.1(SSS Postulate): If the sides of one triangle are congruent to the sides of a
second triangle, then the triangles are congruent.
Postulate 4.2(SAS Postulate): If two sides and the included angle of one triangle are
congruent to two angles and the included angle of another triangle, then the triangles are
congruent.
Postulate 4.3(ASA Postulate): If two angles and the included side of one triangle are
congruent to two angles and the included side of another triangle, the triangles are
congruent.
Theorem 4.5(AAS): If two angles and a noninsured side of one triangle are congruent to
the corresponding two angles and side of a second triangle, the two triangles are
congruent.
Theorem 4.6 (Isosceles Triangle Theorem): If two sides of a triangle are congruent, then
the angles opposite those sides are congruent.
Theorem 4.7: If two angles of a triangle are congruent, then the sides opposite those
angles are congruent.
Corollary 4.3: A triangle is equilateral if and only if it is equiangular.
Corollary 4.4: Each angle of an equilateral triangle measures 60 degree.
Chapter 5-Applying Congruent Triangles
Theorem 5.1: A point on the perpendicular bisector of a segment is equidistant from the
endpoints of the segment.
Theorem 5.2: A point equidistant from the endpoints of a segment lies on the
perpendicular bisector of the segment.
Theorem 5.3: A point on the bisector of an angle is equidistant from the sides of the
angle.
Theorem 5.4: A point on or in the interior of an angle and equidistant from the sides of an
angle lies on the bisector of the angle.
Theorem 5.5(LL): If the legs of one right triangle are congruent to the corresponding legs
of another right triangle, then the triangles are congruent.
Theorem 5.6(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 two triangles are congruent.
Theorem 5.7(LA): If one leg and an acute angle of one right triangle are congruent to the
corresponding leg and acute angle of another right triangle, then the triangles are
congruent.
Postulate 5.1(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 triangles are
congruent.
Theorem 5.8(Exterior angle Inequality Theorem): If an angle is an exterior angle of a
triangle, then its measure is greater than the measure of either of its corresponding remote
interior angles.
Theorem 5.9: If one side of a triangle is longer than another side, then the angle opposite
the longer side has a greater measure than the angle opposite the shorter side.
Theorem 5.10: If one angle of a triangle has a greater measure than another angle, then
the side opposite the greater angle is longer than the side opposite the lesser angle.
Theorem 5.11: The perpendicular segment from a point to a line is the shortest segment
from the point to the line.
Corollary 5.1: The perpendicular segment from a point to a plane is the shortest segment
from the point to the plane.
Theorem 5.12(Triangle Inequality Theorem): The sum of the lengths of any two sides of
a triangle is greater than the length of the third side.
Theorem 5.13(SAS Inequality-Hinge Theorem): If two sides of one triangle are
congruent to two sides of another triangle, and the included angle in one triangle is
greater than the included angle in the other, then the third side of the first triangle is
longer than the third side in the second triangle.
Theorem5.14 (SSS Inequality): If two sides of one triangle are congruent to two sides of
another triangle and the third side in one triangle is longer than the third side in the other,
them the angle between the pair of congruent sides in the first triangle is greater than the
corresponding angle in the second triangle.
Chapter 6-Exploring Quadrilaterals
Theorem6.1: Opposite sides of a parallelogram are congruent.
Theorem6.2: Opposite angles of a parallelogram are congruent.
Theorem6.3: Consecutive angles in a parallelogram are supplementary.
Theorem6.4: The diagonals of a parallelogram bisect each other.
Theorem6.5: If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem6.6: If both pairs of opposite angles of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem6.7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is
a parallelogram.
Theorem6.8: If one pair of opposite sides of a quadrilateral are both parallel and
congruent, then the quadrilateral is a parallelogram.
Theorem6.9: If a parallelogram is a rectangle, then its diagonals are congruent.
Theorem6.10: If the diagonals of a parallelogram are congruent, then the parallelogram is
a rectangle.
Theorem6.11: The diagonals of a rhombus are perpendicular.
Theorem6.12: If the diagonals of a parallelogram are perpendicular, then the
parallelogram is a rhombus.
Theorem6.13: Each diagonal of a rhombus bisects a pair of opposite angles.
Theorem6.14: Both pairs of base angles of an isosceles trapezoid are congruent.
Therorem6.15: The diagonals of an isosceles trapezoid are congruent.
Theorem6.16: The median of a trapezoid is parallel to the bases, and its measure is onehalf the sum of the measures of the bases.
Chapter7-Connecting proportion and Similarity
Postulate7.1 (AA Similarity): If two angles of one triangle are congruent to two angles of
another triangle, then the triangles are similar.
Theorem7.1 (SSS Similarity): If the measures of the corresponding sides of two triangles
are proportional, then the triangles are similar.
Theorem7.2 (SAS Similarity): If the measures of two sides of a triangle are proportional
to the measures of two corresponding sides of another triangle and the included angles
are congruent, then the triangles are similar.
Theorem7.3: Similarity of triangles is reflexive, symmetric, and transitive.
Theorem7.4 (Triangle Proportionality): If a line is parallel to one side of a triangle and
intersects the other two sides in two distinct points, then it separates these sides into
segments of proportional lengths.
Theorem7.5: If a line intersects two sides of a triangle and separates the sides into
corresponding segments of proportional lengths, then the line is parallel to the third side.
Theorem7.6: A segment whose endpoints are the midpoints of two sides of a triangle is
parallel to the third side of the triangle and its length is one-half the length of the third
side.
Corollary7.1: Is three or more parallel lines intersect two transversals, then they cut off
the transversals proportionally.
Corollary7.2: If three or more parallel lines cut off congruent segments on one
transversal, then they cut off congruent segments on every transversal.
Theorem7.7 (Proportional Perimeters): It two triangles are similar, then the perimeters
are proportional to the measures of corresponding sides.
Theorem7.8: If two triangles are similar, then the measures of the corresponding altitudes
are proportional to the measures of the corresponding sides.
Theorem7.9: If two triangles are similar, then the measures of the corresponding angle
bisectors are proportional to the measure of the corresponding sides.
Theorem7.10: If two triangles are similar, then the measures of the corresponding
medians are proportional to the measures of the corresponding sides.
Theorem7.11 (Angle Bisector Theorem): An angle bisector in a triangle separates the
opposite side into segments that have the same ratio as the other two sides.
Chapter8-Applying Right Triangles and Trigonometry
Theorem8.1: If the altitude is drawn from the vertex of the right angle of a right triangle
to its hypotenuse, then the two triangles formed are similar to the given triangle and to
each other.
Theorem8.2: The measure of the altitude drawn from the vertex of the right angle of a
right triangle to its hypotenuse is the geometric mean between the measures of the two
segments of the hypotenuse.
Theorem8.3: If the altitude is drawn to the hypotenuse of a right triangle, then the
measure of a leg of the triangle is the geometric mean between the measures of the
hypotenuse and the segment of the hypotenuse adjacent to the leg.
Theorem8.4 (Pythagorean Theorem): In a right triangle, the sum of the squares of the
measures of the legs equals the square of the measure of the hypotenuse.
Theorem8.5 (Converse of the Pythagorean): If the sum of the squares of the measures of
two sides of a triangle equals the square of the measure of the longest side, then the
triangle is a right triangle.
Theorem8.6: In a 45-45-90 degree triangle, the hypotenuse is root 2 times as long as a
leg.
Theorem8.7: In a 30-60-90 degree triangle, the hypotenuse is twice as long as the shorter
leg, and the longer leg is root 3 times as long as the shorter leg.