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Conjectures
Chapter 2
C-l
Linear Pair Conjecture If two angles form a linear pair, then the nleasures of the
add up
C-2
angies
to 180'. (Lesson 2.5)
Vertical Angles Conjecture If two angles are vertical angles, then they are congruenl
(have equal measures). (Lesson 2.5)
C-3a
Corresponding Angles Conjecture, or CA Conjecture
lf two parallel lines are cut by a
transversal, then corresponding angles are congruent. (Lesson 2.6)
C-3b
Alternate lnterior Angles Conjecture, or AIA Conjecture If two parallel lines are cut by
a transversal, then alternate interior angles are congruent. (Lesson 2.6)
C-3c
Alternate Exterior Angles Conjecture, or AEA Conjecture If two parallel lines are cut
by a transversal, then alternate exterior angles are congruent. (Lesson 2.6)
C-3
Parallel Lines Conjecture if two parallel lines are cut by a transversal, then
corresponding angles are congruent, alternate interior angles are congruent, and alternate
exterior angles are congruent. (Lesson 2.6)
C-4
Converse of the Parallel Lines Conjecture If two lines are cut by a transversal to form
pairs of congruent corresponding angles, congruent alternate interior angles, or congruent
alternate exterior angles, then the lines are parallel. (Lesson 2.6)
Chapter 3
C-5
Perpendicular Bisector Conjecture if a point is on the perpendicular bisector of
segment, then it is equidistant from the endpoints. (Lesson 3.2)
C-6
Converse of the Perpendicular Bisector Conjecture If a point is equidistant from
the endpoints of a segment, then it is on the perpendicular bisector of the segment.
a
(Lesson 3.2)
C-7
Shortest Distance Conjecture The shortest distance from a point to a line is measured
along the perpendicular segment from the point to the line. (Lesson 3.3)
C-8
Angle Bisector Conjecture If a point is on the bisector of an angle, then it is equidistant
from the sides of the angle. (Lesson 3.4)
C-9
Angle Bisector Concurrency Conjecture The three angle bisectors of a triangle
are
concurrent (meet at a point). (Lesson 3.7)
C-10
Perpendicular Bisector Concurrency Conjecture The three perpendicular bisectors of
a triangle are concurrent. (Lesson 3.7)
C-l1
Altitude Concurrency Conjecture The three altitudes (or the lines containing the
altitudes) of a triangle are concurrent. (Lesson 3.7)
C-12
Circumcenter Conjecture The circumcenter of a triangle is equidistant from the vertices.
(Lesson 3.7)
C-l3
lncenter Conjecture The incenter of a triangle is equidistant from the sides. (Lesson 3.7)
C-14
Median Concurrency Conjecture The three medians of a triangle are concurrent.
(Lesson 3.8)
C-l5
122
Centroid Conjecture The centroid of a triangle divides eacl-r median into two parts so
that the distance from the centroid to the vertex is twice the distance from the centroid to
the midpoint of the opposite side. (Lesson 3.8)
coNJECruREs
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C-l6 Centerof GravityConjecture
Thecentroidof atriangleisthecenterof gravityof the
triangular region. (Lesson 3.8)
Chapter 4
C-17
Triangle Sum Conjecture The sum of the measures of the angles in every triangle
is
180'. (Lesson 4,1)
C-18
Third Angle Conjecture If two angles of one triangle are equal in measure to two angles
of another triangle, then the third angle in each triangle is equal in measure to the third
angle in the other triangle, (Lesson 4.1)
C-l9
lsosceles Triangle Conjecture
If a triangle
is isosceles, then its base angles are congruent.
(Lesson 4.2)
C-20
Converse of the lsosceles Triangle Conjecture
then it is an isosceles triangle. (Lesson 4.2)
If a triangle
has two congruent angles,
C-21 Triangle lnequality Conjecture
The sum of the lengths of any two sides of a triangle is
greater than the length of the third side. (Lesson 4.3)
C-22
Side-Angle lnequality Conjecture In a triangle, if one side is ionger than another
side,
then the angle opposite the longer side is larger than the angle opposite the shorter side.
