PLANE GEOMETRY
DEFINITIONS.
Proposition. A proposition is a general statement concerning geometric
relationships.
Postulate. A postulate is a proposition that we accept without deductive
reasoning.
Theorem. A theorem is a proposition that we establish by means of
deductive reasoning.
Angle. The inclination of one line with respect to the other of two
straight lines drawn from a point. The point is called the vertex of
the angle.
Right angle. An angle of 90 degrees.
Straight angle. An angle of 180 degrees.
Supplementary angles. Two angles whose sum is 180 degrees.
Complementary angles. Two angles whose sum is 90 degrees.
Adjacent angles. Two angles that have the same vertex and a common side
between them.
Vertical angles. The opposite angles formed by two intersecting straight
lines.
Acute angle. An angle less than a right angle.
Obtuse angle. An angle greater than a right angle but less than a
straight angle.
Polygon. A closed plane figure bounded by straight lines.
Vertex of a polygon. The point where two sides of a polygon meet.
Triangle. A three-sided polygon.
Quadrilateral. A four-sided polygon.
Equilateral triangle. A triangle with three equal sides.
Isosceles triangle. A triangle with two equal sides.
Scalene triangle. A triangle with no two sides equal.
Right triangle. A triangle one of whose angles is a right angle.
Obtuse triangle. A triangle one of whose angles is an obtuse angle.
Acute triangle. A triangle all of whose angles are acute.
Parallelogram. A quadrilateral with its opposite sides parallel.
Rectangle. A parallelogram with one right angle.
Rhombus. A parallelogram with two adjacent sides equal.
Square. A rectangle with two adjacent sides equal.
Trapezoid. A quadrilateral with two and only two sides parallel.
Isosceles trapezoid. A trapezoid whose nonparallel sides are equal.
Perpendicular lines. Lines that meet each other and form right angles.
Perpendicular bisector of a line. A line which not only bisects the line
but is perpendicular to it.
Altitude of a triangle. A line from any one vertex of the triangle,
perpendicular to the opposite side, and terminated by that side.
Median of a triangle. A line drawn from any vertex to the middle point
of the opposite side.
Orthocenter of a triangle. The point of intersection of the altitudes of
the triangle.
Circumcenter of a triangle. The point of intersection of the
perpendicular bisectors of the sides of the triangle.
Incenter of a triangle. The point of intersection of the bisectors of
the angles of the triangle.
Median of a trapezoid. A line joining the middle points of the
nonparallel sides of the trapezoid.
Transversal. A line which intersects two or more lines.
Circle. A closed curve all points of which are in the same plane and are
equally distant from a point within it called the center.
Radius of a circle. A line from the center of the circle to any point on
the circle.
Diameter of a circle. A straight line through the center of the circle
with its ends on the circle.
Arc of a circle. Any part of the line forming the circle.
Semicircle. Half a circle.
Concentric circles. Two or more circles having the same center but
different radii.
Minor arc of a circle. An arc that is less than a semicircle.
Chord. A line joining any two points on a circle.
Central angle of a circle. An angle with its vertex at the center of the
circle and with radii for its sides.
Polygon inscribed in a circle. A polygon is said to be inscribed in a
circle if all its vertices lie on the circle. In this case, the
circle is said to be circumscribed about the polygon.
Tangent to a circle. A line touching a circle at only one point is called
a tangent to the circle.
Common tangents. A line tangent to each of two circles is called a common
tangent. If the circles lie on opposite sides of the tangent, it is a
common internal tangent. If the circles lie on the same side of the
tangent, it is a common external tangent.
Polygon circumscribed about a circle. A polygon is said to be
circumscribed about a circle if all its sides are tangent to the
circle. In this case the circle is inscribed in the polygon.
Line of centers of two circles. The line joining their centers.
Tangent circles. Two circles are said to be tangent to each other if they
are both tangent to the same line at the same point. They are tangent
internally if one circle lies within the other. They are tangent
externally if each circle lies outside the other.
Inscribed angle; angle inscribed in a circle. An angle whose vertex is
on a circle and whose sides are chords of the circle.
Angle inscribed in an arc. An angle is inscribed in an arc if the vertex
is on the arc and its sides meet the extremities of the arc.
Secant. A line that intersects a circle at two points.
Proportions. An equation that states that two ratios are equal is called
a proportion. In the proportion a/b = c/d a, b, c, and d are
respectively the first, second, third and fourth terms. The first and
fourth terms, a and d, are called the extremes and the second and
third terms, b and c, the means, of the proportion.
