Postulates and Theorems A85
Postulates
Postulates and Theorems
1.1 Ruler Postulate
The points on a line can be matched one to one with
the real numbers. The real number that corresponds to a
point is the coordinate of the point. The distance between
points A and B, written as AB, is the absolute value of the
difference of the coordinates of A and B.
1.2 Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
If AB + BC = AC, then B is between A and C.
1.3 Protractor Postulate
Consider
OB and a point A on one side of
OB . The rays
of the form
OA can be matched one to one with the real
numbers from 0 to 180. The measure of ∠AOB, which can
be written as m∠AOB, is equal to the absolute value of the
difference between the real numbers matched with
OA and
OB on a protractor.
1.4 Angle Addition Postulate
If P is in the interior of ∠RST, then the measure of ∠RST is
equal to the sum of the measures of ∠RSP and ∠PST.
2.1 Two Point Postulate
Through any two points, there exists exactly one line.
2.2 Line-Point Postulate
A line contains at least two points.
2.3 Line Intersection Postulate
If two lines intersect, then their intersection is exactly one
point.
2.4 Three Point Postulate
Through any three noncollinear points, there exists exactly
one plane.
2.5 Plane-Point Postulate
A plane contains at least three noncollinear points.
2.6 Plane-Line Postulate
If two points lie in a plane, then the line containing them
lies in the plane.
2.7 Plane Intersection Postulate
If two planes intersect, then their intersection is a line.
2.8 Linear Pair Postulate
If two angles form a linear pair, then they are
supplementary.
3.1 Parallel Postulate
If there is a line and a point not on the line, then there is
exactly one line through the point parallel to the given line.
3.2 Perpendicular Postulate
If there is a line and a point not on the line, then there is
exactly one line through the point perpendicular to the
given line.
4.1 Translation Postulate
A translation is a rigid motion.
4.2 Re ection Postulate
A re ection is a rigid motion.
4.3 Rotation Postulate
A rotation is a rigid motion.
10.1 Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the
sum of the measures of the two arcs.
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A86 Postulates and Theorems
Theorems
2.1 Properties of Segment Congruence
Segment congruence is re exive, symmetric, and transitive.
Re exive For any segment AB,
—
AB ≅
—
AB .
Symmetric If
—
AB ≅
—
CD , then
—
CD ≅
—
AB .
Transitive If
—
AB ≅
—
CD and
—
CD ≅
—
EF , then
—
AB ≅
—
EF .
2.2 Properties of Angle Congruence
Angle congruence is re exive, symmetric, and transitive.
Re exive For any angle A, ∠A ≅ ∠A.
Symmetric If ∠A ≅ ∠B, then ∠B ≅ ∠A.
Transitive If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.
2.3 Right Angles Congruence Theorem
All right angles are congruent.
2.4 Congruent Supplements Theorem
If two angles are supplementary to the same angle (or to
congruent angles), then they are congruent.
2.5 Congruent Complements Theorem
If two angles are complementary to the same angle (or to
congruent angles), then they are congruent.
2.6 Vertical Angles Congruence Theorem
Vertical angles are congruent.
3.1 Corresponding Angles Theorem
If two parallel lines are cut by a transversal, then the pairs
of corresponding angles are congruent.
3.2 Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs
of alternate interior angles are congruent.
3.3 Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs
of alternate exterior angles are congruent.
3.4 Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs
of consecutive interior angles are supplementary.
3.5 Corresponding Angles Converse
If two lines are cut by a transversal so the corresponding
angles are congruent, then the lines are parallel.
3.6 Alternate Interior Angles Converse
If two lines are cut by a transversal so the alternate interior
angles are congruent, then the lines are parallel.
3.7 Alternate Exterior Angles Converse
If two lines are cut by a transversal so the alternate exterior
angles are congruent, then the lines are parallel.
3.8 Consecutive Interior Angles Converse
If two lines are cut by a transversal so the consecutive
interior angles are supplementary, then the lines are
parallel.
3.9 Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they are
parallel to each other.
3.10 Linear Pair Perpendicular Theorem
If two lines intersect to form a linear pair of congruent
angles, then the lines are perpendicular.
3.11 Perpendicular Transversal Theorem
In a plane, if a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to the other line.
3.12 Lines Perpendicular to a Transversal
Theorem
In a plane, if two lines are perpendicular to the same line,
then they are parallel to each other.
3.13 Slopes of Parallel Lines
In a coordinate plane, two nonvertical lines are parallel if
and only if they have the same slope. Any two vertical lines
are parallel.
3.14 Slopes of Perpendicular Lines
In a coordinate plane, two nonvertical lines are
perpendicular if and only if the product of their slopes is
−1. Horizontal lines are perpendicular to vertical lines.