(Lesson 4.3)
C-23
Triangle Exterior Angle Conjecture The measure of an exterior angle of a triangle
equal to the sum of the measures of the remote interior angles. (Lesson 4.3)
C-24
SSS
is
Congruence Conjecture If the three sides of one triangle are congruent to the three
sides
of another triangle, then the triangles are congruent. (Lesson 4.4)
C-25
5AS Congruence Conjecture If two sides and the included angle of one triangle are
congruent to two sides and the included angle of another triangle, then the triangles are
congruent. (Lesson 4.4)
C-26
ASA Congruence Conjecture lf two angles and the included side of one triangle are
congruent to two angles and the included side of another triangle, then the triangles are
congrLrent. (Lesson 4.5)
C-27
SAA Congruence Conjecture If two angles and a non-included side of one triangle are
congruent to the corresponding angles and side of another triangle, then the triangles are
congruent. (Lesson 4.5)
C-28
Vertex Angle Bisector Conjecture In an isosceles triangle, the bisector of the vertex
angle is also the altitude and the median to the base. (Lesson 4.8)
C-29
Equilateral/Equiangular Triangle Conjecture Every equilateral triangle is equiangular.
Conversely, every equiangular triangle is equilateral. (Lesson 4.8)
Chapter 5
C-30
Quadrilateral Sum Conjecture The sum of the measures of the four angles of any
quadrilateral is 360". (Lesson 5.1)
C-31
Pentagon Sum Conjecture The sunr of the measures of the five angies of any pentagon
is 540". (Lesson 5.1)
C-32
Polygon Sum Conjecture The sum of the measures of the
is 180'(r - 2). (Lesson 5.1)
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interior angles of an /,-gon
coNJECTURES
123
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C-33
Exterior Angle Sum Conjecture For any polygon, the sum of the measures of a set of
exterior angles is 360". (Lesson 5.2)
C'34
Equiangular Polygon Conjecture You can find the measure of each interior angle of an
r80"(L * 2).
equiangular n-gonby using either of these formulas: l80o * {I ur
(Lesson 5.2)
C-35
C-36
C'37
Kite Angles Conjecture The nonvertex angles of a kite are congruent. (Lesson 5.3)
Kite Diagonals Conjecture The diagonals of a kite are perpendicular. (Lesson 5.3)
Kite Diagonal Bisector Conjecture The diagonal connecting the vertex angles of a kite
is the perpendicuiar bisector of the other diagonal. (Lesson 5.3)
C-38
Kite Angle Bisector Conjecture The vertex angles of a kite are bisected by a diagonal.
(Lesson 5.3)
C-39
Trapezoid Consecutive Angles Conjecture The consecutive angles between the
of a trapezoid are supplementary. (Lesson 5.3)
C-40
lsosceles Trapezoid Conjecture The base angles of an isosceles trapezoid are congruent.
bases
(Lesson 5.3)
C-41
lsosceles Trapezoid Diagonals Conjecture The diagonals of an isosceles trapezoid are
congruent. (Lesson 5.3)
C-42
Three Midsegments Conjecture The three midsegments of a triangle divide it into
four congruent triangles. (Lesson 5.4)
C-43
Triangle Midsegment Conjecture A midsegment of a triangle is paraliel to the third
side and half the length of the third side. (Lesson 5.4)
C'44
Trapezoid Midsegment Conjecture The midsegment of a trapezoid is parallel to the
bases and is equal in length to the average of the lengths of the bases. (Lesson 5.4)
C-45
Parallelogram Opposite Angles Conjecture The opposite angles of a parallelogram
are congruent. (Lesson 5.5)
C-46
Parallelogram Consecutive Angles Conjecture The consecutive angles of
a
parallelogram are supplementary. (Lesson 5.5)
C-47
Parallelogram Opposite Sides Conjecture The opposite sides of a parallelogranl
are
congruent. (Lesson 5.5)
C-48
Parallelogram Diagonals Conjecture The diagonals of a parallelogram bisect each other.