Congruent figures. Any two geometric figures that can be made to exactly
coincide (fit exactly on each other).
Similar polygons. Polygons whose corresponding angles are equal and
whose corresponding sides are in proportion.
Equal figures. Figures that have the same area.
Center of a regular polygon. The common center of its inscribed and
circumscribed circles.
Radius of a regular polygon. The radius of its circumscribed circle.
Apothem of a regular polygon. The apothem of a regular polygon is the
radius of its inscribed circle drawn to the point of contact.
Central angle of a regular polygon. The angle between the radii drawn to
adjacent vertices of the polygon.
Sector of a circle. The figure formed by two radii and their intercepted
arc.
Segment of a circle. The figure formed by a chord and its arc. If the
arc is a minor arc, the segment is a minor segment; if the arc is a
major arc, the segment is a major segment.
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ASSUMPTIONS.
Things equal to the same or equal things are equal to each other.
A quantity may be substituted for its equal in any expression or
equation.
The whole equals the sum of its parts.
Any quantity equals itself.
If equals are added to equals, the sums are equal.
If equals are subtracted from equals, the differences are equal.
If equals are multiplied by equals, the products are equal.
If equals are divided by equals, the quotients are equal.
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POSTULATES.
Two triangles are congruent if two sides and the included angle of one
are equal to two sides and the included angle of the other (i.e.
s.a.s. = s.a.s.).
Two triangles are congruent if two angles and the included side of one
are equal to two angles and the included side of the other (i.e.
a.s.a. = a.s.a.).
Two triangles are congruent if three sides of one are equal to three
sides of the other (i.e. s.s.s. = s.s.s.).
All right angles are equal.
All straight angles are equal.
A straight line can be extended in either direction to any desired
length.
Two straight lines cannot intersect in more than one point.
Through two given points one and only one straight line can be drawn.
A straight line is the shortest line that can be drawn between two
points.
One and only one circle can be drawn with any point as center and any
line segment as radius.
At a point on a line or from a point outside a line only one
perpendicular can be drawn to the line.
The shortest distance from a point to a line is the perpendicular from
the point to the line.
An angle has only one bisector.
A line has only one midpoint.
If two right triangles have the hypotenuse and an acute angle of one
equal to the hypotenuse and an acute angle of the other, they are
congruent.
Through a given point not on a given line, one and only one line can be
drawn parallel to the given line.
Two lines perpendicular to a third line all in the same plane are
parallel.
Two circles are equal if their radii or their diameters are equal.
A diameter bisects a circle.
If a line bisects a circle and is terminated the circle, it is a
diameter.
A straight line cannot intersect a circle in more than two points.
If the distance from a point to the center of a circle is equal to a
radius, the point lies on the circle. If the distance from a point to
the center of a circle is greater than a radius, the point lies outside
the circle. If the distance from a point to the center of a circle is
less than a radius, the point lies within the circle.
In the same circle or in equal circles, equal central angles have equal
arcs.
In the same circle or in equal circles, equal arcs have equal central
angles.
A straight line perpendicular to a radius at its outer extremity is
tangent to the circle.
A tangent to a circle is perpendicular to the radius drawn to the point
of contact.
A line perpendicular to a tangent at its point of contact passes through
the center of the circle.
A line from the center of a circle and perpendicular to a tangent passes
through the point of contact.
A central angle has the same number of degrees as its intercepted arc.
A line parallel to one side of a triangle and intersecting the other two
sides divides them into segments which, taken in the same order, have
the same ratio.
The area of a rectangle is equal to the product of its base and altitude.
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THEOREMS (unproved)
If one straight line meets another so as to form adjacent angles, the
angles are supplementary.
If two adjacent angles form a right angle, the angles are
complementary.
If two angles are supplementary to the same angle or to equal angles,
they are equal.
If two angles are complementary to the same angle or to equal angles,
they are equal.
If two straight lines intersect, the vertical angles are equal.
In the same circle or in equal circles, the larger of two central
angles has the longer arc.
In the same circle or in equal circles, the larger of two minor arcs
has the larger central angle.
Three parallel lines cut off on any two transversals segments which
taken in the same order have the same ratio.
A line parallel to one side of a triangle divides the other two sides
so that either side is to one of its segments as the other is to the
corresponding segment.
Corresponding angles of similar polygons are equal.
Corresponding sides of similar polygons are in proportion.
Two polygons are equal if they are composed of respectively congruent
parts.