4.1 Composition Theorem
The composition of two (or more) rigid motions is a rigid
motion.
4.2 Re ections in Parallel Lines Theorem
If lines k and m are parallel, then a re ection in line k
followed by a re ection in line m is the same as a
translation. If A is the image of A, then
1.
—
AA is perpendicular to k and m, and
2. AA = 2d, where d is the distance between k and m.
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Postulates and Theorems A87
Postulates and Theorems
4.3 Re ections in Intersecting Lines Theorem
If lines k and m intersect at point P, then a re ection in
line k followed by a re ection in line m is the same as a
rotation about point P. The angle of rotation is 2x°, where
x° is the measure of the acute or right angle formed by
lines k and m.
5.1 Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle
is 180°.
5.2 Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to
the sum of the measures of the two nonadjacent interior
angles
Corollary 5.1 Corollary to the Triangle Sum
Theorem
The acute angles of a right triangle are complementary.
5.3 Properties of Triangle Congruence
Triangle congruence is re exive, symmetric, and transitive.
Re exive For any triangle ABC, ABC ≅ ABC.
Symmetric If ABC ≅ DEF, then DEF ≅ ABC.
Transitive If ABC ≅ DEF and DEF ≅ JKL, then
ABC ≅ JKL.
5.4 Third Angles Theorem
If two angles of one triangle are congruent to two angles of
another triangle, then the third angles are also congruent.
5.5 Side-Angle-Side (SAS) Congruence
Theorem
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of a second
triangle, then the two triangles are congruent.
5.6 Base Angles Theorem
If two sides of a triangle are congruent, then the angles
opposite them are congruent.
5.7 Converse of the Base Angles Theorem
If two angles of a triangle are congruent, then the sides
opposite them are congruent.
Corollary 5.2 Corollary to the Base Angles
Theorem
If a triangle is equilateral, then it is equiangular.
Corollary 5.3 Corollary to the Converse of the
Base Angles Theorem
If a triangle is equiangular, then it is equilateral.
5.8 Side-Side-Side (SSS) Congruence Theorem
If three sides of one triangle are congruent to three sides of
a second triangle, then the two triangles are congruent.
5.9 Hypotenuse-Leg (HL) Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent
to the hypotenuse and a leg of a second right triangle, then
the two triangles are congruent.
5.10 Angle-Side-Angle (ASA) Congruence
Theorem
If two angles and the included side of one triangle are
congruent to two angles and the included side of a second
triangle, then the two triangles are congruent.
5.11 Angle-Angle-Side (AAS) Congruence
Theorem
If two angles and a non-included side of one triangle
are congruent to two angles and the corresponding
non-included side of a second triangle, then the two
triangles are congruent.
6.1 Perpendicular Bisector Theorem
In a plane, if a point lies on the perpendicular bisector of
a segment, then it is equidistant from the endpoints of the
segment.
6.2 Converse of the Perpendicular Bisector
Theorem
In a plane, if a point is equidistant from the endpoints of
a segment, then it lies on the perpendicular bisector of the
segment.
6.3 Angle Bisector Theorem
If a point lies on the bisector of an angle, then it is
equidistant from the two sides of the angle.
6.4 Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is equidistant
from the two sides of the angle, then it lies on the bisector
of the angle.
6.5 Circumcenter Theorem
The circumcenter of a triangle is equidistant from the
vertices of the triangle.
6.6 Incenter Theorem
The incenter of a triangle is equidistant from the sides of
the triangle.
6.7 Centroid Theorem
The centroid of a triangle is two-thirds of the distance from
each vertex to the midpoint of the opposite side.
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A88 Postulates and Theorems
6.8 Triangle Midsegment Theorem
The segment connecting the midpoints of two sides of a
triangle is parallel to the third side and is half as long as
that side.
6.9 Triangle Longer Side Theorem
If one side of a triangle is longer than another side, then
the angle opposite the longer side is larger than the angle
opposite the shorter side.
6.10 Triangle Larger Angle Theorem
If one angle of a triangle is larger than another angle, then
the side opposite the larger angle is longer than the side
opposite the smaller angle.
6.11 Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.
6.12 Hinge Theorem
If two sides of one triangle are congruent to two sides of
another triangle, and the included angle of the rst is larger
than the included angle of the second, then the third side of
the rst is longer than the third side of the second.
6.13 Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of
another triangle, and the third side of the rst is longer than
the third side of the second, then the included angle of the
rst is larger than the included angle of the second.
7.1 Polygon Interior Angles Theorem
The sum of the measures of the interior angles of a convex
n-gon is (n − 2)
⋅
180°.