(Lesson 5.5)
C-49
Double-Edged Straightedge Conjecture If two parallel lines are intersected by a
of parallel lines that are the same distance apart as the first pair, then the
second pair
parallelogram formed is a rhombus. (Lesson 5.6)
C-50
Rhombus Diagonals Conjecture The diagonals of a rhombus are perpendicuiar, and
they bisect each other. (Lesson 5,6)
C-51
Rhombus Angles Conjecture The diagonals of a rhombus bisect the angles of the
rhombus. (Lesson 5.6)
C-52
Rectangle Diagonals Conjecture The diagonals of a rectangle are consruent and bisect
each other. (Lesson 5.6)
C-53
Square Diagonals Conjecture The diagonals of a sqltare are congruent, perpendicular,
and bisect each other. (Lesson 5.6)
124
coNJECTUREs
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Chapter 5
c-54
' chord central Angles conjecture If two chords in
a circle are congruent, then they
determine two central angles that are
congruent. (Lesson 6.1)
c-55 chord Arcs conjecture
If two chords in a circle are congruent, then their
intercepted
are congruent. (Lesson 6.1)
c-56
c'57
c-58
arcs
Perpendicular to a chord conjecture
The perpendicular from the center of
a circre to
chord is the bisector of the chord. (Lesson
6.li
a
chord Distance to center conjecture Two congruent
chords in a circle are equi<listant
from the center of the circle. (Lesson 6.1)
Perpendicular Bisector of a chord coniecture
The perpendicular bisector of a chord
through the center of the circle. (Lesson 6.1)
Tangent conjecture A tangent to a circle is
perpendicular to the radius drawn to the
point of tangency. (Lesson 6.2)
passes
c-59
c-60
Tangent segments conjecture Thngent segments
to a circle from a point outside the
circle are congruent.
(Lesson 6.2)
c'61
c-62
Inscribed Angle conjecture The measure
of an angle inscribed in a circle is one-half
the measure of the central angle. (Lesson 6.3)
lnscribed Angtes lntercepting Arcs conjecture
Inscribed angles that intercept the
same arc
are congruent. (Lesson 6.3)
c-53
Angles lnscribed in a semicircle conjecture
Angles inscribecl in a semicircie are right
angles. (Lesson
c-64
cyclic Quadrilateral conjecture The opposite
angres of a cyclic quadrilateral
6.3)
supplementary. (Lesson 6.3)
c-65 Parallel Lines lntercepted Arcs conjecture Paraliel lines
a circle. (Lesson 6.3)
c-66
are
intercept congruent arcs on
circumference conjecture If c is the circumference
and d is the diameter of a circle,
then thereisanumberz-suchthat c:
rd. If cr:2rwhere ristheradius,then c :2nr.
(Lesson 6.5)
c-67
Arc Length conjecture The tength of an
arc equals the circumference times the
measure
of the cenrral angle
divided by 36-0". (Lesson 6.7f
Chapter 7
c-58 Reflection Line coniecture The line of reflection is the
perpendicular bisector of every
segment joining a point in the originar
figure with its i.,rug.. (Lesson 7.r)
C-69
Coordinate Transformations Conjecture
The ordered pair rule (x, y) --> (-x, y)
is a reflection over the
7_axis.
The ordered pair rule (x, y) _-s (x, _y)
is a reflection over the x-axis.
The ordered pair rure (x, y)-+(-x, *y)
is a rotation about trre origin.
The ordered pair rule (x, y) _-> (Z x) is a
reflection
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y:
x. (Lesson 7.2)
coNJECTURES
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