The areas of two similar polygons are to each other as the squares of
any two corresponding sides.
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PROPORTION THEOREMS
1.0 In any proportion, the product of the extremes is equal to the
product of the means. Thus if a/b = c/d, then ad = bc.
2.0 If the product of two numbers is equal to the product of two other
numbers, either pair may be made the means and the other pair the
extremes of a proportion. Thus if ad = bc, then a/b = c/d,
b/a = d/c.
3.0 If the numerators of a proportion are equal, the denominators are
equal. If the denominators are equal, the numerators are equal.
4.0 The terms of a proportion are also in proportion by inversion; that
is, the second term is to the first as the fourth is to the third.
Thus if a/b = c/d, then b/a = d/c.
5.0 The terms of a proportion are also in proportion by alternation;
that is, the first term is to the third as the second is to the
fourth. Thus if a/b = c/d, then a/c = b/d.
6.0 The terms of a proportion are also in proportion by addition; that
is, the sum of the first and second terms is to the second term as
the sum of the third and fourth is to the fourth. Thus if
a/b = c/d, then (a+b)/b = (c+d)/d.
7.0 The terms of a proportion are also in proportion by subtraction;
that is, the first term minus the second is to the second as the
third term minus the fourth is to the fourth. Thus if a/b = c/d,
then (a-b)/b = (c-d)/d.
8.0 If three terms of one proportion are equal respectively to the three
corresponding terms of another proportion, then the remaining term
of the first is equal to the remaining term of the second. Thus
if a/b = x/c and a/b = y/c, then x = y.
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GENERAL THEOREMS
1.0 If two sides of a triangle are equal, the angles opposite those
sides are equal.
1.1 An equilateral triangle is also equiangular.
2.0 If two angles of a triangle are equal, the sides opposite those
angles are equal.
2.1 An equiangular triangle is also equilateral.
3.0 The bisector of the vertex angle of an isosceles triangle bisects
the base and is perpendicular to it.
4.0 The line that connects the vertex of an isosceles triangle with the
middle point of the base bisects the vertex angle and is
perpendicular to the base.
5.0 If lines are drawn from any point on the perpendicular bisector of a
line to the extremities of the line, they are equal.
6.0 Two points each equally distant from the extremities of a line
determine the perpendicular bisector of the line.
7.0 If two right triangles have the hypotenuse and another side of one
equal to the hypotenuse and a side of the other, they are
congruent.
8.0 If two straight lines are cut by a transversal so that two alternate
interior angles are equal, the lines are parallel.
9.0 If two straight lines are cut by a transversal so that two
corresponding angles are equal, the lines are parallel.
10.0 If two straight lines are cut by a transversal so that two interior
angles on the same side of the transversal are supplementary, the
lines are parallel.
11.0 If a line is perpendicular to one of two parallel lines, it is
perpendicular to the other also.
12.0 If two parallel lines are cut by a transversal, the alternate
interior angles are equal.
13.0 If two parallel lines are cut by a transversal, the corresponding
angles are equal.
14.0 If two parallel lines are cut by a transversal, the interior angles
on the same side of the transversal are supplementary.
15.0 Two straight lines parallel to the same straight line are parallel
to each other.
16.0 The sum of the angles of a triangle is 180 degrees.
16.1 If two angles of one triangle are equal to two angles of another
triangle, the third angles are equal.
16.2 A triangle can have but one right angle or one obtuse angle.
16.3 If two right triangles have an acute angle of one equal to an acute
angle of the other, the other acute angles are equal.
16.4 The acute angles of a right triangle are complementary.
16.5 Each angle of an equilateral triangle is 60 degrees.
16.6 If two triangles have two angles and a side of one equal
respectively to two angles and the corresponding side of the
other, they are congruent.
16.7 If one side of a triangle is extended, the exterior angle thus
formed is equal to the sum of the two remote interior angles.
17.0 The sum of the angles of a polygon of n sides is (n-2)180 degrees.
17.1 The sum of the exterior angles of a polygon made by extending each
of its sides in succession is equal to 360 degrees.
18.0 The opposite sides of a parallelogram are equal.
18.1 The opposite angles of a parallelogram are equal.
18.2 A diagonal of a parallelogram divides it into two congruent
triangles.
18.3 Parallel lines are at all points the same distance apart.
18.4 The successive angles of a parallelogram are supplementary.
19.0 The diagonals of a parallelogram bisect each other.
20.0 If the opposite sides of a quadrilateral are equal, the figure is a
parallelogram.