Corollary 7.1 Corollary to the Polygon Interior
Angles Theorem
The sum of the measures of the interior angles of a
quadrilateral is 360°.
7.2 Polygon Exterior Angles Theorem
The sum of the measures of the exterior angles of a convex
polygon, one angle at each vertex, is 360°.
7.3 Parallelogram Opposite Sides Theorem
If a quadrilateral is a parallelogram, then its opposite sides
are congruent.
7.4 Parallelogram Opposite Angles Theorem
If a quadrilateral is a parallelogram, then its opposite angles
are congruent.
7.5 Parallelogram Consecutive Angles
Theorem
If a quadrilateral is a parallelogram, then its consecutive
angles are supplementary.
7.6 Parallelogram Diagonals Theorem
If a quadrilateral is a parallelogram, then its diagonals
bisect each other.
7.7 Parallelogram Opposite Sides Converse
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
7.8 Parallelogram Opposite Angles Converse
If both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
7.9 Opposite Sides Parallel and Congruent
Theorem
If one pair of opposite sides of a quadrilateral are congruent
and parallel, then the quadrilateral is a parallelogram.
7.10 Parallelogram Diagonals Converse
If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
Corollary 7.2 Rhombus Corollary
A quadrilateral is a rhombus if and only if it has four
congruent sides.
Corollary 7.3 Rectangle Corollary
A quadrilateral is a rectangle if and only if it has four right
angles.
Corollary 7.4 Square Corollary
A quadrilateral is a square if and only if it is a rhombus and
a rectangle.
7.11 Rhombus Diagonals Theorem
A parallelogram is a rhombus if and only if its diagonals
are perpendicular.
7.12 Rhombus Opposite Angles Theorem
A parallelogram is a rhombus if and only if each diagonal
bisects a pair of opposite angles.
7.13 Rectangle Diagonals Theorem
A parallelogram is a rectangle if and only if its diagonals
are congruent.
7.14 Isosceles Trapezoid Base Angles Theorem
If a trapezoid is isosceles, then each pair of base angles is
congruent.
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Postulates and Theorems A89
Postulates and Theorems
7.15 Isosceles Trapezoid Base Angles Converse
If a trapezoid has a pair of congruent base angles, then it is
an isosceles trapezoid.
7.16 Isosceles Trapezoid Diagonals Theorem
A trapezoid is isosceles if and only if its diagonals are
congruent.
7.17 Trapezoid Midsegment Theorem
The midsegment of a trapezoid is parallel to each base, and
its length is one-half the sum of the lengths of the bases.
7.18 Kite Diagonals Theorem
If a quadrilateral is a kite, then its diagonals are
perpendicular.
7.19 Kite Opposite Angles Theorem
If a quadrilateral is a kite, then exactly one pair of opposite
angles are congruent.
8.1 Perimeters of Similar Polygons
If two polygons are similar, then the ratio of their perimeters
is equal to the ratios of their corresponding side lengths.
8.2 Areas of Similar Polygons
If two polygons are similar, then the ratio of their areas is
equal to the squares of the ratios of their corresponding side
lengths.
8.3 Angle-Angle (AA) Similarity Theorem
If two angles of one triangle are congruent to two angles of
another triangle, then the two triangles are similar.
8.4 Side-Side-Side (SSS) Similarity Theorem
If the corresponding side lengths of two triangles are
proportional, then the triangles are similar.
8.5 Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is congruent to an angle of a
second triangle and the lengths of the sides including these
angles are proportional, then the triangles are similar.
8.6 Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other
two sides, then it divides the two sides proportionally.
8.7 Converse of the Triangle Proportionality
Theorem
If a line divides two sides of a triangle proportionally, then it
is parallel to the third side.
8.8 Three Parallel Lines Theorem
If three parallel lines intersect two transversals, then they
divide the transversals proportionally.
8.9 Triangle Angle Bisector Theorem
If a ray bisects an angle of a triangle, then it divides the
opposite side into segments whose lengths are proportional
to the lengths of the other two sides.
9.1 Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse
is equal to the sum of the squares of the lengths of the legs.
9.2 Converse of the Pythagorean Theorem
If the square of the length of the longest side of a triangle is
equal to the sum of the squares of the lengths of the other
two sides, then the triangle is a right triangle.
9.3 Pythagorean Inequalities Theorem
For any ABC, where c is the length of the longest side, the
following statements are true.
If c
2
< a
2
+ b
2
, then ABC is acute.
If c
2
> a
2
+ b
2
, then ABC is obtuse.
9.4 45-45-90 Triangle Theorem
In a 45°-45°-90° triangle, the hypotenuse is
√
—
2 times as long
as each leg.
9.5 30-60-90 Triangle Theorem
In a 30°-60°-90° triangle, the hypotenuse is twice as long as
the shorter leg, and the longer leg is
√
—
3 times as long as the
shorter leg.
9.6 Right Triangle Similarity Theorem
If the altitude is drawn to the hypotenuse of a right triangle,
then the two triangles formed are similar to the original
triangle and to each other.
9.7 Geometric Mean (Altitude) Theorem
In a right triangle, the altitude from the right angle to the
hypotenuse divides the hypotenuse into two segments. The
length of the altitude is the geometric mean of the lengths of
the two segments of the hypotenuse.
9.8 Geometric Mean (Leg) Theorem
In a right triangle, the altitude from the right angle to the
hypotenuse divides the hypotenuse into two segments. The
length of each leg of the right triangle is the geometric mean
of the lengths of the hypotenuse and the segment of the
hypotenuse that is adjacent to the leg.
9.9 Law of Sines
The Law of Sines can be written in either of the following
forms for ABC with sides of length a, b, and c.
sin A
—
a
=
sin B
—
b
=
sin C
—
c
a
—
sin A
=
b
—
sin B
=
c
—
sin C
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A90 Postulates and Theorems
9.10 Law of Cosines
If ABC has sides of length a, b, and c, then the following
are true.
a
2
= b
2
+ c
2
− 2bc cos A
b
2
= a
2
+ c
2
− 2ac cos B
c
2
= a
2
+ b
2
− 2ab cos C
10.1 Tangent Line to Circle Theorem
In a plane, a line is tangent to a circle if and only if the line
is perpendicular to a radius of the circle at its endpoint on the
circle.
10.2 External Tangent Congruence Theorem
Tangent segments from a common external point are
congruent.
10.3 Congruent Circles Theorem
Two circles are congruent circles if and only if they have the
same radius.
10.4 Congruent Central Angles Theorem
In the same circle, or in congruent circles, two minor arcs are
congruent if and only if their corresponding central angles
are congruent.
10.5 Similar Circles Theorem
All circles are similar.
10.6 Congruent Corresponding Chords Theorem
In the same circle, or in congruent circles, two minor arcs
are congruent if and only if their corresponding chords are
congruent.
10.7 Perpendicular Chord Bisector Theorem
If a diameter of a circle is perpendicular to a chord, then the
diameter bisects the chord and its arc.
10.8 Perpendicular Chord Bisector Converse
If one chord of a circle is a perpendicular bisector of another
chord, then the rst chord is a diameter.
10.9 Equidistant Chords Theorem
In the same circle, or in congruent circles, two chords are
congruent if and only if they are equidistant from the center.
10.10 Measure of an Inscribed Angle Theorem
The measure of an inscribed angle is one-half the measure of
its intercepted arc.
10.11 Inscribed Angles of a Circle Theorem
If two inscribed angles of a circle intercept the same arc, then
the angles are congruent.
10.12 Inscribed Right Triangle Theorem
If a right triangle is inscribed in a circle, then the hypotenuse
is a diameter of the circle. Conversely, if one side of an
inscribed triangle is a diameter of the circle, then the triangle
is a right triangle and the angle opposite the diameter is the
right angle.
10.13 Inscribed Quadrilateral Theorem
A quadrilateral can be inscribed in a circle if and only if its
opposite angles are supplementary.
10.14 Tangent and Intersected Chord Theorem
If a tangent and a chord intersect at a point on a circle, then
the measure of each angle formed is one-half the measure of
its intercepted arc.
10.15 Angles Inside the Circle Theorem
If two chords intersect inside a circle, then the measure of
each angle is one-half the sum of the measures of the arcs
intercepted by the angle and its vertical angle.
10.16 Angles Outside the Circle Theorem
If a tangent and a secant, two tangents, or two secants
intersect outside a circle, then the measure of the angle
formed is one-half the difference of the measures of the
intercepted arcs.
10.17 Circumscribed Angle Theorem
The measure of a circumscribed angle is equal to 180° minus
the measure of the central angle that intercepts the same arc.
10.18 Segments of Chords Theorem
If two chords intersect in the interior of a circle, then the
product of the lengths of the segments of one chord is equal
to the product of the lengths of the segments of the other
chord.
10.19 Segments of Secants Theorem
If two secant segments share the same endpoint outside a
circle, then the product of the lengths of one secant segment
and its external segment equals the product of the lengths of
the other secant segment and its external segment.
10.20 Segments of Secants and Tangents
Theorem
If a secant segment and a tangent segment share an endpoint
outside a circle, then the product of the lengths of the secant
segment and its external segment equals the square of the
length of the tangent segment.
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