21.0 If two sides of a quadrilateral are parallel and equal, the figure
is a parallelogram.
22.0 If the diagonals of a quadrilateral bisect each other, the figure is
a parallelogram.
22.1 Each angle of a rectangle is a right angle.
22.2 The diagonals of a rectangle are equal.
22.3 If the diagonals of a parallelogram are equal, it is a rectangle.
22.4 The diagonals of a rhombus are perpendicular to each other.
22.5 The diagonals of a rhombus bisect its angles.
22.6 If the diagonals of a parallelogram are perpendicular to each other,
it is a rhombus.
23.0 If a line joins the middle points of two sides of a triangle, it is
parallel to the third side and equal to half of it.
24.0 The median of a trapezoid is parallel to the bases and equal to one
half their sum.
25.0 If three or more parallel lines cut off equal segments on one
transversal, they cut off equal segments on every transversal.
25.1 If a line is parallel to one side of a triangle and bisects another
side, it bisects the third side also.
25.2 If a line is parallel to the bases of a trapezoid and bisects one of
the nonparallel sides, it bisects the other also.
26.0 In the same circle or in equal circles, equal chords have equal
arcs.
27.0 In the same circle or in equal circles, equal arcs have equal
chords.
28.0 If a line through the center of a circle is perpendicular to a
chord, it bisects the chord and its arc.
28.1 A line through the center of a circle that bisects a chord (not a
diameter) is perpendicular to it.
29.0 In the same circle or in equal circles, equal chords are equally
distant from the center.
30.0 In the same circle or in equal circles, chords equally distant from
the center are equal.
31.0 The tangents to a circle from a point outside the circle are equal.
31.1 If two tangents are drawn to a circle from an outside point, the
line from the point to the center bisects the angle between the
tangents.
32.0 If two circles are tangent to each other, their line of centers
passes through the point of contact.
33.0 If two sides of a triangle are unequal, the angles opposite those
sides are unequal in the same order.
34.0 If two angles of a triangle are unequal, the sides opposite those
sides are unequal in the same order.
35.0 If two sides of one triangle are equal to two sides of another
triangle and the included angle of the first is greater than the
included angle of the second, then the third side of the first is
greater than the third side of the second.
36.0 If two triangles have two sides of one equal to two sides of the
other and the third side of the first is greater than the third
side of the second, then the angle opposite the third side of the
first is greater than the angle opposite the third side of the
second.
37.0 In the same circle or in equal circles, the longer of two minor arcs
has the longer chord.
38.0 In the same circle or in equal circles, the longer of two chords has
the longer minor arc.
39.0 In the same circle or in equal circles, unequal chords are unequally
distant from the center, the longer chord being the nearer.
40.0 In the same circle or in equal circles, chords unequally distant
from the center are unequal, the nearer being the longer.
41.0 An inscribed angle has half as many degrees as the intercepted arc.
41.1 Angles inscribed in the same arc are equal.
41.2 An angle inscribed in a semicircle is a right angle.
42.0 An angle formed by a tangent and a chord meeting it at the point of
contact has half as many degrees as the intercepted arc.
43.0 An angle formed by two chords intersecting within a circle has half
as many degrees as the sum of the two arcs intercepted by it and
by the vertical angle.
44.0 An angle formed by two secants intersecting outside a circle has
half as many degrees as the difference between the two intercepted
arcs.
44.1 The angle formed by a secant and a tangent or two tangents meeting
outside the circle has half as many degrees as the difference
between the two intercepted arcs.
45.0 Parallel lines intercept equal arcs on a circle.
46.0 The opposite angles of an inscribed quadrilateral are
supplementary.
47.0 If a circle is divided into any number of equal arcs, the chords of
these arcs form a regular polygon of that number of sides.
47.1 An equilateral polygon inscribed in a circle is a regular polygon.
47.2 A regular quadrilateral is a square.
47.3 The side of a regular hexagon inscribed in a circle is equal to the
radius of the circle.
47.4 Chords joining the alternate vertices of a regular inscribed hexagon
form an equilateral triangle.
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LOCUS THEOREMS
1.0 The locus of a point at a given distance from a given point is a
circle with the point as center and the given distance as radius.
2.0 The locus of a point at a given distance from a given line is a pair
of lines, one on each side of the given line, parallel to it at
the given distance from it.
3.0 The locus of a point equally distant from two parallel lines is a line
parallel to them and midway between them.
4.0 The locus of a point equally distant from two points is the
perpendicular bisector of the line joining the two points.
4.1 The perpendicular bisector of a chord passes through the center of
the circle.
5.0 The locus of a point equally distant from the sides of an angle is the
bisector of the angle.
6.0 The locus of the vertex of the right angle with a fixed hypotenuse is
a circle whose diameter is the hypotenuse.
7.0 The locus of the vertex of a triangle having a fixed base and a given
vertex angle is the arc of a circle drawn to the extremities of
the base and passing through any one position of the vertex.
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48.0 The perpendicular bisectors of the sides of a triangle meet at a
point which is equally distant from the vertices of the triangle.
49.0 The bisectors of the angles of a triangle meet at a point that is
equally distant from the sides of the triangle.
50.0 The altitudes of a triangle meet at a point.
51.0 The medians of a triangle meet at a point which is two thirds of the
distance from each vertex to the midpoint of the opposite side.
52.0 The bisector of an angle of a triangle divides the opposite side
into segments which have the same ratio as the other two sides.
53.0 If a line divides two sides of a triangle proportionally, it is
parallel to the third side.
53.1 If a line divides two sides of a triangle so that either side is to
one of its segments as the other side is to the corresponding
segment, it is parallel to the third side.
54.0 Two triangles are similar if two angles of one are equal to two
angles of the other.
54.1 Two right triangles are similar if an acute angle of one is equal to
an acute angle of the other.
55.0 Two triangles are similar if they have an angle of one equal to an
angle of the other, and the sides including these angles are in
proportion.
56.0 Two triangles are similar if their corresponding sides are in
proportion.
57.0 Corresponding altitudes of similar triangles have the same ratio as
any two corresponding sides.
58.0 If two chords intersect within a circle, the product of the segments
of one is equal to the product of the segments of the other.
59.0 If from a point outside a circle two secants are drawn, the product
of one secant and its external segment is equal to the product of
the other secant and its external segment.
60.0 If from a point outside a circle a secant and a tangent are drawn,
the product of the secant and its external segment is equal to the
square of the tangent.
61.0 In any right triangle the altitude upon the hypotenuse is the mean
proportional between the segments of the hypotenuse.
62.0 If the altitude upon the hypotenuse of a right triangle is drawn,
either arm is the mean proportional between the whole hypotenuse
and the segment of the hypotenuse adjacent to the arm.
63.0 In any right triangle, the square of the hypotenuse is equal to the
sum of the square of the arms.
64.0 In a right triangle, if one angle is 30 degrees, the hypotenuse is
twice the side opposite the 30 degree angle.
65.0 In a right triangle, if the hypotenuse is twice one of the arms the
angle opposite that arm is 30 degrees.
66.0 The area of a parallelogram is equal to the product of its base and
altitude.
66.1 Parallelograms with equal bases and equal altitudes are equal.
67.0 The area of a triangle is equal to one half the product of its base
and altitude.
67.1 Triangles with equal bases and equal altitudes are equal.
67.2 Triangles which have equal bases in the same straight line and
vertices in a line parallel to the bases are equal.
68.0 The area of a trapezoid equals one half its altitude times the sum
of its bases.
69.0 The areas of two similar triangles are to each other as the squares
of any two corresponding sides.
70.0 A circle can be circumscribed about any regular polygon.
71.0 A circle can be inscribed in any regular polygon.
71.1 The central angle of a regular polygon of n sides is equal to 360/n
degrees.
71.2 The apothem of a regular polygon is the perpendicular bisector of
its side.
71.3 The radius of a regular polygon bisects the angle to whose vertex it
is drawn.
72.0 If a circle is divided into three or more equal arcs, the tangents
at the points of division form a regular circumscribed polygon.
73.0 Regular polygons of the same number of sides are similar.
74.0 In a series of equal ratios the sum of the numerators is to the sum
of the denominators as any numerator is to its denominator. Thus
if a/b = c/d = e/f = r, then (a+c+e)/(b+d+f) = r = a/b = c/d =
e/f.
75.0 The perimeters of similar polygons are to each other as any two
corresponding sides.
76.0 The perimeters of two regular polygons of the same number of sides
are to each other as their radii or as their apothems.
77.0 The area of a regular polygon is half the product of its apothem and
its perimeter.
77.1 The areas of two regular polygons of the same number of sides are to
each other as the squares of their radii or as the squares of
their apothems.
78.0 The circumferences of two circles are to each other as their radii.
79.0 The areas of two circles are to each other as the squares of their
radii.
Source: Clark, Smith, Schorling. Modern-School Geometry.